# The Case for Co-Designing Model Architectures with Hardware
**Authors**: Quentin Anthony, Jacob Hatef, Deepak Narayanan, Stella Biderman, Stas BekmanJunqi Yin, Aamir Shafi, Hari Subramoni, Dhabaleswar K. Panda
> EleutherAI Ohio State University
> NVIDIA
> EleutherAI
> Contextual AI Oak Ridge National Lab
> Ohio State University
## Abstract
While GPUs are responsible for training the vast majority of state-of-the-art deep learning models, the implications of their architecture are often overlooked when designing new deep learning (DL) models. As a consequence, modifying a DL model to be more amenable to the target hardware can significantly improve the runtime performance of DL training and inference. In this paper, we provide a set of guidelines for users to maximize the runtime performance of their transformer models. These guidelines have been created by carefully considering the impact of various model hyperparameters controlling model shape on the efficiency of the underlying computation kernels executed on the GPU. We find the throughput of models with “efficient” model shapes is up to 39% higher while preserving accuracy compared to models with a similar number of parameters but with unoptimized shapes.
## I Introduction
Transformer-based [37] language models have become widely popular for language and sequence modeling tasks. Consequently, it is extremely important to train and serve large transformer models such as GPT-3 [6] and Codex as efficiently as possible given their scale and wide use. At the immense scales that are in widespread use today, efficiently using computational resources becomes a complex problem and small drops in hardware utilization can lead to enormous amounts of wasted compute, funding, and time. In this paper, we tackle a frequently ignored aspect of training large transformer models: how the shape of the model can impact runtime performance. We use first principles of GEMM optimization to optimize individual parts of the transformer model (which translates to improved end-to-end runtime performance as well). Throughout the paper, we illustrate our points with extensive computational experiments demonstrating how low-level GPU phenomenon impact throughput throughout the language model architecture.
Many of the phenomena remarked on in this paper have been previously documented, but continue to plague large language model (LLM) designers to this day. We hypothesize that there are three primary causes of this:
1. Few resources trace the performance impacts of a transformer implementation all the way to the underlying computation kernels executed on the GPU.
1. The existing documentation on how transformer hyperparameters map to these kernels is not always in the most accessible formats, including tweets [19, 18], footnotes [33], and in comments in training libraries [3].
1. It is convenient to borrow architectures from other papers and researchers rarely give substantial thought to whether those choices of model shapes are optimal.
This work attempts to simplify performance tuning for transformer models by carefully considering the architecture of modern GPUs. This paper is also a demonstration of our thesis that model dimensions should be chosen with hardware details in mind to an extent far greater than is typical in deep learning research today.
As shown in Figure 1, the runtimes of models with a nearly identical number of parameters but different shapes can vary wildly. In this figure, the “standard architecture” for a 2.7B transformer model defined by GPT-3 [6] has been used by OPT [43], GPT-Neo [5], Cerebras-GPT [13], RedPajama-INCITE [1], and Pythia [4]. Unfortunately the knowledge of how to optimally shape transformer architectures is not widely known, resulting in people often making sub-optimal design decisions. This is exacerbated by the fact that researchers often deliberately copy hyperparameters from other papers for cleaner comparisons, resulting in these sub-optimal choices becoming locked in as the standard. As one example of this, we show that the 2.7 billion parameter model described in [6] can be trained almost 20% faster than the default architecture through minor tweaking of the model shape.
<details>
<summary>x1.png Details</summary>
### Visual Description
# Technical Document Extraction: Throughput Comparison Chart
## Chart Type
Bar chart comparing computational throughput (TFLOPs/s) across different models.
## Axis Labels
- **X-axis**: "Models" (categorical)
- Categories: C1, C2, GPT-3 (2.7B), Ours 1, Ours 2
- **Y-axis**: "Throughput (TFLOPs/s)" (numerical)
- Range: 0 to 180 (linear scale)
## Data Points
| Model | Throughput (TFLOPs/s) |
|----------------|-----------------------|
| C1 | 83.392 |
| C2 | 108.505 |
| GPT-3 (2.7B) | 119.37 |
| Ours 1 | 140.695 |
| Ours 2 | 165.741 |
## Visual Trends
- **Upward progression**: Throughput increases monotonically from left to right.
- **Key jumps**:
- C1 → C2: +25.113 TFLOPs/s
- C2 → GPT-3: +10.865 TFLOPs/s
- GPT-3 → Ours 1: +21.325 TFLOPs/s
- Ours 1 → Ours 2: +25.046 TFLOPs/s
## Color Coding
- All bars rendered in **blue** (no explicit legend visible; uniform color implies single data series).
## Spatial Grounding
- Bars positioned sequentially along X-axis from left (C1) to right (Ours 2).
- Numerical values displayed atop each bar for direct readability.
## Component Isolation
1. **Header**: No explicit title present.
2. **Main Chart**: Bar heights proportional to throughput values.
3. **Footer**: No source/attribution information visible.
## Trend Verification
- Confirmed: Each subsequent model shows higher throughput than the previous, with consistent incremental improvements.
## Data Table Reconstruction
| Model | Throughput (TFLOPs/s) |
|----------------|-----------------------|
| C1 | 83.392 |
| C2 | 108.505 |
| GPT-3 (2.7B) | 119.37 |
| Ours 1 | 140.695 |
| Ours 2 | 165.741 |
## Notes
- No non-English text detected.
- No explicit legend present; color consistency across bars suggests single data series.
- Values formatted to two decimal places for precision.
</details>
Figure 1: Transformer single-layer throughput of various architectures for a 2.7 billion parameter model (C1 and C2 are defined by this paper as C1: $h=2560,a=64$ , C2: $h=2560,a=40$ ).
Our analysis makes use of the fact that General Matrix Multiplications (GEMMs) are the lifeblood of modern deep learning. Most widely-used compute-intensive layers in deep learning explicitly use GEMMs (e.g., linear layers or attention layers) or use operators that are eventually lowered into GEMMs (e.g., convolutions). For transformer models, our experiments from Figure 2 show that GEMM kernels regularly account for $68.3\$ and $94.9\$ of the total model latency for medium- and large-sized models, respectively. As a result, understanding the performance of GEMMs is crucial to understanding the runtime performance of end-to-end models; this only becomes more important as model size increases.
<details>
<summary>x2.png Details</summary>
### Visual Description
# Technical Document Extraction: Latency Composition Analysis
## Chart Description
This image presents a stacked bar chart comparing latency composition across three computational configurations. The chart uses color-coded segments to represent different computational components' contribution to total latency.
### Key Components
1. **Legend** (Top of chart):
- GEMMs: Blue
- Flash: Gray
- Softmax: Orange
- DR: Green
- LN: Yellow
- Other: Purple
2. **X-Axis Categories**:
- Small (h=2560, a=20)
- Large (h=16384, a=128)
- Large + Flash (h=16384, a=128)
3. **Y-Axis**:
- Label: "Percentage of Latency (%)"
- Range: 0-100%
## Data Analysis
### Spatial Grounding Verification
- Legend position: Top-center
- Color consistency confirmed:
- Blue = GEMMs (dominant in all categories)
- Gray = Flash (appears only in Large + Flash)
- Orange = Softmax (visible in Small and Large)
- Green = DR (small presence in all)
- Yellow = LN (minimal in all)
- Purple = Other (consistent 1% across all)
### Trend Verification
1. **Small Configuration**:
- GEMMs: 68% (blue)
- Softmax: 12% (orange)
- DR: 6% (green)
- LN: 2% (yellow)
- Flash: 1% (gray)
- Other: 1% (purple)
2. **Large Configuration**:
- GEMMs: 94% (blue)
- Softmax: 3% (orange)
- DR: 1% (green)
- LN: 1% (yellow)
- Flash: 1% (gray)
- Other: 1% (purple)
3. **Large + Flash Configuration**:
- GEMMs: 92% (blue)
- Flash: 3% (gray)
- Softmax: 2% (orange)
- DR: 1% (green)
- LN: 1% (yellow)
- Other: 1% (purple)
## Technical Observations
1. **Dominant Component**: GEMMs consistently represent >90% of latency in Large configurations
2. **Flash Impact**: Addition of Flash in Large + Flash configuration reduces GEMMs' share by 2% while maintaining total latency
3. **Softmax Reduction**: Softmax contribution decreases from 12% (Small) to 2% (Large + Flash)
4. **Stable Components**: DR, LN, and Other maintain <3% contribution across all configurations
## Data Table Reconstruction
| Configuration | GEMMs (%) | Flash (%) | Softmax (%) | DR (%) | LN (%) | Other (%) |
|--------------------|-----------|-----------|-------------|--------|--------|-----------|
| Small (h=2560) | 68 | 1 | 12 | 6 | 2 | 1 |
| Large (h=16384) | 94 | 1 | 3 | 1 | 1 | 1 |
| Large + Flash | 92 | 3 | 2 | 1 | 1 | 1 |
## Language Analysis
- All text appears in English
- No non-English content detected
## Critical Findings
1. **Latency Bottleneck**: GEMMs dominate computational latency in large-scale operations
2. **Hardware Impact**: Flash integration shows minimal latency contribution (3%) but enables GEMM optimization
3. **Algorithmic Efficiency**: Softmax and DR components show significant reduction in larger configurations
</details>
Figure 2: The proportion of latency from each transformer component for one layer of various model sizes
On account of their parallel architecture, GPUs are a natural hardware platform for GEMMs. However, the observed throughput for these GEMMs depends on the matrix dimensions due to how the computation is mapped onto the execution units of the GPU (called streaming multiprocessors or SMs for short). As a result, GPU efficiency is sensitive to the model depth and width, which control the arithmetic efficiency of the computation, SM utilization, kernel choice, and the usage of tensor cores versus slower cuda cores. This work tries to determine how best to size models to ensure good performance on GPUs, taking these factors into account. Optimizing model shapes for efficient GEMMs will increase throughput for the entire lifetime of the model, decreasing training time and inference costs We expect best results when the inference GPU is the same as the training GPU, but the guidelines we present could also be useful when the two are different. for production models.
### I-A Contributions
Our contributions are as follows:
- We map the transformer model to its underlying matrix multiplications / GEMMs, and show how each component of the transformer model can suffer from using sub-optimal transformer dimensions.
- We compile a list of GPU performance factors into one document and explain how to choose optimal GEMM dimensions.
- We define rules to ensure transformer models are composed of efficient GEMMs.
## II Related Work
### II-A GPU Characterization of DNNs
DL model training involves the heavy use of GPU kernels, and the characterization of such kernel behavior constitutes a large body of prior work that this paper builds upon. GPU kernels, especially GEMM kernels, are key to improving DL training and inference performance. Therefore, characterizing [20] and optimizing [42, 2, 16, 15] these kernels have received a lot of attention in recent work [22].
Beyond GPU kernels, new algorithms and DL training techniques have been developed to optimize I/O [10, 9] and leverage hardware features like Tensor Cores [40, 31] as efficiently as possible. In addition to the above studies for DL training, exploiting Tensor Core properties has also shown excellent speedups for scientific applications such as iterative solvers [17] and sparse linear algebra subroutines [36].
### II-B Comparison Across DL Accelerators
In recent years, there has emerged a range of acceleration strategies such as wafer-scale (Cerebras), GPUs (AMD and NVIDIA), and tensor processing units (Google). Given this diverse array of new AI accelerators, many pieces of work perform cross-generation and cross-accelerator comparison that have helped elucidate the strengths and weaknesses of each accelerator. Cross-accelerator studies such as [14, 39, 23] enable HPC and cloud customers to choose an appropriate accelerator for their DL workload. We seek to extend this particular line of work by evaluating across various datacenter-class NVIDIA (V100, A100, and H100) and AMD GPUs (MI250X).
### II-C DL Training Performance Guides
The most similar effort to our work is a GPU kernel characterization study for RNNs and CNNs performed in [41]. Since the transformer architecture differs greatly compared to RNNs and CNNs, we believe that our work provides a timely extension. Further, our focus on creating a practical performance guide is similar in nature to the 3D-parallelism optimization for distributed GPU architectures presented in [25].
From the above discussion, one can posit that while many papers exist to optimize DL performance on GPUs [22], such papers tend to neglect the fundamental effects that GPU properties (e.g. Tensor Cores, tiling, wave quantization, etc.) have on model training. Because of this omission, many disparate DL training groups have rediscovered a similar set of model sizing takeaways [19, 18, 33, 3]. We seek to provide explanations for these takeaways from the perspective of fundamental GPU first-principles, and to aggregate these explanations into a concise set of takeaways for efficient transformer training and inference.
## III Background
We will now discuss some of the necessary prerequisite material to understand the performance characteristics of the GPU kernels underlying transformer models.
### III-A GPU Kernels
General Matrix Multiplications (GEMMs) serve as a crucial component for many functions in neural networks, including fully-connected layers, recurrent layers like RNNs, LSTMs, GRUs, and convolutional layers. If $A$ is an $m× k$ matrix and $B$ is a $k× n$ matrix, then the matrix product $AB$ is a simple GEMM. We can then generalize this to $C=α AB+β C$ (in the previous example, $α$ is 1, and $β$ is 0). In a fully-connected layer’s forward pass, the weight matrix would be argument $A$ and input activations would be argument $B$ ( $α$ and $β$ would typically be 1 and 0 as before; $β$ can be 1 in certain scenarios, such as when adding a skip-connection with a linear operation).
Matrix-matrix multiplication is a fundamental operation in numerous scientific and engineering applications, particularly in the realm of deep learning. It is a computationally intensive task that requires significant computational resources for large-scale problems. To address this, various algorithms and computational techniques have been developed to optimize matrix-matrix multiplication operations.
Matrix multiplication variants like batched matrix-matrix (BMM) multiplication kernels have also been introduced to improve the throughput of certain common DL operators like attention [37]. A general formula for a BMM operation is given by Equation 1 below, where $\{A_i\}$ and $\{B_i\}$ are a batch of matrix inputs, $α$ and $β$ are scalar inputs, and $\{C_i\}$ is a batch of output matrices.
$$
C_i=α A_iB_i+β C_i, i=1,...N \tag{1}
$$
### III-B NVIDIA GEMM Implementation and Performance Factors
There are a number of performance factors to consider when analyzing GEMMs on NVIDIA GPU architectures. NVIDIA GPUs divide the output matrix into regions or tiles as shown in Figure 3 and schedule them to one of the available streaming multiprocessors (SM) on the GPU (e.g., A100 GPUs have 108 SMs). Each tile or thread block is processed in a Tensor Core, which NVIDIA introduced for fast tensor operations. NVIDIA Tensor Cores are only available for GEMMs with appropriate dimensions. Tensor Cores can be fully utilized when GEMM dimensions $m$ , $k$ , and $n$ are multiples of 16 bytes and 128 bytes for V100 and A100 GPUs, respectively. Since a FP16 element is 2 bytes, this corresponds to dimension sizes that are multiples of 8 and 64 elements, respectively. If these dimension sizes are not possible, Tensor Cores perform better with larger multiples of 2 bytes.
<details>
<summary>extracted/5378885/figures/tiling.png Details</summary>
### Visual Description
# Technical Document Extraction: Matrix Diagram Analysis
## Overview
The image depicts a block matrix decomposition across three matrices (A, B, C) with labeled dimensions and highlighted tiles. All text is in English.
---
### Matrix A
- **Dimensions**: M (rows) × K (columns)
- **Highlighted Tile**:
- **Color**: Blue
- **Label**: `Mtile` (rows) × `Ktile` (columns)
- **Position**: Bottom-right quadrant of Matrix A
---
### Matrix B
- **Dimensions**: K (rows) × N (columns)
- **Highlighted Tile**:
- **Color**: Yellow
- **Label**: `Ktile` (rows) × `Ntile` (columns)
- **Position**: Top-right quadrant of Matrix B
---
### Matrix C
- **Dimensions**: N (rows) × N (columns)
- **Highlighted Tile**:
- **Color**: Green
- **Label**: `Mtile` (rows) × `Ntile` (columns)
- **Position**: Bottom-right quadrant of Matrix C
- **Additional Label**: `Block_m,n` (top-right quadrant)
---
### Legend & Color Mapping
- **Blue**: `Mtile` (Matrix A)
- **Yellow**: `Ktile` (Matrix B)
- **Green**: `Mtile` (Matrix C)
- **Spatial Grounding**:
- Legend colors match tile colors exactly.
- No explicit legend box; colors are inferred from tile highlights.
---
### Key Observations
1. **Tile Consistency**:
- `Mtile` appears in both Matrix A and C, suggesting a shared submatrix dimension.
- `Ktile` and `Ntile` are unique to their respective matrices.
2. **Block Structure**:
- Matrices A and B share the `Ktile` dimension, implying a partitioning of the K-axis.
- Matrix C’s `Block_m,n` suggests a hierarchical decomposition of the N×N matrix.
---
### Data Extraction Summary
| Matrix | Dimensions | Highlighted Tile | Color | Labels |
|--------|------------|------------------|--------|--------------|
| A | M × K | Mtile × Ktile | Blue | Mtile, Ktile |
| B | K × N | Ktile × Ntile | Yellow | Ktile, Ntile |
| C | N × N | Mtile × Ntile | Green | Mtile, Ntile, Block_m,n |
---
### Notes
- No numerical data or trends are present; the image focuses on structural decomposition.
- All axis labels (M, K, N) and tile labels (`Mtile`, `Ktile`, `Ntile`) are explicitly defined.
- No omitted labels; all textual components are transcribed.
</details>
Figure 3: GEMM tiling [26].
There are multiple tile sizes that the kernel can choose from. If the GEMM size does not divide evenly into the tile size, there will be wasted compute, where the thread block must execute fully on the SM, but only part of the output is necessary. This is called the tile quantization effect, as the output is quantized into discrete tiles.
Another quantization effect is called wave quantization. As the thread blocks are scheduled to SMs, only 108 thread blocks at a time may be scheduled. If, for example, 109 thread blocks must be scheduled, two rounds, or waves, of thread blocks must be scheduled to GPU. The first wave will have 108 thread blocks, and the second wave will have 1. The second wave will have almost the same latency as the first, but with a small fraction of the useful compute. As the matrix size increases, the last or tail wave grows. The throughput will increase, until a new wave is required. Then, the throughput will drop.
### III-C Transformer Models
In this study, we examine a decoder-only transformer architecture popularized by GPT-2 [29]. We focus on this architecture due to its popularity for training very large models [6, 7, 34] , but most of our conclusions also apply to encoder-only models [12, 21]. Due to the nature of the transition between the encoder and decoder, our analysis will largely not apply to encoder-decoder models [37, 30].
<details>
<summary>x3.png Details</summary>
### Visual Description
# Technical Diagram Analysis: Transformer Layer vs Decoder-only LLM
## Diagram Overview
The image compares two neural network architectures through labeled components and directional flow arrows. Two primary blocks are depicted:
1. **Transformer Layer** (left)
2. **Decoder-only LLM** (right)
---
## Transformer Layer Components
### Spatial Layout
- **Vertical Stack** of four processing blocks
- **Bidirectional Connections** between components
### Component Breakdown
1. **Multi-Head Attention (MHA)**
- Position: Bottom-most block
- Connections:
- 3 upward arrows to **Add & Norm** (left)
- 1 rightward arrow to **Add & Norm** (center)
2. **MLP (Multi-Layer Perceptron)**
- Position: Middle block
- Connections:
- 1 upward arrow from **Add & Norm** (center)
- 1 downward arrow to **Add & Norm** (left)
3. **Add & Norm Blocks**
- **Left Block**:
- Receives input from MHA (3 arrows)
- Outputs to MHA (1 arrow)
- **Center Block**:
- Receives input from MLP (1 arrow)
- Outputs to MLP (1 arrow)
- **Right Block**:
- Receives input from MHA (1 arrow)
- Outputs to MHA (1 arrow)
4. **Residual Connections**
- All Add & Norm blocks implement residual connections
- Normalization layers follow addition operations
---
## Decoder-only LLM Architecture
### Component Stack
1. **Embedding Layer**
- Position: Bottom-most
- Function: Input token conversion
2. **Decoder Stack**
- **Four Identical Decoder Blocks** (stacked vertically)
- Each block contains:
- Self-attention mechanism
- Cross-attention mechanism
- Feed-forward network
- Residual connections
3. **Unembedding Layer**
- Position: Top-most
- Function: Output token reconstruction
### Data Flow
- **Bottom-to-Top** processing sequence
- Embedding → Decoder 1 → Decoder 2 → Decoder 3 → Decoder 4 → Unembedding
---
## Key Architectural Differences
| Feature | Transformer Layer | Decoder-only LLM |
|------------------------|----------------------------------|---------------------------------|
| **Directionality** | Bidirectional flow | Unidirectional flow |
| **Component Repetition**| 2 Add & Norm blocks | 4 Decoder blocks |
| **Attention Type** | Multi-head attention | Self/cross-attention |
| **Normalization** | Explicit Add & Norm blocks | Implicit in decoder blocks |
---
## Technical Notes
1. **Transformer Layer**:
- Follows standard transformer block architecture
- Contains pre-layer normalization (Add & Norm before MHA/MLP)
2. **Decoder-only LLM**:
- Resembles GPT-style architecture
- Lacks encoder components present in full transformers
- Uses tied embeddings (embedding/unembedding weight sharing)
3. **Arrow Conventions**:
- Black arrows: Data flow direction
- Dashed lines: Residual connections
- Solid lines: Primary data paths
---
## Missing Elements
- No numerical data or performance metrics
- No explicit parameter counts
- No activation function specifications
- No positional encoding details
This diagram focuses on architectural comparison rather than quantitative analysis.
</details>
Figure 4: The transformer architecture [29].
For a mapping from variables to their definitions, see Table I. Initially, the network takes in raw input tokens which are then fed into a word embedding table of size $v× h$ . These token embeddings are then merged with learned positional embeddings of size $s× h$ . The output from the embedding layer, which serves as the input for the transformer block, is a 3-D tensor of size $s× b× h$ . Each layer of the transformer comprises a self-attention block with attention heads, followed by a two-layer multi-layer perceptron (MLP) that expands the hidden size to $4h$ before reducing it back to $h$ . The input and output sizes for each transformer layer remain consistent at $s× b× h$ . The final output from the last transformer layer is projected back into the vocabulary dimension to compute the cross-entropy loss.
| a b h | Number of attention heads Microbatch size Hidden dimension size | s t v | Sequence length Tensor-parallel size Vocabulary size |
| --- | --- | --- | --- |
| L | Number of transformer layers | | |
TABLE I: Variable names.
Each transformer layer consists of the following matrix multiplication operators:
1. Attention key, value, query transformations: These can be expressed as a single matrix multiplication of size: $(b· s,h)×(h,\frac{3h}{t})$ . Output is of size $(b· s,\frac{3h}{t})$ .
1. Attention score computation: $b· a/t$ batched matrix multiplications (BMMs), each of size $(s,\frac{h}{a})×(\frac{h}{a},s)$ . Output is of size $(\frac{b· a}{t},s,s)$ .
1. Attention over value computation: $\frac{b· a}{t}$ batched matrix multiplications of size $(s,s)×(s,\frac{h}{a})$ . Output is of size $(\frac{b· a}{t},s,\frac{h}{a})$ .
1. Post-attention linear projection: a single matrix multiplication of size $(b· s,\frac{h}{t})×(\frac{h}{t},h)$ . Output is of size $(b· s,h)$ .
1. Matrix multiplications in the MLP block of size $(b· s,h)×(h,\frac{4h}{t})$ and $(b· s,\frac{4h}{t})×(\frac{4h}{t},h)$ . Outputs are of size $(b· s,\frac{4h}{t})$ and $(b· s,h)$ .
The total number of parameters in a transformer can be calculated using the formula $P=12h^2L+13hL+(v+s)h$ . This is commonly approximated as $P=12h^2L$ , omitting the lower-order terms.
| Module Input Embedding Layer Norm 1 | GEMM Size — — | Figure — — |
| --- | --- | --- |
| $QKV$ Transform | $(b· s,h)×(h,\frac{3h}{t})$ | 16 |
| Attention Score | $(\frac{b· a}{t},s,\frac{h}{a})×(\frac{b· a}{t},\frac{h}{a},s)$ | 7a 8 |
| Attn over Value | $(\frac{b· a}{t},s,s)×(\frac{b· a}{t},s,\frac{h}{a})$ | 7b 9 |
| Linear Projection | $(b· s,\frac{h}{t})×(\frac{h}{t},h)$ | 19 |
| Layer Norm 2 | — | — |
| MLP $h$ to $4h$ | $(b· s,h)×(h,\frac{4h}{t})$ | 10a |
| MLP $4h$ to $h$ | $(b· s,\frac{4h}{t})×(\frac{4h}{t},h)$ | 10b |
| Linear Output | $(b· s,v)×(v,h)$ | 20 |
TABLE II: Summary of operators in the transformer layer considered in this paper, along with the size of the GEMMs used to execute these operators.
Here, we make the assumption that the projection weight dimension in the multi-headed attention block is $h/a$ , which is the default in existing implementations like Megatron [33] and GPT-NeoX [3].
The total number of compute operations needed to perform a forward pass for training is then $24bsh^2+4bs^2h=24bsh^2≤ft(1+\frac{s}{6h}\right)$ .
Parallelization Across GPUs. Due to the extreme size of modern transformer models, and the additional buffers and activations needed for training, it is common to split transformers across multiple GPUs using tensor and pipeline parallelism [33, 25]. Since this paper focuses on the computations being done on a single GPU, we will largely ignore parallelism. When we speak of the hidden size of a model, that should be understood to mean the hidden size per GPU. For example, with $t$ -way tensor parallelism, the hidden size per GPU is typically $h/t$ . We leave an analysis of the implications of pipeline and sequence parallelism on optimal model shapes to future work.
| AWS p4d ORNL Summit SDSC Expanse | NVIDIA NVIDIA NVIDIA | 8x(A100 40GB) 6x(V100 16GB) 4x(V100 32GB) | Intel Cascade Lake 8275CL IBM POWER9 AMD EPYC 7742 | Amazon EFA [400 Gbps] InfiniBand EDR [200 Gbps] InfiniBand HDR [200 Gbps] | NVLINK [600 GBps] NVLINK (2x3) [100 GBps] NVLINK [100 GBps] |
| --- | --- | --- | --- | --- | --- |
TABLE III: Hardware systems used in this paper.
## IV Experimental Setup
### IV-A Hardware Setup
All experimental results were measured on one of the systems described in Table III. We used compute from a wide variety of sources such as Oak Ridge National Laboratory (ORNL), the San Diego Supercomputing Center (SDSC), and cloud providers such as AWS and Cirrascale. In order to increase the coverage of our takeaways as much as possible, we have included a diverse range of systems in this study.
### IV-B Software Setup
Each hardware setup has used slightly different software. For the V100 experiments, we used PyTorch 1.12.1 and CUDA 11.3. For the A100 experiments, we used PyTorch 1.13.1, CUDA 11.7. For H100 experiments, we used PyTorch 2.1.0 and CUDA 12.2.2. For MI250X experiments, we used PyTorch 2.1.1 and ROCM 5.6.0. All transformer implementations are ported from GPT-NeoX [3].
## V GEMM Results
<details>
<summary>extracted/5378885/figures/mm/basicGemmMSweep.png Details</summary>
### Visual Description
# Technical Document Extraction: GPU Throughput Analysis
## Chart Description
This image is a **line chart** comparing the throughput performance of two GPU models (A100 and V100) across varying computational loads (denoted as "m"). The chart visualizes throughput in teraflops per second (TFLOP/s) against the parameter "m".
---
### Axis Labels and Markers
- **X-axis (Horizontal):**
- Label: `m`
- Markers: `0`, `2000`, `4000`, `6000`, `8000`
- **Y-axis (Vertical):**
- Label: `Throughput (TFLOP/s)`
- Markers: `0`, `50`, `100`, `150`, `200`
---
### Legend
- **Placement:** Right side of the chart (outside the plot area).
- **Labels and Colors:**
- `A100` (Blue line with circular markers)
- `V100` (Orange line with circular markers)
---
### Data Series and Trends
#### A100 (Blue Line)
- **Trend:** Steep upward curve, indicating exponential growth in throughput as "m" increases.
- **Data Points (x, y):**
- (0, 0)
- (1000, 30)
- (2000, 120)
- (4000, 170)
- (8000, 210)
#### V100 (Orange Line)
- **Trend:** Gradual upward slope, indicating linear growth in throughput as "m" increases.
- **Data Points (x, y):**
- (0, 0)
- (1000, 10)
- (2000, 60)
- (4000, 80)
- (8000, 90)
---
### Key Observations
1. **Performance Gap:**
- A100 consistently outperforms V100 across all values of "m".
- At "m = 8000", A100 achieves **210 TFLOP/s**, while V100 achieves **90 TFLOP/s**.
2. **Scalability:**
- A100 demonstrates superior scalability, with throughput increasing by **~180 TFLOP/s** from "m = 0" to "m = 8000".
- V100 shows minimal improvement, with throughput rising only **~80 TFLOP/s** over the same range.
---
### Spatial Grounding
- **Legend Position:** Right-aligned, outside the plot area.
- **Data Point Verification:**
- Blue markers (A100) align with the blue line.
- Orange markers (V100) align with the orange line.
---
### Notes
- No additional text, tables, or embedded diagrams are present.
- All textual information is in **English**.
- The chart focuses on quantitative performance metrics without qualitative annotations.
</details>
(a) $(m,4096)×(4096,m)$
<details>
<summary>extracted/5378885/figures/mm/basicGemmKSweep.png Details</summary>
### Visual Description
# Technical Document Analysis: GPU Throughput Comparison
## Image Description
The image is a **line chart** comparing the throughput (in TFLOPs per second) of two GPUs, **A100** and **V100**, plotted against a variable **k** (x-axis). The chart includes labeled axes, a legend, and two distinct data series.
---
## Key Components
### Axis Labels
- **Y-axis**: "Throughput (TFLOP/s)"
- Range: 0 to 150 (increments of 50).
- **X-axis**: "k"
- Range: 0 to 500 (increments of 100).
### Legend
- Located on the **right side** of the chart.
- **Blue line**: A100 GPU.
- **Orange line**: V100 GPU.
---
## Data Series Analysis
### A100 (Blue Line)
- **Trend**:
- Starts at (0, 0).
- Sharp upward slope to ~100 TFLOP/s at **k=50**.
- Continues rising with fluctuations, peaking at **~170 TFLOP/s** around **k=250**.
- Dips to **~140 TFLOP/s** at **k=300**, then rises again to **~160 TFLOP/s** at **k=400**, and finally reaches **~180 TFLOP/s** at **k=500**.
- **Key Data Points**:
- (0, 0)
- (50, 100)
- (250, 170)
- (300, 140)
- (400, 160)
- (500, 180)
### V100 (Orange Line)
- **Trend**:
- Starts at (0, 0).
- Gradual upward slope to **~50 TFLOP/s** at **k=200**.
- Dips to **~40 TFLOP/s** at **k=300**, then rises to **~70 TFLOP/s** at **k=400**, and reaches **~80 TFLOP/s** at **k=500**.
- **Key Data Points**:
- (0, 0)
- (200, 50)
- (300, 40)
- (400, 70)
- (500, 80)
---
## Spatial Grounding
- **Legend Position**: Right side of the chart.
- **Color Consistency**:
- Blue data points (A100) match the legend.
- Orange data points (V100) match the legend.
---
## Observations
1. **A100 outperforms V100** across all values of **k**, with a significantly higher throughput.
2. **A100 exhibits volatility** (e.g., dip at **k=300**), while **V100 shows a smoother trend** with a mid-chart dip.
3. **No additional text or data tables** are present in the image.
---
## Conclusion
The chart demonstrates a clear performance gap between the A100 and V100 GPUs, with A100 achieving higher throughput values for all tested **k** values. The trends suggest A100’s throughput is more sensitive to changes in **k**, while V100’s performance remains relatively stable but lower.
</details>
(b) $(27648,4096)×(4096,k)$
<details>
<summary>extracted/5378885/figures/mm/basicGemmLargeKSweep.png Details</summary>
### Visual Description
# Technical Document Analysis: GPU Throughput Comparison
## Chart Type
Line chart comparing throughput performance of two GPU models over a variable parameter `k`.
## Axis Labels
- **X-axis**: `k` (ranges from 0 to 6000, increments of 1000)
- **Y-axis**: `Throughput (TFLOP/s)` (ranges from 0 to 150, increments of 50)
## Legend
- **Location**: Right side of the chart
- **Labels**:
- `A100` (blue line with circular markers)
- `V100` (orange line with circular markers)
## Data Trends
### A100 (Blue Line)
- **Initial Value**: Starts at ~10 TFLOP/s at `k=0`
- **Trend**: Steadily increases with minor fluctuations
- **Final Value**: Reaches ~170 TFLOP/s at `k=6000`
- **Key Observations**:
- Sharp rise from `k=0` to `k=1000` (10 → ~120 TFLOP/s)
- Gradual ascent with periodic dips (e.g., ~150 → ~140 at `k=3000`)
- Consistent upward trajectory after `k=4000`
### V100 (Orange Line)
- **Initial Value**: Starts at 0 TFLOP/s at `k=0`, jumps to ~60 TFLOP/s
- **Trend**: Plateaus at ~80 TFLOP/s with minor oscillations
- **Final Value**: Remains ~80 TFLOP/s at `k=6000`
- **Key Observations**:
- Immediate spike at `k=0` (0 → 60 TFLOP/s)
- Stable performance with slight fluctuations (e.g., 80 → 85 → 78 between `k=2000` and `k=4000`)
- No significant growth beyond `k=1000`
## Spatial Grounding
- **Legend Position**: Right-aligned, outside the plot area
- **Data Point Colors**:
- Blue markers correspond to `A100` (confirmed)
- Orange markers correspond to `V100` (confirmed)
## Component Isolation
1. **Header**: No explicit header text
2. **Main Chart**:
- Dual-line plot with distinct markers
- Y-axis scaled logarithmically? (No, linear scale)
3. **Footer**: No footer text
## Critical Data Points
| `k` Value | A100 Throughput (TFLOP/s) | V100 Throughput (TFLOP/s) |
|-----------|---------------------------|---------------------------|
| 0 | 10 | 60 |
| 1000 | ~120 | ~70 |
| 2000 | ~140 | ~75 |
| 3000 | ~150 | ~80 |
| 4000 | ~160 | ~82 |
| 5000 | ~170 | ~80 |
| 6000 | ~170 | ~80 |
## Notes
- The `A100` demonstrates superior scalability with `k`, achieving ~17x higher throughput than `V100` at `k=6000`.
- `V100` shows no meaningful improvement beyond `k=1000`, suggesting potential hardware limitations.
- Both lines use circular markers, but `A100` exhibits more pronounced variability in its growth pattern.
</details>
(c) $(2304,4096)×(4096,k)$ .
Figure 5: Throughput (in teraFLOP/s) for matrix multiplication computations of various sizes.
Figure 5 shows the throughput (in teraFLOP/s) of matrix multiplication computations of various sizes on two types of NVIDIA GPUs. As the GEMM size increases, the operation becomes more computationally intensive and uses memory more efficiently (GEMMs are memory-bound for small matrices). As shown in Figure 5 a, throughput of the GEMM kernel increases with matrix size as the kernel becomes compute-bound. However, wave quantization inefficiencies reduce the throughput when the GEMM size crosses certain thresholds. The effects of wave quantization can be seen clearly in Figure 5 b. Additionally, when the size of the GEMM is sufficiently large, PyTorch may automatically choose a tile size that decreases quantization effects. In Figure 5 c, the effects of wave quantization are lessened, as PyTorch is able to better balance the improvements from GEMM parallelization and inefficiencies from wave quantization to improve throughput.
<details>
<summary>extracted/5378885/figures/bmm/v100/b_sweep.png Details</summary>
### Visual Description
# Technical Document Analysis: Throughput vs. b (Log Scale)
## Chart Description
The image is a line graph titled **"Throughput vs. b (Log Scale)"**, depicting the relationship between throughput (measured in TFLOP/s) and the variable **b** (on a logarithmic scale). The graph includes four data series, each represented by a distinct line with unique markers and colors. Below is a detailed breakdown of the components, trends, and data points.
---
### **Axis Labels and Scales**
- **Y-Axis**:
- Label: **"Throughput (TFLOP/s)"**
- Range: **0 to 100** (in increments of 25).
- **X-Axis**:
- Label: **"b"**
- Scale: **Logarithmic** (base 2).
- Markers: **2¹, 2³, 2⁵** (x-values).
---
### **Legend**
The legend is positioned on the **right side** of the graph and maps colors/markers to **m** values:
1. **Blue line with circles**: **m = 1024**
2. **Green line with circles**: **m = 4096**
3. **Orange line with squares**: **m = 2048**
4. **Red line with circles**: **m = 8192**
---
### **Data Series and Trends**
#### 1. **m = 1024 (Blue Line with Circles)**
- **Trend**: Steady upward slope.
- **Data Points**:
- At **2¹**: ~50 TFLOP/s
- At **2³**: ~60 TFLOP/s
- At **2⁵**: ~70 TFLOP/s
#### 2. **m = 4096 (Green Line with Circles)**
- **Trend**: Sharp decline at **2¹**, followed by a slight downward trend.
- **Data Points**:
- At **2¹**: ~90 TFLOP/s
- At **2³**: ~75 TFLOP/s
- At **2⁵**: ~70 TFLOP/s
#### 3. **m = 2048 (Orange Line with Squares)**
- **Trend**: Gradual increase at **2¹**, then plateaus.
- **Data Points**:
- At **2¹**: ~70 TFLOP/s
- At **2³**: ~75 TFLOP/s
- At **2⁵**: ~75 TFLOP/s
#### 4. **m = 8192 (Red Line with Circles)**
- **Trend**: Sharp decline at **2¹**, followed by stabilization.
- **Data Points**:
- At **2¹**: ~100 TFLOP/s
- At **2³**: ~75 TFLOP/s
- At **2⁵**: ~75 TFLOP/s
---
### **Key Observations**
1. **Logarithmic X-Axis**: The x-axis (b) increases exponentially (2¹ → 2³ → 2⁵), which explains the non-linear spacing of data points.
2. **Performance Trends**:
- Higher **m** values (e.g., **m = 8192**) show steeper initial declines in throughput.
- Lower **m** values (e.g., **m = 1024**) exhibit more gradual improvements.
3. **Stabilization**: All lines converge to similar throughput values (~70–75 TFLOP/s) at **2⁵**, suggesting diminishing returns at higher **b** values.
---
### **Footer Notes**
- **Source**: "Example Data Source"
- **Note**: "Logarithmic scale on x-axis (b)"
---
### **Spatial Grounding**
- **Legend Position**: Right side of the graph.
- **Data Point Verification**:
- Colors and markers for each **m** value match the legend exactly.
- Example: At **2¹**, the red circle (m = 8192) aligns with the legend.
---
### **Conclusion**
The graph illustrates how throughput varies with **b** (logarithmically scaled) for different **m** values. Higher **m** values initially achieve higher throughput but degrade more sharply as **b** increases, while lower **m** values show more stable performance. All data points and trends are consistent with the logarithmic scale and legend annotations.
</details>
(a) $(b,m,m)×(b,m,m)$ BMM on V100 GPU.
<details>
<summary>extracted/5378885/figures/bmm/a100/b_sweep.png Details</summary>
### Visual Description
# Technical Document Extraction: Throughput vs. Parameter 'b'
## Chart Description
This image is a **line chart** visualizing the relationship between computational throughput (measured in TFLOP/s) and a parameter labeled 'b' (on a logarithmic scale). The chart includes four distinct data series, each represented by a unique line style, color, and marker.
---
### **Axis Labels and Scale**
- **X-axis (Horizontal):**
- Label: `b`
- Values: Logarithmic scale with markers at `2¹`, `2³`, and `2⁵` (i.e., 2, 8, 32).
- **Y-axis (Vertical):**
- Label: `Throughput (TFLOP/s)`
- Range: 0 to 300 (linear scale).
---
### **Legend**
- **Position:** Right side of the chart.
- **Labels and Colors:**
- `m=1024` → **Blue line** with **circle markers** (`●`).
- `m=2048` → **Orange line** with **square markers** (`■`).
- `m=4096` → **Green line** with **triangle markers** (`▲`).
- `m=8192` → **Red line** with **diamond markers** (`◆`).
---
### **Data Series and Trends**
#### 1. **m=1024 (Blue Line)**
- **Trend:** Starts at ~60 TFLOP/s at `b=2¹`, increases steadily to ~140 TFLOP/s at `b=2³`, then rises sharply to ~220 TFLOP/s at `b=2⁵`.
- **Key Points:**
- `b=2¹`: ~60 TFLOP/s
- `b=2³`: ~140 TFLOP/s
- `b=2⁵`: ~220 TFLOP/s
#### 2. **m=2048 (Orange Line)**
- **Trend:** Begins at ~180 TFLOP/s at `b=2¹`, rises slightly to ~230 TFLOP/s at `b=2³`, then plateaus at ~240 TFLOP/s at `b=2⁵`.
- **Key Points:**
- `b=2¹`: ~180 TFLOP/s
- `b=2³`: ~230 TFLOP/s
- `b=2⁵`: ~240 TFLOP/s
#### 3. **m=4096 (Green Line)**
- **Trend:** Starts at ~240 TFLOP/s at `b=2¹`, remains stable at ~250 TFLOP/s at `b=2³`, then drops to ~200 TFLOP/s at `b=2⁵`.
- **Key Points:**
- `b=2¹`: ~240 TFLOP/s
- `b=2³`: ~250 TFLOP/s
- `b=2⁵`: ~200 TFLOP/s
#### 4. **m=8192 (Red Line)**
- **Trend:** Begins at ~260 TFLOP/s at `b=2¹`, dips slightly to ~210 TFLOP/s at `b=2³`, then stabilizes at ~210 TFLOP/s at `b=2⁵`.
- **Key Points:**
- `b=2¹`: ~260 TFLOP/s
- `b=2³`: ~210 TFLOP/s
- `b=2⁵`: ~210 TFLOP/s
---
### **Spatial Grounding and Validation**
- **Legend Accuracy:**
- All line colors and markers match the legend labels (e.g., blue line = `m=1024`).
- **Data Point Consistency:**
- Confirmed that line colors and markers align with their respective `m` values across all `b` values.
---
### **Summary of Observations**
- **Throughput Trends:**
- Lower `m` values (e.g., `m=1024`) show increasing throughput with higher `b`.
- Higher `m` values (e.g., `m=4096`, `m=8192`) exhibit diminishing returns or declines at larger `b`.
- **Critical Insight:**
- The optimal `b` value for maximum throughput varies by `m`. For example, `m=4096` peaks at `b=2³`, while `m=1024` benefits most at `b=2⁵`.
---
### **Additional Notes**
- No textual blocks, tables, or non-English content are present.
- The chart focuses solely on quantitative relationships between `b` and throughput, with no qualitative annotations.
</details>
(b) $(b,m,m)×(b,m,m)$ BMM on A100 GPU.
<details>
<summary>extracted/5378885/figures/bmm/v100/BmmMSweep.png Details</summary>
### Visual Description
# Technical Document Extraction: Line Chart Analysis
## 1. **Axis Labels and Titles**
- **X-axis**: Labeled `m`, with values marked at `2^6`, `2^8`, `2^10`, and `2^12`.
- **Y-axis**: Labeled `Throughput (TFLOP/s)`, with a linear scale from `0` to `100`.
## 2. **Legend and Data Series**
- **Legend Position**: Top-left corner of the chart.
- **Data Series**:
- **Blue Line**: `b=1` (Throughput values).
- **Orange Line**: `b=4` (Throughput values).
- **Green Line**: `b=16` (Throughput values).
## 3. **Key Trends and Data Points**
### a. **Green Line (`b=16`)**
- **Trend**: Steadily increases from near `0` at `m=2^6` to approximately `75 TFLOP/s` at `m=2^12`.
- **Data Points**:
- `m=2^6`: ~`0 TFLOP/s`
- `m=2^8`: ~`45 TFLOP/s`
- `m=2^10`: ~`70 TFLOP/s`
- `m=2^12`: ~`75 TFLOP/s`
### b. **Orange Line (`b=4`)**
- **Trend**: Sharp rise from near `0` at `m=2^6` to ~`75 TFLOP/s` at `m=2^10`, then plateaus.
- **Data Points**:
- `m=2^6`: ~`0 TFLOP/s`
- `m=2^8`: ~`20 TFLOP/s`
- `m=2^10`: ~`70 TFLOP/s`
- `m=2^12`: ~`75 TFLOP/s`
### c. **Blue Line (`b=1`)**
- **Trend**: Gradual increase from near `0` at `m=2^6`, accelerating sharply after `m=2^10` to ~`95 TFLOP/s` at `m=2^12`.
- **Data Points**:
- `m=2^6`: ~`0 TFLOP/s`
- `m=2^8`: ~`10 TFLOP/s`
- `m=2^10`: ~`60 TFLOP/s`
- `m=2^12`: ~`95 TFLOP/s`
## 4. **Spatial Grounding and Color Verification**
- **Legend Colors**:
- Blue (`b=1`) matches the blue line.
- Orange (`b=4`) matches the orange line.
- Green (`b=16`) matches the green line.
- **Data Point Accuracy**: All data points align with their respective legend colors.
## 5. **Component Isolation**
- **Main Chart**: Line plot with logarithmic x-axis (`m`) and linear y-axis (`Throughput`).
- **No Additional Components**: No headers, footers, or embedded text blocks.
## 6. **Summary of Observations**
- **Performance Scaling**:
- `b=16` (green) achieves the highest throughput early and maintains a steady increase.
- `b=4` (orange) shows rapid growth but plateaus near `75 TFLOP/s`.
- `b=1` (blue) lags initially but surpasses other series at `m=2^12`.
- **Implications**: Higher `b` values correlate with better throughput, though diminishing returns are observed for `b=4` and `b=16` at larger `m`.
## 7. **Final Notes**
- The chart uses a **logarithmic scale** for the x-axis (`m`), as indicated by the exponential notation (`2^6`, `2^8`, etc.).
- No textual data or tables are present in the image.
- All trends and data points are visually consistent with the legend and axis labels.
</details>
(c) $(b,m,4096)×(b,4096,m)$ BMM on V100 GPU.
<details>
<summary>extracted/5378885/figures/bmm/a100/BmmMSweep.png Details</summary>
### Visual Description
# Technical Document Extraction: Line Chart Analysis
## Labels and Axis Titles
- **X-axis**: Labeled `m` with values: `2^6`, `2^8`, `2^10`, `2^12`.
- **Y-axis**: Labeled `Throughput (TFLOP/s)` with values: `0`, `50`, `100`, `150`, `200`, `250`, `300`.
## Legend
- **Location**: Right side of the chart.
- **Entries**:
- `b=1` (Blue line with circle markers).
- `b=4` (Orange line with square markers).
- `b=16` (Green line with diamond markers).
## Data Series and Trends
### Series 1: `b=1` (Blue)
- **Trend**: Gradual upward slope with consistent growth.
- **Data Points**:
- `m=2^6`: ~0 TFLOP/s.
- `m=2^8`: ~10 TFLOP/s.
- `m=2^9`: ~50 TFLOP/s.
- `m=2^10`: ~130 TFLOP/s.
- `m=2^11`: ~220 TFLOP/s.
- `m=2^12`: ~250 TFLOP/s.
### Series 2: `b=4` (Orange)
- **Trend**: Accelerated upward slope, surpassing `b=1` after `m=2^10`.
- **Data Points**:
- `m=2^6`: ~0 TFLOP/s.
- `m=2^8`: ~10 TFLOP/s.
- `m=2^9`: ~60 TFLOP/s.
- `m=2^10`: ~180 TFLOP/s.
- `m=2^11`: ~240 TFLOP/s.
- `m=2^12`: ~250 TFLOP/s.
### Series 3: `b=16` (Green)
- **Trend**: Steep initial rise, peaks at `m=2^11`, then declines.
- **Data Points**:
- `m=2^6`: ~0 TFLOP/s.
- `m=2^7`: ~10 TFLOP/s.
- `m=2^8`: ~50 TFLOP/s.
- `m=2^9`: ~170 TFLOP/s.
- `m=2^10`: ~230 TFLOP/s.
- `m=2^11`: ~240 TFLOP/s.
- `m=2^12`: ~210 TFLOP/s.
## Spatial Grounding
- **Legend Position**: Right-aligned, outside the plot area.
- **Color Consistency**:
- Blue (`b=1`) matches all blue data points.
- Orange (`b=4`) matches all orange data points.
- Green (`b=16`) matches all green data points.
## Key Observations
1. **Performance Scaling**: Higher `b` values (e.g., `b=16`) achieve higher throughput earlier but plateau or decline at larger `m`.
2. **Convergence**: All series converge near `m=2^12` (~250 TFLOP/s), suggesting diminishing returns at extreme scales.
3. **Efficiency**: `b=4` and `b=16` outperform `b=1` significantly at mid-to-high `m` values.
## Notes
- No non-English text or additional components (e.g., tables, heatmaps) are present.
- The chart focuses on computational throughput scaling with parameter `b` and input size `m`.
</details>
(d) $(b,m,4096)×(b,4096,m)$ BMM on A100 GPU.
Figure 6: Throughput (in teraFLOP/s) for batched matrix multiplication (BMM) computations with various dimensions.
Figure 6 shows the throughput (in teraFLOP/s) of batched matrix multiplication (BMMs) computations of various sizes. Since BMMs are composed of GEMMs, the same wave quantization effects would apply (though they do not for these BMM sizes and on these GPU architectures). BMM throughput also increases as the size of the BMM and arithmetic intensity increases.
## VI Transformer Results
### VI-A Transformer as a Series of GEMMs
The settings of the various hyperparameters in the transformer layer controlling its shape all have an impact on its observed end-to-end throughput. Some of these hyperparameters can affect performance in subtle ways. The purpose of this section is to map GEMM performance to transformer throughput, use these mappings to explain the performance effects of relevant hyperparameters, and finally to boil down these effects into a series of practical takeaways.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a32.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Analysis
## Chart Overview
The image depicts a line chart titled **"Attention Key Query Score, a=32"**, analyzing the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOP/s)** (y-axis). The chart includes seven data series representing different **h/a** (hidden size to attention head ratio) values, with trends visualized across a logarithmic scale of Hidden Size.
---
### Key Components
1. **Title**:
- *"Attention Key Query Score, a=32"*
- Indicates the analysis focuses on attention mechanisms with a fixed parameter `a=32`.
2. **Axes**:
- **X-axis (Hidden Size)**:
- Logarithmic scale ranging from `0` to `32768`.
- Markers at intervals: `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`.
- **Y-axis (Throughput)**:
- Linear scale from `0` to `200` TFLOP/s.
- Markers at intervals: `0`, `50`, `100`, `150`, `200`.
3. **Legend**:
- Located in the **top-right corner**.
- Colors map to **h/a ratios**:
- `1` (blue), `2` (orange), `4` (green), `8` (red), `16` (purple), `32` (brown), `64` (pink).
- Confirmed spatial grounding: All line colors match legend entries exactly.
---
### Data Series Trends
1. **h/a = 1 (Blue Line)**:
- **Trend**: Gradual upward slope with minor fluctuations.
- **Key Points**:
- At Hidden Size `4096`: ~30 TFLOP/s.
- At Hidden Size `32768`: ~90 TFLOP/s.
2. **h/a = 2 (Orange Line)**:
- **Trend**: Steeper initial rise, then plateaus.
- **Key Points**:
- At Hidden Size `4096`: ~50 TFLOP/s.
- At Hidden Size `32768`: ~140 TFLOP/s.
3. **h/a = 4 (Green Line)**:
- **Trend**: Consistent upward trajectory with minor dips.
- **Key Points**:
- At Hidden Size `4096`: ~70 TFLOP/s.
- At Hidden Size `32768`: ~145 TFLOP/s.
4. **h/a = 8 (Red Line)**:
- **Trend**: Sharp rise, followed by stabilization with oscillations.
- **Key Points**:
- At Hidden Size `4096`: ~90 TFLOP/s.
- At Hidden Size `32768`: ~200 TFLOP/s.
5. **h/a = 16 (Purple Line)**:
- **Trend**: Rapid ascent, then sustained high throughput with minor dips.
- **Key Points**:
- At Hidden Size `4096`: ~120 TFLOP/s.
- At Hidden Size `32768`: ~200 TFLOP/s.
6. **h/a = 32 (Brown Line)**:
- **Trend**: Highest throughput, peaking early and maintaining near-maximum.
- **Key Points**:
- At Hidden Size `4096`: ~150 TFLOP/s.
- At Hidden Size `32768`: ~210 TFLOP/s.
7. **h/a = 64 (Pink Line)**:
- **Trend**: Highest throughput overall, with slight fluctuations.
- **Key Points**:
- At Hidden Size `4096`: ~160 TFLOP/s.
- At Hidden Size `32768`: ~210 TFLOP/s.
---
### Observations
- **Scaling Behavior**: Higher **h/a ratios** (e.g., 32, 64) achieve significantly higher throughput, especially at larger Hidden Sizes.
- **Efficiency**: Lower ratios (e.g., 1, 2) show diminishing returns as Hidden Size increases.
- **Stability**: Lines for h/a ≥ 8 exhibit smoother trends compared to lower ratios, which show more volatility.
---
### Notes
- No non-English text detected.
- No embedded data tables or heatmaps present.
- All textual elements (labels, titles, legend) are in English.
This analysis confirms that throughput scales non-linearly with Hidden Size, with higher h/a ratios achieving optimal performance at larger scales.
</details>
(a) Attention key-query score GEMM throughput for 32 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a32.png Details</summary>
### Visual Description
# Technical Document Extraction: Line Graph Analysis
## Image Description
The image is a line graph titled **"Attention over Values, a=32"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOPs/s)** (y-axis). The graph includes six data series, each represented by a distinct colored line, corresponding to different **h/a** ratios. The legend is positioned on the right side of the graph.
---
## Key Components
### 1. **Title**
- **Text**: "Attention over Values, a=32"
- **Purpose**: Indicates the focus of the analysis (attention mechanisms) and a fixed parameter value (a=32).
### 2. **Axes**
- **X-axis (Hidden Size)**:
- **Label**: "Hidden Size"
- **Range**: 0 to 32768
- **Tick Marks**: 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768
- **Y-axis (Throughput)**:
- **Label**: "Throughput (TFLOPs/s)"
- **Range**: 0 to 200
- **Tick Marks**: 0, 50, 100, 150, 200
### 3. **Legend**
- **Location**: Right side of the graph
- **Entries**:
- **Blue**: h/a = 1
- **Orange**: h/a = 2
- **Green**: h/a = 4
- **Red**: h/a = 8
- **Purple**: h/a = 16
- **Pink**: h/a = 64
---
## Data Series and Trends
### 1. **h/a = 1 (Blue Line)**
- **Trend**: Starts at 0 and increases steadily with minor fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~50 TFLOPs/s
- At Hidden Size = 8192: ~70 TFLOPs/s
- At Hidden Size = 12288: ~80 TFLOPs/s
- At Hidden Size = 16384: ~90 TFLOPs/s
- At Hidden Size = 20480: ~95 TFLOPs/s
- At Hidden Size = 24576: ~100 TFLOPs/s
- At Hidden Size = 28672: ~110 TFLOPs/s
- At Hidden Size = 32768: ~120 TFLOPs/s
### 2. **h/a = 2 (Orange Line)**
- **Trend**: Similar to h/a = 1 but with slightly higher throughput and minor fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~60 TFLOPs/s
- At Hidden Size = 8192: ~90 TFLOPs/s
- At Hidden Size = 12288: ~110 TFLOPs/s
- At Hidden Size = 16384: ~120 TFLOPs/s
- At Hidden Size = 20480: ~130 TFLOPs/s
- At Hidden Size = 24576: ~140 TFLOPs/s
- At Hidden Size = 28672: ~150 TFLOPs/s
- At Hidden Size = 32768: ~160 TFLOPs/s
### 3. **h/a = 4 (Green Line)**
- **Trend**: Higher throughput than h/a = 2, with more pronounced fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~80 TFLOPs/s
- At Hidden Size = 8192: ~120 TFLOPs/s
- At Hidden Size = 12288: ~140 TFLOPs/s
- At Hidden Size = 16384: ~150 TFLOPs/s
- At Hidden Size = 20480: ~160 TFLOPs/s
- At Hidden Size = 24576: ~170 TFLOPs/s
- At Hidden Size = 28672: ~180 TFLOPs/s
- At Hidden Size = 32768: ~190 TFLOPs/s
### 4. **h/a = 8 (Red Line)**
- **Trend**: Higher throughput than h/a = 4, with significant peaks and troughs.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~100 TFLOPs/s
- At Hidden Size = 8192: ~150 TFLOPs/s
- At Hidden Size = 12288: ~180 TFLOPs/s
- At Hidden Size = 16384: ~190 TFLOPs/s
- At Hidden Size = 20480: ~200 TFLOPs/s
- At Hidden Size = 24576: ~210 TFLOPs/s
- At Hidden Size = 28672: ~220 TFLOPs/s
- At Hidden Size = 32768: ~230 TFLOPs/s
### 5. **h/a = 16 (Purple Line)**
- **Trend**: Highest throughput among all series, with sharp peaks and troughs.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~120 TFLOPs/s
- At Hidden Size = 8192: ~170 TFLOPs/s
- At Hidden Size = 12288: ~200 TFLOPs/s
- At Hidden Size = 16384: ~210 TFLOPs/s
- At Hidden Size = 20480: ~220 TFLOPs/s
- At Hidden Size = 24576: ~230 TFLOPs/s
- At Hidden Size = 28672: ~240 TFLOPs/s
- At Hidden Size = 32768: ~250 TFLOPs/s
### 6. **h/a = 64 (Pink Line)**
- **Trend**: Highest throughput with the most pronounced fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~140 TFLOPs/s
- At Hidden Size = 8192: ~200 TFLOPs/s
- At Hidden Size = 12288: ~220 TFLOPs/s
- At Hidden Size = 16384: ~230 TFLOPs/s
- At Hidden Size = 20480: ~240 TFLOPs/s
- At Hidden Size = 24576: ~250 TFLOPs/s
- At Hidden Size = 28672: ~260 TFLOPs/s
- At Hidden Size = 32768: ~270 TFLOPs/s
---
## Observations
1. **Throughput Correlation**: Higher **h/a** ratios (e.g., 64) generally correspond to higher throughput, though with increased variability.
2. **Fluctuations**: Lines with higher **h/a** values (e.g., 16, 64) exhibit more pronounced peaks and troughs compared to lower ratios (e.g., 1, 2).
3. **Consistency**: All lines start at 0 TFLOPs/s and show a general upward trend as Hidden Size increases.
---
## Notes
- **Language**: All text in the image is in English.
- **Data Integrity**: Legend colors and line placements were cross-verified for accuracy.
- **Spatial Grounding**: The legend is positioned on the right side of the graph, ensuring clear association with the data series.
---
## Conclusion
The graph demonstrates that increasing the **h/a** ratio correlates with higher throughput, though with varying degrees of stability. The **h/a = 64** (pink line) achieves the highest throughput but with the most fluctuations, while **h/a = 1** (blue line) shows the most stable but lowest performance.
</details>
(b) Attention over value GEMM throughput for 32 attention heads.
Figure 7: Attention GEMM performance on A100 GPUs. Each plot is a single series (i.e. if we didn’t split, there would be three regions with spikes), but split by the largest power of two that divides $h/a$ to demonstrate that more powers of two leads to better performance up to $h/a=64$ .
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_ha64_hdim16384.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Chart
## Title
**Attention Key Query Score (h/a = 64)**
---
## Axes
- **X-axis (Horizontal):**
- Label: `Hidden Size`
- Range: `0` to `16384` (logarithmic scale)
- Tick Marks: `0`, `2048`, `4096`, `6144`, `8192`, `10240`, `12288`, `14336`, `16384`
- **Y-axis (Vertical):**
- Label: `Throughput (TFLOP/s)`
- Range: `50` to `225`
- Tick Marks: `50`, `75`, `100`, `125`, `150`, `175`, `200`, `225`
---
## Legend
- **Location:** Top-right corner
- **Color-Coded Labels:**
- `a: 12` → Blue
- `a: 24` → Orange
- `a: 32` → Green
- `a: 40` → Red
- `a: 64` → Purple
- `a: 80` → Brown
- `a: 96` → Pink
---
## Data Series & Trends
1. **Blue Line (a: 12)**
- **Trend:** Steep upward slope from `50 TFLOP/s` (hidden size 0) to `230 TFLOP/s` (hidden size 16384).
- **Key Points:**
- `2048`: ~90 TFLOP/s
- `4096`: ~190 TFLOP/s
- `8192`: ~220 TFLOP/s
- `16384`: ~230 TFLOP/s
2. **Orange Line (a: 24)**
- **Trend:** Rapid increase from `60 TFLOP/s` (hidden size 2048) to `210 TFLOP/s` (hidden size 16384).
- **Key Points:**
- `4096`: ~110 TFLOP/s
- `8192`: ~150 TFLOP/s
- `12288`: ~205 TFLOP/s
3. **Green Line (a: 32)**
- **Trend:** Gradual rise from `60 TFLOP/s` (hidden size 2048) to `200 TFLOP/s` (hidden size 16384).
- **Key Points:**
- `6144`: ~130 TFLOP/s
- `10240`: ~170 TFLOP/s
- `16384`: ~200 TFLOP/s
4. **Red Line (a: 40)**
- **Trend:** Moderate increase from `60 TFLOP/s` (hidden size 2048) to `190 TFLOP/s` (hidden size 16384).
- **Key Points:**
- `8192`: ~120 TFLOP/s
- `12288`: ~170 TFLOP/s
- `16384`: ~190 TFLOP/s
5. **Purple Line (a: 64)**
- **Trend:** Slow upward trajectory from `60 TFLOP/s` (hidden size 2048) to `150 TFLOP/s` (hidden size 16384).
- **Key Points:**
- `10240`: ~110 TFLOP/s
- `14336`: ~145 TFLOP/s
6. **Brown Line (a: 80)**
- **Trend:** Linear increase from `60 TFLOP/s` (hidden size 2048) to `135 TFLOP/s` (hidden size 16384).
- **Key Points:**
- `12288`: ~120 TFLOP/s
- `16384`: ~135 TFLOP/s
7. **Pink Line (a: 96)**
- **Trend:** Gentle slope from `60 TFLOP/s` (hidden size 2048) to `125 TFLOP/s` (hidden size 16384).
- **Key Points:**
- `10240`: ~100 TFLOP/s
- `16384`: ~125 TFLOP/s
---
## Spatial Grounding
- **Legend Position:** Top-right corner (outside the plot area).
- **Color Consistency Check:**
- All line colors match legend labels (e.g., blue = a:12, orange = a:24).
---
## Additional Observations
- **Shaded Regions:**
- Green (`0-1B`), Blue (`1B-10B`), and Pink (`10B-300B`) highlight hidden size ranges but do not directly correlate with data series.
- **h/a Ratio:** Constant value of `64` (title annotation).
---
## Summary
The chart illustrates the relationship between **hidden size** and **throughput (TFLOP/s)** for varying **attention key query scores (a)**. Higher `a` values (e.g., 96) yield lower throughput, while lower `a` values (e.g., 12) achieve higher throughput. Throughput increases non-linearly with hidden size, with steeper growth observed for smaller `a` values.
</details>
Figure 8: Attention key-query score GEMM throughput assuming fixed ratio of $\frac{h}{a}=64$ on A100 GPU
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_ha64_hdim16384.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values (h/a = 64)
## Chart Overview
This line chart visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOP/s)** (y-axis) across multiple data series. The chart includes shaded regions and a legend with color-coded lines representing different parameter values.
---
### **Key Components**
1. **Title**:
`Attention over Values (h/a = 64)`
- Indicates a fixed ratio of `h/a = 64` for all data series.
2. **Axes**:
- **X-axis (Hidden Size)**:
- Range: `0` to `16384`
- Tick marks: `0, 2048, 4096, 6144, 8192, 10240, 12288, 14336, 16384`
- **Y-axis (Throughput (TFLOP/s))**:
- Range: `75` to `225`
- Tick marks: `75, 125, 175, 225`
3. **Legend**:
- Located in the upper-left corner.
- Color-coded lines represent different `a` values:
- `a:12` (blue)
- `a:24` (orange)
- `a:32` (green)
- `a:40` (red)
- `a:64` (purple)
- `a:80` (brown)
- `a:96` (pink)
4. **Shaded Regions**:
- **Green (0-1B)**: Covers `Hidden Size` from `0` to `2048`.
- **Blue (1B-10B)**: Covers `Hidden Size` from `2048` to `10240`.
- **Pink (10B-30B)**: Covers `Hidden Size` from `10240` to `16384`.
---
### **Data Series Analysis**
#### 1. **Blue Line (a:12)**
- **Trend**: Starts at `75 TFLOP/s` at `Hidden Size = 0`, peaks at `225 TFLOP/s` around `Hidden Size = 10240`, then fluctuates downward.
- **Key Points**:
- `Hidden Size = 0`: `75 TFLOP/s`
- `Hidden Size = 2048`: `150 TFLOP/s`
- `Hidden Size = 4096`: `175 TFLOP/s`
- `Hidden Size = 6144`: `225 TFLOP/s`
- `Hidden Size = 8192`: `200 TFLOP/s`
- `Hidden Size = 10240`: `225 TFLOP/s`
- `Hidden Size = 12288`: `210 TFLOP/s`
- `Hidden Size = 14336`: `220 TFLOP/s`
- `Hidden Size = 16384`: `215 TFLOP/s`
#### 2. **Orange Line (a:24)**
- **Trend**: Gradual increase with fluctuations, peaking at `225 TFLOP/s` near `Hidden Size = 12288`.
- **Key Points**:
- `Hidden Size = 0`: `75 TFLOP/s`
- `Hidden Size = 2048`: `125 TFLOP/s`
- `Hidden Size = 4096`: `175 TFLOP/s`
- `Hidden Size = 6144`: `210 TFLOP/s`
- `Hidden Size = 8192`: `160 TFLOP/s`
- `Hidden Size = 10240`: `200 TFLOP/s`
- `Hidden Size = 12288`: `225 TFLOP/s`
- `Hidden Size = 14336`: `190 TFLOP/s`
- `Hidden Size = 16384`: `200 TFLOP/s`
#### 3. **Green Line (a:32)**
- **Trend**: Steady upward trajectory with minor dips.
- **Key Points**:
- `Hidden Size = 0`: `75 TFLOP/s`
- `Hidden Size = 2048`: `125 TFLOP/s`
- `Hidden Size = 4096`: `150 TFLOP/s`
- `Hidden Size = 6144`: `175 TFLOP/s`
- `Hidden Size = 8192`: `225 TFLOP/s`
- `Hidden Size = 10240`: `150 TFLOP/s`
- `Hidden Size = 12288`: `180 TFLOP/s`
- `Hidden Size = 14336`: `210 TFLOP/s`
- `Hidden Size = 16384`: `225 TFLOP/s`
#### 4. **Red Line (a:40)**
- **Trend**: Sharp initial rise, followed by volatility.
- **Key Points**:
- `Hidden Size = 0`: `75 TFLOP/s`
- `Hidden Size = 2048`: `80 TFLOP/s`
- `Hidden Size = 4096`: `130 TFLOP/s`
- `Hidden Size = 6144`: `160 TFLOP/s`
- `Hidden Size = 8192`: `200 TFLOP/s`
- `Hidden Size = 10240`: `220 TFLOP/s`
- `Hidden Size = 12288`: `160 TFLOP/s`
- `Hidden Size = 14336`: `180 TFLOP/s`
- `Hidden Size = 16384`: `190 TFLOP/s`
#### 5. **Purple Line (a:64)**
- **Trend**: Consistent upward slope with minor fluctuations.
- **Key Points**:
- `Hidden Size = 0`: `75 TFLOP/s`
- `Hidden Size = 2048`: `100 TFLOP/s`
- `Hidden Size = 4096`: `125 TFLOP/s`
- `Hidden Size = 6144`: `150 TFLOP/s`
- `Hidden Size = 8192`: `175 TFLOP/s`
- `Hidden Size = 10240`: `200 TFLOP/s`
- `Hidden Size = 12288`: `225 TFLOP/s`
- `Hidden Size = 14336`: `210 TFLOP/s`
- `Hidden Size = 16384`: `220 TFLOP/s`
#### 6. **Brown Line (a:80)**
- **Trend**: Gradual increase with a plateau near the end.
- **Key Points**:
- `Hidden Size = 0`: `75 TFLOP/s`
- `Hidden Size = 2048`: `100 TFLOP/s`
- `Hidden Size = 4096`: `125 TFLOP/s`
- `Hidden Size = 6144`: `150 TFLOP/s`
- `Hidden Size = 8192`: `175 TFLOP/s`
- `Hidden Size = 10240`: `200 TFLOP/s`
- `Hidden Size = 12288`: `220 TFLOP/s`
- `Hidden Size = 14336`: `210 TFLOP/s`
- `Hidden Size = 16384`: `215 TFLOP/s`
#### 7. **Pink Line (a:96)**
- **Trend**: Moderate upward slope with a sharp rise at the end.
- **Key Points**:
- `Hidden Size = 0`: `75 TFLOP/s`
- `Hidden Size = 2048`: `100 TFLOP/s`
- `Hidden Size = 4096`: `125 TFLOP/s`
- `Hidden Size = 6144`: `150 TFLOP/s`
- `Hidden Size = 8192`: `175 TFLOP/s`
- `Hidden Size = 10240`: `200 TFLOP/s`
- `Hidden Size = 12288`: `225 TFLOP/s`
- `Hidden Size = 14336`: `210 TFLOP/s`
- `Hidden Size = 16384`: `220 TFLOP/s`
---
### **Shaded Region Correlation**
- **Green (0-1B)**: All lines show low throughput (`75–125 TFLOP/s`).
- **Blue (1B-10B)**: Throughput increases significantly (`125–225 TFLOP/s`).
- **Pink (10B-30B)**: Throughput stabilizes or fluctuates (`150–225 TFLOP/s`).
---
### **Critical Observations**
1. **Performance Trends**:
- Higher `a` values (e.g., `a:64`, `a:96`) generally achieve higher throughput at larger `Hidden Size` values.
- Lines with `a ≥ 64` dominate the upper regions of the chart.
2. **Anomalies**:
- The red line (`a:40`) exhibits a sharp drop at `Hidden Size = 12288` before recovering.
- The blue line (`a:12`) has the most pronounced fluctuations.
3. **Legend Validation**:
- All line colors match the legend labels (e.g., blue = `a:12`, green = `a:32`).
---
### **Conclusion**
The chart demonstrates that throughput increases with `Hidden Size`, with higher `a` values achieving better performance. The shaded regions highlight performance tiers, with the `10B-30B` range (pink) showing the most variability.
</details>
Figure 9: Attention over value GEMM throughput assuming fixed ratio of $\frac{h}{a}=64$ on A100 GPU.
<details>
<summary>extracted/5378885/figures/transformer/mlp_h_to_4h.png Details</summary>
### Visual Description
# Technical Document Extraction: GPU Throughput Analysis
## Chart Description
This image presents a line chart comparing the computational throughput (in TFLOP/s) of two GPU architectures (A100 and V100) across varying hidden sizes. The chart demonstrates performance scaling characteristics of these architectures in a computational task.
### Key Components
1. **Axes**
- **X-axis (Hidden Size)**:
- Range: 2048 to 32768
- Increment: Powers of 2 (2048, 4096, 6144, 8192, 10240, 12288, 14336, 16384, 18432, 20480, 22528, 24576, 26624, 28672, 30720, 32768)
- **Y-axis (Throughput (TFLOP/s))**:
- Range: 0 to 250
- Increment: 50
2. **Legend**
- Position: Right side of chart
- Color coding:
- Blue: A100
- Orange: V100
3. **Data Series**
- **A100 (Blue Line)**:
- Initial value: 10 TFLOP/s at 2048 hidden size
- Peak: 250 TFLOP/s at 4096 hidden size
- Plateau: Maintains ~250 TFLOP/s from 4096 to 32768
- **V100 (Orange Line)**:
- Initial value: 80 TFLOP/s at 10240 hidden size
- Plateau: Maintains ~80 TFLOP/s from 10240 to 30720
- Decline: Drops to 75 TFLOP/s at 32768
### Trend Analysis
1. **A100 Performance**
- Exhibits exponential growth pattern
- Achieves maximum throughput at 4096 hidden size
- Maintains peak performance across all larger hidden sizes
- Spatial grounding: Blue data points consistently align with legend
2. **V100 Performance**
- Flat performance curve across most hidden sizes
- Shows slight degradation at maximum hidden size (32768)
- Spatial grounding: Orange data points match legend color throughout
### Technical Observations
- A100 demonstrates superior scalability, achieving 250 TFLOP/s at 4096 hidden size
- V100 maintains consistent but lower performance (80 TFLOP/s) across tested range
- Both architectures show performance plateauing beyond certain hidden size thresholds
- No crossover points observed between A100 and V100 performance
## Data Points Table
| Hidden Size | A100 (TFLOP/s) | V100 (TFLOP/s) |
|-------------|----------------|----------------|
| 2048 | 10 | - |
| 4096 | 250 | - |
| 6144 | 250 | - |
| 8192 | 250 | - |
| 10240 | 250 | 80 |
| 12288 | 250 | 80 |
| 14336 | 250 | 80 |
| 16384 | 250 | 80 |
| 18432 | 250 | 80 |
| 20480 | 250 | 80 |
| 22528 | 250 | 80 |
| 24576 | 250 | 80 |
| 26624 | 250 | 80 |
| 28672 | 250 | 80 |
| 30720 | 250 | 80 |
| 32768 | 250 | 75 |
## Spatial Grounding Verification
- Legend position: Right side (confirmed)
- Color consistency:
- All blue points match A100 legend
- All orange points match V100 legend
- Axis alignment:
- X-axis values increase left-to-right
- Y-axis values increase bottom-to-top
## Trend Verification
1. A100 line shows:
- Steep upward slope from 2048→4096
- Horizontal plateau from 4096→32768
2. V100 line shows:
- Horizontal plateau from 10240→30720
- Slight downward slope at 32768
## Component Isolation
1. Header: Chart title (implied by axes)
2. Main Chart: Dual-line plot with data points
3. Footer: Legend and axis labels
## Language Analysis
- Primary language: English
- No non-English text detected
## Critical Findings
1. A100 achieves 25x higher throughput than V100 at 4096 hidden size
2. V100 maintains consistent performance across 20480→30720 range
3. Both architectures show performance saturation beyond specific hidden size thresholds
4. No performance degradation observed in A100 across full tested range
</details>
(a) MLP h to 4h Block
<details>
<summary>extracted/5378885/figures/transformer/mlp_4h_to_h.png Details</summary>
### Visual Description
# Technical Document Extraction: GPU Throughput Analysis
## Chart Description
This image is a **line chart** comparing the throughput (in TFLOP/s) of two GPU architectures (A100 and V100) across varying hidden sizes. The chart emphasizes performance trends as hidden size increases.
---
### Labels and Axis Titles
- **Y-Axis**: "Throughput (TFLOP/s)"
- Scale: 0 to 250 (increments of 50)
- **X-Axis**: "Hidden Size"
- Scale: 0 to 32768 (increments of 2048)
---
### Legend
- **Location**: Right side of the chart
- **Labels**:
- **Blue (solid line with circles)**: A100
- **Orange (dashed line with circles)**: V100
---
### Data Trends
#### A100 (Blue)
- **Initial Behavior**:
- Starts at **20 TFLOP/s** at hidden size **2048**.
- Sharp increase to **~250 TFLOP/s** by hidden size **4096**.
- **Stabilization**:
- Maintains **~250 TFLOP/s** plateau from hidden size **4096** to **32768**.
- Minor dips observed at hidden sizes **12288**, **16384**, and **24576**, but no sustained deviation from the plateau.
#### V100 (Orange)
- **Initial Behavior**:
- Starts at **0 TFLOP/s** at hidden size **0**.
- Jumps to **~80 TFLOP/s** at hidden size **10240**.
- **Mid-Range Behavior**:
- Maintains **~80 TFLOP/s** plateau from hidden size **10240** to **16384**.
- **Drop**:
- Declines to **~60 TFLOP/s** at hidden size **20480**.
- **Final Behavior**:
- Stabilizes at **~60 TFLOP/s** from hidden size **20480** to **32768**.
---
### Key Observations
1. **Performance Gap**:
- A100 consistently outperforms V100 across all hidden sizes.
- At hidden size **32768**, A100 achieves **~250 TFLOP/s**, while V100 achieves **~60 TFLOP/s**.
2. **Scalability**:
- A100 demonstrates near-linear scaling up to hidden size **4096**, then plateaus.
- V100 shows limited scalability, with performance peaking at hidden size **10240** and declining thereafter.
3. **Efficiency**:
- A100 maintains high throughput even at the largest hidden size (**32768**).
- V100’s performance drops significantly at hidden sizes beyond **16384**, suggesting architectural limitations.
---
### Spatial Grounding
- **Legend Position**: Right-aligned, outside the plot area.
- **Data Point Colors**:
- All A100 data points match the **blue** legend marker.
- All V100 data points match the **orange** legend marker.
---
### Conclusion
The chart highlights A100’s superior throughput and scalability compared to V100, particularly at larger hidden sizes. V100’s performance plateaus early and declines at higher hidden sizes, indicating potential inefficiencies in handling larger computational loads.
</details>
(b) MLP 4h to h Block
Figure 10: Throughput (in teraFLOP/s) for multilayer perceptrons (MLP) for each transformer layer as a function of hidden dimension for $a=128$ .
For example, let us consider the attention block on an A100 GPU. The number of attention heads affects the number of independent matrix multiplications in the BMM, as well as the size of each matrix multiplication computation. Figure 6 shows the effect of the number of attention heads and the hidden size on the throughput of the BMM used in attention key-query score computation and attention over value computation. NVIDIA Tensor Cores are more efficient when the dimensions of the matrices $m$ , $n$ , and $k$ are multiples of 128 bytes for A100 GPUs. Therefore, efficiency is maximized when matrix sizes are multiples of 64 FP16 elements. If this cannot be achieved, sizes that are multiples of larger powers of 2 perform better, as shown in Figures 7a and 7b, where the matrix dimension of interest is of size $h/a$ . Figures 8 and 9 show how decreasing the number of attention heads for any given hidden size results in more efficient GEMMs. Because a decrease in $a$ is an increase in $h/a$ and these two GEMMs are memory bound, an increase in component matrices size creates much more efficient GEMMs. Figure 9 also clearly shows the effects of wave quantization in the peaks and valleys within any given line. Since each line moves in steps of $64h/a$ , the BMMs corresponding to each line grow at different rates. This causes the period of the wave quantization effect to appear different for each $a$ value.
<details>
<summary>x4.png Details</summary>
### Visual Description
# Technical Document Extraction: Latency Analysis Chart
## Chart Type
Stacked bar chart comparing latency distribution across three scenarios:
1. Small (h=2560, a=20)
2. Large (h=16384, a=128)
3. Large + Flash (h=16384, a=128)
## Axis Labels
- **Y-axis**: "Percentage of Latency (%)" (0-100 scale)
- **X-axis**: Scenario labels with hardware parameters
## Legend
[Top-right placement] Color-coded components:
1. QKV (blue)
2. Flash (cyan)
3. Score (orange)
4. AOV (green)
5. Linproj (yellow)
6. MLP h to 4h (purple)
7. MLP 4h to h (gray)
8. Non-GEMM (red)
## Key Trends
1. **Small Scenario (h=2560, a=20)**:
- Non-GEMM dominates (45%)
- MLP h to 4h (20%) and MLP 4h to h (15%) significant
- QKV (10%) and Flash (5%) minimal
2. **Large Scenario (h=16384, a=128)**:
- QKV becomes dominant (35%)
- MLP h to 4h (25%) and MLP 4h to h (20%) persist
- Non-GEMM reduced to 10%
3. **Large + Flash Scenario**:
- QKV dominates (40%)
- Flash component appears (5%)
- MLP h to 4h (25%) and MLP 4h to h (20%) remain
- Non-GEMM minimal (5%)
## Data Point Verification
| Scenario | QKV | Flash | Score | AOV | Linproj | MLP h→4h | MLP 4h→h | Non-GEMM |
|-------------------|-----|-------|-------|-----|---------|----------|----------|----------|
| Small | 10% | 5% | 10% | 5% | 5% | 20% | 15% | 45% |
| Large | 35% | 0% | 2% | 1% | 5% | 25% | 20% | 10% |
| Large + Flash | 40% | 5% | 2% | 1% | 5% | 25% | 20% | 5% |
## Spatial Grounding
- Legend positioned at [x=0.85, y=0.95] (top-right corner)
- Color consistency verified: All legend colors match bar segments exactly
## Component Isolation
1. **Header**: Chart title "Percentage of Latency (%)"
2. **Main Chart**:
- Three vertical bars with segmented color coding
- Each segment's height corresponds to percentage contribution
3. **Footer**:
- X-axis labels with hardware parameters
- Y-axis scale (0-100%)
## Language Note
All text appears in English. No non-English content detected.
## Trend Verification Logic
- Non-GEMM dominance in Small scenario confirmed by red segment height
- QKV growth in Large/Large+Flash scenarios verified by increasing blue segment
- Flash component only appears in Large+Flash scenario (cyan segment)
- MLP components maintain consistent proportions across scenarios
</details>
Figure 11: The proportion of latency of each GEMM module for one layer of various model sizes.
Figure 11 shows the proportion of latency spent in each transformer GEMM; consequently, it also shows the most relevant GEMMs to optimize in the transformer module. As the size of the model grows, it is even more important to optimize GEMM operations. For the largest models, the $QKV$ transformation in the attention block along with the MLP block are the most prevalent GEMMs. Therefore, the overall latency of the model would benefit most from optimizing these kernels. Attention over value (AOV) computation is the smallest GEMM computation in large transformer models; however, optimizing attention key-query score computation will have similar benefits to attention over value computation, so both can be optimized at the same time.
### VI-B Analysis
To recap, we have the following requirements to efficiently run GEMMs on NVIDIA GPUs:
- Tensor Core Requirement: Ensure the inner and outer dimension of the GEMM is divisible by 128 bytes (64 FP16 elements).
- Tile Quantization: To use the most efficient tile size ensure that the output matrix is divisible into $128× 256$ blocks.
- Wave Quantization: Ensure that the number of blocks that the output matrix is divided into is divisible by the number of streaming multiprocessors (80 for V100s, 108 for A100s, and 144 for H100s).
While tile quantization is relevant to GEMM performance, tile quantization is hard to observe by the user. If the GEMM does not divide evenly into the tile size, a tile without a full compute load will execute. However, this tile will execute concurrently with other tiles in the same wave. In effect, the kernel will run with the same latency as a kernel with a larger problem size.
Wave quantization is more easily observable. There will be no wave quantization inefficiency when a matrix of size $(X,Y)$ satisfies the following constraints on its size (assuming a tile size of $t_1× t_2$ ):
$$
≤ft\lceil{\frac{X}{t_1}}\right\rceil·≤ft\lceil{\frac{Y}{t_2}}
\right\rceil≡ 0 or ≤ft\lceil{\frac{X}{t_2}}\right\rceil·
≤ft\lceil{\frac{Y}{t_1}}\right\rceil≡ 0±od{\#SMs}
$$
Assuming a tile size of $128× 256$ which is the most efficient, there is not a transformer configuration with GEMMs that fill tensor core requirements without wave quantization inefficiency. Further, PyTorch’s linear algebra backend can use different tile sizes for each GEMM. Therefore, PyTorch is unable to efficiently overcome the effects of wave quantization.
Therefore to ensure the best performance from transformer models, ensure:
- The vocabulary size should be divisible by $64$ .
- The microbatch size $b$ should be as large as possible [24].
- $b· s$ , $\frac{h}{a}$ , and $\frac{h}{t}$ should be divisible by a power of two, though there is no further benefit to going beyond $64$ .
- $(b· a)/t$ should be an integer.
- $t$ should be as small as possible [25].
Importantly, the microbatch size $b$ does not itself need to be divisible by a large power of 2 since the sequence length $s$ is a large power of two.
Whether it is optimal to train using pipeline parallelism depends on additional details of the computing set-up, most notable the speed and bandwidth of internode connections. We note that this is further evidence for our thesis that model dimensions should be chosen with hardware details in mind, but leave an analysis of this phenomenon to future work. In all cases it is optimal for the number of layers to be divisible by the number of pipeline parallel stages.
Using these recommendations we can achieve a 1.18 $×$ speed-up on a widely used model architecture introduced by [6]. GPT-3 2.7B’s architecture was copied for many other models including GPT-Neo 2.7B [5], OPT 2.7B [43], RedPajama 3B [8], and Pythia 2.8B [4], but possesses an inefficiency. It features $32$ attention heads and a hidden dimension of $2560$ , resulting in a head dimension of $h/a=2560/32=80$ which is not a multiple of $64$ . This can be addressed either by increasing the side of the hidden dimension to $4096$ or by decreasing the number of heads to $20$ . Increasing the hidden dimension to $4096$ doubles the number of parameters to $6.7$ billion, so instead we decrease the number of heads. These results are shown in Figure 1.
To raise $h/a$ , the easiest solution is to decrease $a$ , but decreasing $a$ may lead to a drop in model accuracy. Fortunately, as shown in Figure 11, only a small portion of the latency of large models is the attention score computation and attention over value computation GEMMs, so an increase in the latency of these components will have only a small effect on the end-to-end model performance. Therefore, we recommend either using FlashAttention v2 (see Section VI-C 3) for small models to mitigate these effects, or increasing $h$ as much as possible to reach the saturation point shown in Figures 10a and 10b.
### VI-C Architectural Modifications
While decoder-only architectures are largely standardized and follow the GPT-2 architecture [29] described in the previous section, there are some architectural modifications that are popular in recent work. Here we briefly describe them and how they affect our overall discussion.
#### VI-C 1 Parallel Layers
Parallel attention and MLP layers were introduced by [38]. Instead of computing attention and MLPs sequentially ( $y=x+MLP(Norm(x+Attn(Norm(x))))$ ), the transformer block is formulated as:
$$
y=x+MLP(Norm(x))+Attn(Norm(x)).
$$
While this computation is represented as being in parallel, in practice the two branches are not computed simultaneously. Instead, a speed-up is achieved by fusing the MLP and Attention blocks into a single kernel. We recommend using parallel attention as the default best practice, though it does not impact our analysis at all.
#### VI-C 2 Alternative Positional Embeddings
While the original positional embeddings used in transformers are pointwise operations [37], today other approaches such as Rotary [35] and ALiBi [28] embeddings are more popular. While point-wise operations are slightly faster than the GEMM necessary for Rotary and ALiBi embeddings, the improved model accuracy that Rotary or ALiBi embeddings bring are generally considered well worth it. Recently, custom kernels for rotary embeddings have been introduced, further reducing their costs. We recommend using rotary or ALiBi embeddings as best practice. Using these embeddings again does not impact our analysis.
#### VI-C 3 FlashAttention
<details>
<summary>extracted/5378885/figures/transformer/flash_v2.png Details</summary>
### Visual Description
# Technical Document Extraction: Throughput vs. Hidden Size Analysis
## Chart Description
The image presents a line chart analyzing the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOP/s)** (y-axis). The data is represented by blue circular markers connected by a solid blue line.
---
### Axis Labels and Markers
- **X-Axis (Hidden Size)**:
- Range: `0` to `32768`
- Incremental markers: `0`, `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`
- Label: "Hidden Size" (bold, black text)
- **Y-Axis (Throughput)**:
- Range: `0` to `250`
- Incremental markers: `0`, `50`, `100`, `150`, `200`, `250`
- Label: "Throughput (TFLOP/s)" (bold, black text)
---
### Legend
- **Label**: `a: 128`
- **Color**: Blue (matches the data line)
- **Position**: Top-left corner of the chart
---
### Data Points and Trend Analysis
The chart shows a **non-linear relationship** between Hidden Size and Throughput:
1. **Initial Growth Phase**:
- From Hidden Size `0` to `8192`, Throughput increases steeply.
- Key data points:
- `[0, 0]`
- `[4096, ~50]`
- `[8192, ~150]`
2. **Saturation Phase**:
- From Hidden Size `12288` to `32768`, Throughput plateaus with minor fluctuations.
- Key data points:
- `[12288, ~200]`
- `[16384, ~210]`
- `[20480, ~220]`
- `[24576, ~225]`
- `[28672, ~230]`
- `[32768, ~235]`
**Visual Trend**: The line exhibits a sigmoidal pattern, with rapid growth followed by diminishing returns.
---
### Spatial Grounding
- **Legend Placement**: Top-left corner (coordinates: `[x=0, y=0]` relative to chart boundaries).
- **Data Line Color**: Blue (confirmed to match legend label `a: 128`).
---
### Component Isolation
1. **Main Chart**:
- Focus: Line plot of Throughput vs. Hidden Size.
- No additional sub-charts or annotations.
2. **Legend**:
- Single entry (`a: 128`) with no sub-categories.
---
### Conclusion
The chart demonstrates that Throughput scales with Hidden Size up to a critical threshold (~8192), after which performance stabilizes. The legend confirms the parameter `a=128` governs this relationship. No other languages or textual elements are present.
</details>
Figure 12: Sweep over hidden dimension for FlashAttention (v2) [9] on NVIDIA A100 GPU.
FlashAttention [10] and FlashAttention 2 [9] are novel attention kernels that are widely popular for training large language models. In order to see its impact on the attention calculation sizing, we set $a=128$ and sweep over the hidden dimension in Figure 12. We find that FlashAttention follows a roofline model, which simplifies our attention takeaways to only require that $h$ be as large as possible; the takeaways for MLPs remain unchanged.
#### VI-C 4 SwiGLU and $8h/3$ MLPs
Models such as PaLM, LLaMA, and Mistral use the SwiGLU [32] activation function in place of the more common GLU activation function. While the choice of activation function is generally irrelevant to our analysis, this activation function has an extra parameter compared to other commonly used options. Consequently, its common to adjust the projection factor for the MLP block from $dim_MLP=4· dim_Attn$ to $dim_MLP=\frac{8}{3}· dim_Attn$ to preserve the ratio of the total number of parameters in the attention and MLP blocks. This change has substantial implications for our analysis, which we discuss it detail in Section VII-B.
## VII Case Studies
Finally, we present a series of case studies illustrating how we use the principles described in this paper in practice. These demonstrate real-world challenges we have encountered in training large language models with tens of billions of parameters.
### VII-A 6-GPU Nodes
While the most common data-center scale computing set-up is to have 8 GPUs per node, some machines such as Oak Ridge National Lab’s Summit supercomputer feature six. This presents a multi-layer challenge to training language models when the tensor parallel degree is equal to the number of GPUs on a single node, which is commonly the most efficient 3D-parallelism scheme [25]. This often causes $h/t$ to no longer have a factor of some power of two, which greatly improves performance as we demonstrated above. Therefore:
1. Model architectures common on 8-GPU nodes may not be possible on 6-GPU nodes.
1. Even when they are possible, model architectures common on 8-GPU nodes may not be efficient on 6-GPU nodes.
1. If concessions are made to ameliorate #1 and #2, they may cause problems in deployment if downstream users wish to use the model designed for a 6-GPU node on a 2-GPU, 4-GPU, or 8-GPU node.
Several large transformers on Summit have been trained, such as the INCITE RedPajama 3B and 7B [8], and such model designers must make a choice. Does one choose the most efficient hyperparameters for pretraining only (which would involve a tensor-parallel degree of 6 and therefore a hidden dimension divisible by 6 and 64), or should the pretraining team choose a set of hyperparameters that are more amenable to the node architectures commonly used for finetuning or inference?
### VII-B SwiGLU Activation Functions
Recently the SwiGLU activation function has become popular for training language models. The SwiGLU function contains an additional learned matrix in its activation function, so that now the MLP block contains 3 matrices instead of the original 2. To preserve the total number of parameters in the MLP block the paper that introduces SwiGLU proposes to use $d_ff=\frac{8}{3}h$ instead of the typical $d_ff=4h$ .
If you followed the recommendations in this paper for finding the value of $h$ that would lead to the best $matmul$ performance, you will realize that $\frac{8}{3}h$ is likely to result in a much slower MLP block, because $\frac{1}{3}$ will break all the alignments.
In order to overcome this problem one only needs to realize that the $\frac{8}{3}$ coefficient is only a suggestion and thus it’s possible to find other coefficients that would lead to better-shaped MLP matrices. In fact if you look at the publicly available LLama-2 models, its 7B variant uses $\frac{11008}{4096}=2.6875$ as a coefficient, which is quite close to $\frac{8}{3}=2.667$ , and its 70B variant uses a much larger $\frac{28672}{8192}=3.5$ coefficient. Here the 70B variant ended up with an MLP block that contains significantly more parameters than a typical transformer block that doesn’t use SwiGLU.
Now that we know the recommended coefficient isn’t exact and since a good $h$ has already been chosen, one can now search for a good nearby number that still leads to high-performance GEMMs in the MLP. Running a brute-force search reveals that Llama-2-7B’s intermediate size is indeed one of the best performing sizes in its range.
### VII-C Inference
In order to demonstrate that 1) models trained efficiently on a given GPU will also infer efficiently on the same GPU, since the underlying forward-pass GEMMs are the same, and 2) our sizing recommendations are kernel-invariant, we have run inference benchmarks using DeepSpeed-MII [11] and the Pythia [4] suite. We show in Figure 13 that Pythia-1B is significantly more efficient at inference time than Pythia-410M due to its fewer attention heads and layers than Pythia-410M, and a larger hidden dimension. Despite these architectural changes, the test loss of Pythia-1B is on-trend with the rest of the suite while having significantly higher training and inference throughput.
<details>
<summary>extracted/5378885/figures/inference-line.png Details</summary>
### Visual Description
# Technical Document Extraction: Line Chart Analysis
## 1. Chart Type and Structure
- **Chart Type**: Line chart
- **Axes**:
- **X-axis**: `#Params (B)` (Parameter count in billions)
- **Y-axis**: `Inference Latency (s)` (Inference latency in seconds)
- **Legend**: Located on the right side of the chart, with color-coded labels for each data series.
## 2. Data Series and Labels
The chart contains six data series, each representing a different model with varying parameter counts and inference latencies. The legend explicitly maps colors to model names:
| Model Name | Parameter Count (B) | Inference Latency (s) | Color (Legend) |
|------------------|---------------------|------------------------|----------------|
| Pythia-70m | 0.0 | 0.0 | Blue |
| Pythia-160m | 0.16 | 0.3 | Blue |
| Pythia-410m | 0.41 | 1.25 | Blue |
| Pythia-1B | 1.0 | 0.9 | Blue |
| Pythia-1.4B | 1.4 | 1.3 | Blue |
| Pythia-2.8B | 2.8 | 1.9 | Blue |
## 3. Key Trends and Observations
- **Trend 1**: The line starts at the origin (0.0B, 0.0s) for Pythia-70m.
- **Trend 2**: A sharp increase in latency occurs between 0.16B (Pythia-160m) and 0.41B (Pythia-410m), rising from 0.3s to 1.25s.
- **Trend 3**: A slight decrease in latency is observed at 1.0B (Pythia-1B), dropping to 0.9s.
- **Trend 4**: A gradual increase in latency resumes from 1.0B to 2.8B, peaking at 1.9s for Pythia-2.8B.
## 4. Spatial Grounding and Color Verification
- **Legend Position**: Right side of the chart.
- **Color Consistency**: All data points are plotted in blue, matching the legend's color coding. No discrepancies observed.
## 5. Component Isolation
- **Main Chart**: The line graph dominates the image, with no additional headers, footers, or annotations.
- **Data Points**: Each model's parameter count and latency are explicitly labeled on the chart.
## 6. Textual Information
- **Axis Labels**:
- X-axis: `#Params (B)`
- Y-axis: `Inference Latency (s)`
- **Legend Labels**:
- Pythia-70m, Pythia-160m, Pythia-410m, Pythia-1B, Pythia-1.4B, Pythia-2.8B
- **Data Point Labels**:
- Pythia-70m (0.0B, 0.0s)
- Pythia-160m (0.16B, 0.3s)
- Pythia-410m (0.41B, 1.25s)
- Pythia-1B (1.0B, 0.9s)
- Pythia-1.4B (1.4B, 1.3s)
- Pythia-2.8B (2.8B, 1.9s)
## 7. Language and Transcription
- **Primary Language**: English
- **No Additional Languages**: All text is in English.
## 8. Final Notes
- The chart illustrates a non-linear relationship between parameter count and inference latency, with a notable dip at 1.0B parameters.
- The data points are explicitly labeled, ensuring clarity in the relationship between model size and performance.
</details>
Figure 13: Inference latency of Pythia suite using DeepSpeed-MII [11]. Pythia-410M / Pythia-1B are off-trend due to their sizing.
## VIII Discussion
In the current landscape of AI hardware, Transformer workloads stand out as a pivotal target. They constitute a significant component (e.g., BERT, GPT-3) of the MLCommons benchmarks, capturing the attention of major hardware vendors and data centers. Notably, these benchmarks have been integrated as a crucial metric for procurement [27] in the upcoming Exascale supercomputer at Oak Ridge Leadership Computing Facility. Our analysis strongly suggests that leveraging representative GEMM kernels holds promise as a reliable performance indicator for Transformer-based workloads. Consequently, these kernels should be embraced as a benchmarking tool for hardware co-design. The advantages stem from several key points:
1. The optimizations made at the GEMM level exhibit a demonstrable transferability to various applications, as evidenced in Sec. VI.
1. Benchmarking at the kernel level proves to be more cost-effective and time-efficient.
1. This approach remains model-agnostic, accommodating diverse architectures like GPT-NeoX, Pythia, and OPT, as long as they are based on the Transformer architecture.
This assertion finds partial validation in the observed correlation between MLCommons benchmarks and our findings. To illustrate, consider the performance of BERT benchmarks, which demonstrates a consistent 3:1 ratio between H100- and A100-based systems. Notably, this aligns harmoniously with our observed kernel throughput for the respective hardware configurations (see Sec. V).
## IX Conclusion
State-of-the-art deep learning (DL) models are driving breakthroughs in existing fields and paving the way towards new areas of study. However, while the transformer model is at the forefront of this DL explosion, few transformer architectures consider their underlying hardware. We believe that instead of creating new designs to improve efficiency, many practitioners would be better served by slightly modifying their existing architectures to maximally utilize the underlying hardware. Well informed hyperparameter choices improve training and inference throughput throughout a model’s lifetime. We demonstrate that minor modifications to the model architecture improve GPU throughput by up to 38.9% while maintaining accuracy. Since we have explained how to motivate model hyperparameters from a GPU architecture standpoint, this paper can be used to guide future model design while clarifying the relevant first principles necessary to extend such hyperparameter choices to future architectures.
## X Acknowledgements
We are grateful to Stability AI for providing the compute required for A100 evaluations, Oak Ridge National Lab (ORNL) for providing the compute required for 6-V100 special-case evaluations, and the San Diego Supercomputing Center (SDSC) for providing the compute required for general V100 evalutions.
We thank Horace He and various members of the EleutherAI Discord Server for their feedback.
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## Appendix A Misc
When using PyTorch to invoke GEMMs, we use torch.nn.functional.linear. This function accepts 2 tensors as parameters, where one tensor can be 3 dimensional. Figure 14 shows how the ordering of a tensor’s dimensions impacts performance. We benchmark GEMMs of size $(2048,4,n)×(n,3n)$ , $(4,2048,n)×(n,3n)$ , and $(8192,n)×(n,3n)$ . This shows that the ordering of the batched dimension does not affect performance. The batched implementation is also the same speed as a 2-dimensional GEMM, so these implementation details do not affect performance. Therefore we can represent GEMMs between 3 and 2 dimensional tensors as GEMMs between two 2-dimensional tensors.
<details>
<summary>extracted/5378885/figures/mm/nnlinear_eq_mm.png Details</summary>
### Visual Description
# Technical Document Extraction: Line Chart Analysis
## Chart Overview
The image depicts a line chart illustrating the relationship between variable `n` (x-axis) and computational throughput (y-axis). Three distinct data series are plotted, differentiated by line style and legend labels.
---
### Axis Labels and Markers
- **X-axis**:
- Label: `n`
- Scale: Logarithmic progression from `2^8` to `2^14` (values: `256, 512, 1024, 2048, 4096, 8192, 16384`)
- Tick intervals: Exponential spacing (powers of 2)
- **Y-axis**:
- Label: `Throughput (TFLOP/S)`
- Range: `0` to `300`
- Increment: `50` units
---
### Legend and Data Series
**Legend Position**: Right-aligned, outside the plot area.
**Color-Series Mapping**:
1. **Blue Line** (`m=4, b=2048`):
- Data Points:
- `(2^8, 20)`
- `(2^9, 60)`
- `(2^10, 180)`
- `(2^11, 210)`
- `(2^12, 240)`
- `(2^13, 250)`
- `(2^14, 250)`
- Trend: Steep initial growth, plateauing at `n=2^12` and beyond.
2. **Orange Line** (`m=2048, b=4`):
- Data Points:
- `(2^8, 25)`
- `(2^9, 70)`
- `(2^10, 175)`
- `(2^11, 215)`
- `(2^12, 245)`
- `(2^13, 255)`
- `(2^14, 255)`
- Trend: Similar to blue line but consistently higher throughput across all `n`.
3. **Green Line** (`m=8192, b=0`):
- Data Points:
- `(2^8, 20)`
- `(2^9, 65)`
- `(2^10, 190)`
- `(2^11, 225)`
- `(2^12, 250)`
- `(2^13, 255)`
- `(2^14, 255)`
- Trend: Highest throughput throughout, with gradual convergence to orange/blue lines at `n=2^14`.
---
### Key Observations
1. **Convergence**: All three series plateau at `n=2^14`, suggesting diminishing returns beyond this scale.
2. **Parameter Impact**:
- Higher `m` values (e.g., `m=8192`) correlate with increased throughput.
- Non-zero `b` values (e.g., `b=2048`) slightly reduce performance compared to `b=0`.
3. **Scalability**: Throughput grows exponentially with `n` up to `2^12`, after which gains stabilize.
---
### Spatial Grounding
- **Legend**: Positioned at `[x=1.05, y=0.5]` (normalized coordinates, outside the plot area).
- **Data Point Validation**:
- Blue line markers (`●`) match legend label `m=4, b=2048`.
- Orange line markers (`●`) match `m=2048, b=4`.
- Green line markers (`●`) match `m=8192, b=0`.
---
### Conclusion
The chart demonstrates that computational throughput scales with `n` but plateaus at `n=2^14`. Parameter tuning (specifically `m` and `b`) significantly impacts performance, with `m=8192, b=0` yielding optimal results.
</details>
Figure 14: GEMMs with different ordering of dimensions.
A series of benchmarks are shown in Figure 16 through Figure 20. These figures show the performance of transformer GEMMs listed in Table II. In each of the figures, throughput for a transformer with 128 attention heads is plotted against hidden size. Performance generally increases with hidden size, as the size of each GEMM is growing. However, in GEMMs where one dimension is of size $h/a$ , Attention Score Computation and Attention Over Value, throughput depends on the highest power of 2 that divides $h/a$ , as described in secion VI.A.
<details>
<summary>extracted/5378885/figures/transformer/attention_key_value_query_transform.png Details</summary>
### Visual Description
# Technical Document Analysis: GPU Throughput vs. Hidden Size
## Chart Description
The image is a line chart comparing the throughput performance of two GPUs (A100 and V100) across varying hidden sizes. The chart uses two distinct data series with color-coded markers and lines.
---
### **Axis Labels and Scales**
- **X-Axis (Hidden Size):**
- Label: "Hidden Size"
- Range: 0 to 32,768
- Increment: 2,048 (e.g., 0, 2048, 4096, ..., 32768)
- Format: Numerical values with exponential notation (e.g., 2048, 4096).
- **Y-Axis (Throughput):**
- Label: "Throughput (TFLOP/s)"
- Range: 0 to 250
- Increment: 50 (e.g., 0, 50, 100, ..., 250).
---
### **Legend**
- **Location:** Top-right corner of the chart.
- **Labels and Colors:**
- `A100`: Blue line with circular markers (`●`).
- `V100`: Orange line with circular markers (`●`).
---
### **Data Series Analysis**
#### **A100 (Blue Line)**
- **Trend:**
- Starts at 0 TFLOP/s for hidden size = 0.
- Sharp increase to ~200 TFLOP/s at hidden size = 8192.
- Plateaus at ~250 TFLOP/s for hidden sizes ≥ 4096.
- **Key Data Points:**
| Hidden Size | Throughput (TFLOP/s) |
|-------------|----------------------|
| 0 | 0 |
| 2048 | 50 |
| 4096 | 100 |
| 6144 | 150 |
| 8192 | 200 |
| 10240+ | 250 (plateau) |
#### **V100 (Orange Line)**
- **Trend:**
- Flat line at ~80 TFLOP/s across all hidden sizes ≥ 1024.
- No data points below hidden size = 1024.
- **Key Data Points:**
| Hidden Size | Throughput (TFLOP/s) |
|-------------|----------------------|
| 1024 | 80 |
| 12288 | 80 |
| 16384 | 80 |
| 20480 | 80 |
| 24576 | 80 |
| 32768 | 80 |
---
### **Spatial Grounding**
- **Legend Placement:** Top-right corner (coordinates: [x, y] = [right, top]).
- **Data Point Colors:**
- Blue markers (`●`) correspond to A100.
- Orange markers (`●`) correspond to V100.
---
### **Trend Verification**
- **A100:**
- Initial slope: Steep upward trend from 0 to 8192 hidden size.
- Plateau: Stable at 250 TFLOP/s beyond 4096 hidden size.
- **V100:**
- Constant throughput of ~80 TFLOP/s across all visible hidden sizes.
---
### **Component Isolation**
1. **Header:** No explicit header text.
2. **Main Chart:**
- Line plot with two data series (A100 and V100).
- Gridlines implied by axis increments.
3. **Footer:** No footer text.
---
### **Textual Information Extraction**
- **Axis Titles:**
- X: "Hidden Size"
- Y: "Throughput (TFLOP/s)"
- **Legend Labels:**
- A100 (blue)
- V100 (orange)
- **Numerical Values:**
- Hidden sizes: 0, 2048, 4096, ..., 32768.
- Throughput: 0, 50, 100, ..., 250.
---
### **Conclusion**
The chart demonstrates that the A100 GPU significantly outperforms the V100 in throughput, particularly at larger hidden sizes. The A100 achieves a maximum throughput of 250 TFLOP/s, while the V100 remains capped at ~80 TFLOP/s. No textual data in languages other than English is present.
</details>
Figure 15: Attention $QKV$ transform.
<details>
<summary>extracted/5378885/figures/transformer/tp_sweep.png Details</summary>
### Visual Description
# Technical Document Extraction: Throughput vs. Hidden Size Analysis
## Chart Overview
The image is a line chart comparing computational throughput (in TFLOP/s) across different hidden sizes for four thread pool (TP) configurations. The chart uses distinct colors and markers to differentiate data series.
---
### **Axis Labels & Scales**
- **X-axis (Hidden Size):**
- Range: `0` to `22528`
- Increment: `2048` (e.g., `2048`, `4096`, `6144`, ..., `22528`)
- Label: "Hidden Size"
- **Y-axis (Throughput):**
- Range: `0` to `250`
- Unit: TFLOP/s
- Label: "Throughput (TFLOP/s)"
---
### **Legend & Data Series**
The legend is positioned on the **right side** of the chart. Colors and markers are explicitly mapped as follows:
1. **Blue (●):** TP: 1
2. **Orange (●):** TP: 2
3. **Green (●):** TP: 4
4. **Red (●):** TP: 8
**Spatial Grounding:**
- Legend items are vertically stacked, with TP:1 (blue) at the top and TP:8 (red) at the bottom.
- All data points match their legend labels (e.g., blue markers correspond to TP:1).
---
### **Key Trends & Data Points**
#### **TP: 1 (Blue Line)**
- **Trend:** Steep upward slope, plateauing near `220 TFLOP/s` at maximum hidden size.
- **Notable Points:**
- At `Hidden Size = 2048`: ~20 TFLOP/s
- At `Hidden Size = 22528`: ~220 TFLOP/s
#### **TP: 2 (Orange Line)**
- **Trend:** Gradual increase, plateauing near `110 TFLOP/s`.
- **Notable Points:**
- At `Hidden Size = 2048`: ~10 TFLOP/s
- At `Hidden Size = 22528`: ~110 TFLOP/s
#### **TP: 4 (Green Line)**
- **Trend:** Slow rise, plateauing near `60 TFLOP/s`.
- **Notable Points:**
- At `Hidden Size = 2048`: ~5 TFLOP/s
- At `Hidden Size = 22528`: ~60 TFLOP/s
#### **TP: 8 (Red Line)**
- **Trend:** Minimal increase, plateauing near `30 TFLOP/s`.
- **Notable Points:**
- At `Hidden Size = 2048`: ~3 TFLOP/s
- At `Hidden Size = 22528`: ~30 TFLOP/s
---
### **Observations**
1. **Inverse Relationship:** Higher TP values (e.g., TP:8) correlate with lower throughput, contrary to typical expectations.
2. **Scalability:** TP:1 achieves the highest throughput, suggesting optimal performance at single-threaded configurations.
3. **Data Consistency:** All lines originate at `(0, 0)` and maintain monotonic trends without intersections.
---
### **Conclusion**
The chart illustrates throughput performance across varying hidden sizes and thread pool configurations. While TP:1 dominates in throughput, the inverse relationship with TP values warrants further investigation into system constraints or resource allocation strategies.
</details>
Figure 16: Attention $QKV$ transform with different TP sizes.
<details>
<summary>extracted/5378885/figures/transformer/attention_key_query_prob.png Details</summary>
### Visual Description
# Technical Document Extraction: GPU Throughput Analysis
## Chart Overview
The image presents a line chart comparing the computational throughput (in TFLOPs/s) of two GPU architectures (A100 and V100) across varying hidden sizes. The chart emphasizes performance scaling behavior relative to model complexity (hidden size).
---
## Axis Labels & Scales
- **X-axis (Hidden Size)**:
- Range: 0 to 32,768
- Increment: Logarithmic scale (powers of 2: 2048, 4096, 8192, ..., 32768)
- **Y-axis (Throughput (TFLOPs/s))**:
- Range: 0 to 150
- Increment: Linear scale (0, 50, 100, 150)
---
## Legend & Data Series
- **Legend Position**: Top-right quadrant
- **Color Coding**:
- **Blue (A100)**: Higher-end GPU architecture
- **Orange (V100)**: Legacy GPU architecture
---
## Key Trends & Data Points
### A100 (Blue Line)
- **Trend**: Consistent upward slope with minimal variance
- **Performance Progression**:
- 2048: ~10 TFLOPs/s
- 4096: ~20 TFLOPs/s
- 8192: ~35 TFLOPs/s
- 16384: ~70 TFLOPs/s
- 32768: ~120 TFLOPs/s
- **Error Bars**: Small vertical error bars (≤5% of throughput value)
### V100 (Orange Line)
- **Trend**: Bimodal pattern with peak at mid-range hidden sizes
- **Performance Progression**:
- 2048: ~20 TFLOPs/s
- 4096: ~30 TFLOPs/s
- 8192: ~50 TFLOPs/s
- 16384: ~60 TFLOPs/s (peak)
- 32768: ~40 TFLOPs/s (decline)
- **Error Bars**: Larger variance at higher hidden sizes (10-15% of throughput)
---
## Critical Observations
1. **Scaling Efficiency**:
- A100 maintains linear scaling up to 32,768 hidden size
- V100 exhibits diminishing returns beyond 16,384 hidden size
2. **Performance Gap**:
- At 32,768 hidden size:
- A100: 120 TFLOPs/s
- V100: 40 TFLOPs/s
- **Gap**: 3x performance difference
3. **Practical Implications**:
- A100 enables larger model training (e.g., 32k hidden size models)
- V100 becomes throughput-limited at mid-range model sizes
---
## Data Integrity Verification
- **Color Consistency**: All blue data points match A100 legend; all orange match V100
- **Axis Alignment**: All data points fall within axis ranges
- **Trend Logic**: Visual slopes match numerical progression (e.g., A100's 10→120 TFLOPs/s over 2048→32768 hidden size)
---
## Missing Elements
- No secondary y-axis or annotations
- No gridlines or reference markers
- No temporal or experimental condition metadata
---
## Conclusion
The chart demonstrates A100's superior scaling efficiency compared to V100, with throughput increasing proportionally to hidden size up to 32,768. V100's performance plateaus at mid-range model complexities, making it unsuitable for modern large-scale AI workloads.
</details>
Figure 17: Attention key-query score computation ( $KQ^T$ ).
<details>
<summary>extracted/5378885/figures/transformer/attention_prob_times_values.png Details</summary>
### Visual Description
# Technical Document Extraction: GPU Throughput Analysis
## Chart Overview
The image presents a line chart comparing the computational throughput (in TFLOP/s) of two GPU architectures (A100 and V100) across varying hidden sizes. The chart demonstrates performance scaling relationships between model complexity (hidden size) and hardware capability.
### Axis Labels
- **Y-axis**: Throughput (TFLOP/s)
Scale: 0–200 in increments of 50
- **X-axis**: Hidden Size
Discrete values: 2048, 4096, 6144, 8192, 10240, 12288, 14336, 16384, 18432, 20480, 22528, 24576, 26624, 28672, 30720, 32768
### Legend
- **Location**: Top-right quadrant
- **Labels**:
- `A100` (blue line)
- `V100` (orange line)
### Data Series Analysis
#### A100 (Blue Line)
- **Trend**: Linear upward slope with consistent growth
- **Key Data Points**:
- 2048: 0 TFLOP/s
- 4096: ~20 TFLOP/s
- 6144: ~40 TFLOP/s
- 8192: ~60 TFLOP/s
- 10240: ~80 TFLOP/s
- 12288: ~100 TFLOP/s
- 14336: ~120 TFLOP/s
- 16384: ~140 TFLOP/s
- 18432: ~150 TFLOP/s
- 20480: ~160 TFLOP/s
- 22528: ~170 TFLOP/s
- 24576: ~180 TFLOP/s
- 26624: ~190 TFLOP/s
- 28672: ~200 TFLOP/s
- 30720: ~210 TFLOP/s
- 32768: ~220 TFLOP/s
#### V100 (Orange Line)
- **Trend**: Sublinear growth with plateauing behavior
- **Key Data Points**:
- 2048: 0 TFLOP/s
- 4096: ~10 TFLOP/s
- 6144: ~20 TFLOP/s
- 8192: ~30 TFLOP/s
- 10240: ~40 TFLOP/s
- 12288: ~50 TFLOP/s
- 14336: ~50 TFLOP/s
- 16384: ~50 TFLOP/s
- 18432: ~50 TFLOP/s
- 20480: ~50 TFLOP/s
- 22528: ~50 TFLOP/s
- 24576: ~50 TFLOP/s
- 26624: ~50 TFLOP/s
- 28672: ~50 TFLOP/s
- 30720: ~50 TFLOP/s
- 32768: ~50 TFLOP/s
### Error Bars
- Both lines feature vertical error bars indicating measurement variability
- A100 error bars show increasing spread at higher hidden sizes
- V100 error bars remain relatively constant after 12288 hidden size
### Spatial Grounding
- Legend positioned at [x=0.85, y=0.95] (normalized coordinates)
- Color consistency verified:
- Blue markers correspond to A100 data points
- Orange markers correspond to V100 data points
### Technical Implications
1. A100 demonstrates linear scaling efficiency up to 32768 hidden size
2. V100 exhibits diminishing returns beyond 12288 hidden size
3. Performance gap widens exponentially at larger hidden sizes (e.g., 32768: A100=220 vs V100=50)
## Data Table Reconstruction
| Hidden Size | A100 (TFLOP/s) | V100 (TFLOP/s) |
|-------------|----------------|----------------|
| 2048 | 0 | 0 |
| 4096 | 20 | 10 |
| 6144 | 40 | 20 |
| 8192 | 60 | 30 |
| 10240 | 80 | 40 |
| 12288 | 100 | 50 |
| 14336 | 120 | 50 |
| 16384 | 140 | 50 |
| 18432 | 150 | 50 |
| 20480 | 160 | 50 |
| 22528 | 170 | 50 |
| 24576 | 180 | 50 |
| 26624 | 190 | 50 |
| 28672 | 200 | 50 |
| 30720 | 210 | 50 |
| 32768 | 220 | 50 |
## Chart Type
Line chart with error bars, comparing two continuous data series across a categorical x-axis.
## Language Notes
- All text in English
- No non-English content detected
</details>
Figure 18: Attention score times values.
<details>
<summary>extracted/5378885/figures/transformer/attention_linear_projection.png Details</summary>
### Visual Description
# Technical Document Analysis: Throughput vs. Hidden Size
## Image Type
Line chart comparing throughput performance of two devices (A100 and V100) across varying hidden sizes.
---
## Labels and Axis Titles
- **X-Axis**: "Hidden Size"
- Values: `2048, 4096, 6144, 8192, 10240, 12288, 14336, 16384, 18432, 20480, 22528, 24576, 26624, 28672, 30720, 32768`
- Increment pattern: Doubles from 2048 → 4096, then increments by 2048 thereafter.
- **Y-Axis**: "Throughput (TFLOP/s)"
- Range: `0` to `250`
- Increment: `50`
---
## Legend
- **Placement**: Right side of the chart
- **Labels**:
- `A100` (blue line with circular markers)
- `V100` (orange line with circular markers)
---
## Key Trends and Data Points
### A100 (Blue Line)
- **Trend**:
- Starts at `0 TFLOP/s` for hidden size `2048`.
- Sharp linear increase to `250 TFLOP/s` by hidden size `12288`.
- Plateaus at `250 TFLOP/s` for all larger hidden sizes (`14336` to `32768`).
- **Data Points**:
| Hidden Size | Throughput (TFLOP/s) |
|-------------|----------------------|
| 2048 | 0 |
| 4096 | 50 |
| 6144 | 100 |
| 8192 | 150 |
| 10240 | 200 |
| 12288 | 250 |
| 14336 | 250 |
| ... | ... |
| 32768 | 250 |
### V100 (Orange Line)
- **Trend**:
- Flat line at approximately `80 TFLOP/s` across all hidden sizes.
- **Data Points**:
| Hidden Size | Throughput (TFLOP/s) |
|-------------|----------------------|
| 2048 | 80 |
| 4096 | 80 |
| 6144 | 80 |
| ... | ... |
| 32768 | 80 |
---
## Spatial Grounding and Cross-Reference
- **Legend Colors**:
- Blue (`A100`) matches the blue line.
- Orange (`V100`) matches the orange line.
- **Legend Position**: Right-aligned, outside the plot area.
---
## Observations
1. **A100 Performance**:
- Throughput scales linearly with hidden size up to `10240`, then saturates at `250 TFLOP/s`.
- No further gains observed beyond `12288` hidden size.
2. **V100 Performance**:
- Constant throughput of `80 TFLOP/s` regardless of hidden size.
- No scaling behavior observed.
---
## Language Notes
- **Primary Language**: English (all labels, axis titles, and annotations are in English).
- **No Additional Languages Detected**.
---
## Conclusion
The chart demonstrates that the A100 device exhibits strong scalability with hidden size up to a critical point (`12288`), after which throughput plateaus. In contrast, the V100 maintains a fixed throughput across all tested hidden sizes, indicating no scalability in this parameter range.
</details>
Figure 19: Post-attention linear projection.
Figure 20 shows how the size of the vocab and the hidden dimension affects the logit layer, which is a linear layer at the end of the transformer model. The performance of the logit layer is maximized when $v$ is a multiple of 64, therefore it is best to pad the vocab size to the nearest multiple of 64. Likewise, the layer also performs best with a hidden size that is a multiple of 64.
<details>
<summary>extracted/5378885/figures/transformer/vocab_v_sweep.png Details</summary>
### Visual Description
# Technical Document Extraction: Throughput Analysis Chart
## Chart Type
Line chart with vertical spikes representing throughput measurements.
## Axes Labels
- **X-axis**: Labeled "V" with numerical markers at intervals of 50 (51150, 51175, 51200, 51225, 51250).
- **Y-axis**: Labeled "Throughput (TFLOP/s)" with numerical markers at intervals of 50 (0, 50, 100, 150, 200, 250, 300).
## Legend
- **Location**: Top-right corner of the chart.
- **Label**: "Throughput (TFLOP/s)" in blue text.
- **Color Match**: Blue line and data points correspond to the legend.
## Data Trends
1. **Baseline Throughput**:
- Consistent baseline value of approximately **100 TFLOP/s** across all X-axis intervals.
- Minor fluctuations observed (e.g., 95–105 TFLOP/s) but no significant deviation from the baseline.
2. **Spike Pattern**:
- **Vertical spikes** occur at every X-axis marker (51150, 51175, 51200, etc.).
- **Spike Height**: Uniformly reaches **~250 TFLOP/s** at each interval.
- **Structure**: Each spike consists of a sharp upward line terminating in a circular data point at the peak.
3. **Periodicity**:
- Spikes repeat at regular intervals of **25 units** along the X-axis (e.g., 51150 → 51175 → 51200).
- No irregularities or anomalies in spike timing or magnitude.
## Key Observations
- The chart suggests a **cyclical or periodic process** where throughput spikes to 250 TFLOP/s at fixed intervals of "V" while maintaining a stable baseline.
- The consistent spike height and baseline indicate a controlled or repeatable system behavior.
- No data points fall outside the 100–250 TFLOP/s range, confirming bounded variability.
## Spatial Grounding
- **Legend Position**: Top-right corner (coordinates: [x: 0.95, y: 0.95] relative to chart boundaries).
- **Data Point Alignment**: All spikes align vertically with X-axis markers, confirming precise measurement intervals.
## Conclusion
The chart visualizes a periodic throughput pattern with deterministic spikes at regular intervals of "V". The system maintains a stable baseline of 100 TFLOP/s, with no deviations observed in the measured range.
</details>
(a) Sweep over vocabulary size
<details>
<summary>extracted/5378885/figures/transformer/vocab_h_sweep.png Details</summary>
### Visual Description
# Technical Document Extraction: Line Chart Analysis
## 1. Axis Labels and Markers
- **Y-Axis**:
- Label: `Throughput (TFLOP/s)`
- Range: `0` to `300`
- Increment: `50` (visible grid lines at 0, 50, 100, 150, 200, 250, 300)
- **X-Axis**:
- Label: `h`
- Range: `14275` to `14400`
- Increment: `25` (values at 14275, 14300, 14325, 14350, 14375, 14400)
## 2. Legend
- **Position**: Top-right corner of the chart
- **Label**: `Throughput`
- **Color**: Blue (matches all data points and connecting lines)
## 3. Data Points and Trends
- **X-Axis Values**:
- Data points are plotted at intervals of `25` (e.g., 14275, 14300, 14325, ..., 14400).
- **Y-Axis Behavior**:
- **Peaks**: Consistent at approximately `250 TFLOP/s` for all data points.
- **Troughs**: Consistent at approximately `175 TFLOP/s` for all data points.
- **Visual Trend**:
- The line exhibits a **repeating sinusoidal pattern** with peaks and troughs.
- No deviation from the pattern across the x-axis range.
## 4. Component Isolation
- **Header**: No explicit header text.
- **Main Chart**:
- Blue line with circular markers (dots) at each data point.
- Grid lines for both axes to aid readability.
- **Footer**: No footer text.
## 5. Spatial Grounding
- **Legend Placement**: Top-right corner (standard for line charts).
- **Data Point Alignment**:
- All blue dots and connecting lines align with the x-axis increments.
## 6. Key Observations
- The chart represents a **stable oscillatory throughput** over the x-axis range (`h`).
- No anomalies or outliers detected in the data series.
- The pattern suggests a **periodic process** (e.g., batch processing, cyclic workload).
## 7. Missing Elements
- No additional legends, annotations, or textual data present.
- No secondary y-axis or secondary data series.
## 8. Conclusion
The chart depicts a **consistent, periodic throughput pattern** with no variation in amplitude or frequency across the observed range of `h`.
</details>
(b) Zoomed-in sweep over vocabulary size
Figure 20: Vocabulary embedding transformation.
## Appendix B A100 Results
Figures 21 through 47 show the performance of the Attention Key-Query Score and Attention Over Value computations for various numbers of attention heads. In each of these figures, we highlight the trend observed when using tensor cores. Each color is represented in the legend as a power of 2, which designates the highest power of 2 that divides $h/a$ . This shows how using a value of $h/a$ where the highest power of 2 multiple is 3 or less can impact performance greatly. Figures 34 and 9 show that in general, throughput increases with hidden size and decreases with the number of attention heads. Some of these figures also show the effects of wave quantization.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a8.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Chart
## Chart Title
**Attention Key Query Score, a=8**
## Axes
- **X-axis**: Hidden Size (ranges from 0 to 32768)
- **Y-axis**: Throughput (TFLOPs/s) (ranges from 0 to 250)
## Legend
- **Location**: Right side of the chart
- **Color-Coded Labels**:
- **Blue**: h/a = 1
- **Orange**: h/a = 2
- **Green**: h/a = 4
- **Red**: h/a = 8
- **Purple**: h/a = 16
- **Brown**: h/a = 32
- **Pink**: h/a = 64
## Key Trends
1. **h/a = 1 (Blue Line)**:
- Starts at the origin (0,0).
- Gradually increases, plateauing around **90 TFLOPs/s** by x=32768.
- Lowest throughput among all series.
2. **h/a = 2 (Orange Line)**:
- Starts at the origin.
- Rises sharply, plateauing around **160 TFLOPs/s** by x=32768.
- Second-lowest throughput.
3. **h/a = 4 (Green Line)**:
- Starts at the origin.
- Increases steadily, plateauing near **160 TFLOPs/s** by x=32768.
- Overlaps with h/a=2 at higher x-values.
4. **h/a = 8 (Red Line)**:
- Starts at the origin.
- Rises sharply, plateauing near **240 TFLOPs/s** by x=32768.
- Slightly below h/a=16 and h/a=32.
5. **h/a = 16 (Purple Line)**:
- Starts at the origin.
- Increases rapidly, plateauing near **250 TFLOPs/s** by x=32768.
- Highest throughput among mid-range ratios.
6. **h/a = 32 (Brown Line)**:
- Starts at the origin.
- Rises sharply, plateauing near **250 TFLOPs/s** by x=32768.
- Overlaps with h/a=16 at higher x-values.
7. **h/a = 64 (Pink Line)**:
- Starts at the origin.
- Increases sharply, plateauing near **250 TFLOPs/s** by x=32768.
- Highest throughput overall.
## Spatial Grounding
- **Legend Position**: Right-aligned, outside the chart boundary.
- **Color Consistency**: All lines match their legend labels (e.g., red = h/a=8, pink = h/a=64).
## Component Isolation
- **Header**: Chart title centered at the top.
- **Main Chart**: Line plot with seven data series.
- **Footer**: No additional text or components.
## Data Table (Reconstructed)
| Hidden Size | h/a=1 | h/a=2 | h/a=4 | h/a=8 | h/a=16 | h/a=32 | h/a=64 |
|-------------|-------|-------|-------|-------|--------|--------|--------|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4096 | ~60 | ~80 | ~100 | ~120 | ~140 | ~160 | ~180 |
| 8192 | ~85 | ~120 | ~140 | ~180 | ~210 | ~220 | ~230 |
| 12288 | ~90 | ~140 | ~150 | ~220 | ~235 | ~240 | ~245 |
| 16384 | ~90 | ~150 | ~160 | ~230 | ~240 | ~245 | ~248 |
| 20480 | ~90 | ~160 | ~165 | ~240 | ~245 | ~248 | ~250 |
| 24576 | ~90 | ~160 | ~165 | ~245 | ~248 | ~250 | ~250 |
| 28672 | ~90 | ~160 | ~165 | ~245 | ~248 | ~250 | ~250 |
| 32768 | ~95 | ~165 | ~170 | ~245 | ~250 | ~250 | ~250 |
## Notes
- All lines exhibit a sigmoidal growth pattern, with higher h/a ratios achieving higher throughput plateaus.
- No textual data or embedded diagrams beyond the chart itself.
- No non-English text present.
</details>
Figure 21: Attention key-query score GEMM throughput for 8 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a12.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Chart
## Chart Title
**Attention Key Query Score, a=12**
## Axes Labels
- **X-axis**: Hidden Size (ranges from 0 to 32768)
- **Y-axis**: Throughput (TFLOPs/s) (ranges from 0 to 250)
## Legend
- **Location**: Right side of the chart
- **Color-Coded Labels**:
- Blue: h/a = 1
- Orange: h/a = 2
- Green: h/a = 4
- Red: h/a = 8
- Purple: h/a = 16
- Brown: h/a = 32
- Pink: h/a = 64
## Key Trends and Data Points
### Line Analysis
1. **Blue (h/a=1)**
- **Trend**: Gradual upward slope, plateauing near 100 TFLOPs/s.
- **Key Points**:
- At hidden size 4096: ~70 TFLOPs/s
- At hidden size 8192: ~90 TFLOPs/s
- At hidden size 32768: ~95 TFLOPs/s
2. **Orange (h/a=2)**
- **Trend**: Slightly steeper than blue, plateauing near 150 TFLOPs/s.
- **Key Points**:
- At hidden size 4096: ~100 TFLOPs/s
- At hidden size 8192: ~130 TFLOPs/s
- At hidden size 32768: ~150 TFLOPs/s
3. **Green (h/a=4)**
- **Trend**: Moderate slope, plateauing near 160 TFLOPs/s.
- **Key Points**:
- At hidden size 4096: ~120 TFLOPs/s
- At hidden size 8192: ~140 TFLOPs/s
- At hidden size 32768: ~160 TFLOPs/s
4. **Red (h/a=8)**
- **Trend**: Steeper ascent, plateauing near 220 TFLOPs/s.
- **Key Points**:
- At hidden size 4096: ~180 TFLOPs/s
- At hidden size 8192: ~210 TFLOPs/s
- At hidden size 32768: ~230 TFLOPs/s
5. **Purple (h/a=16)**
- **Trend**: Sharp rise, plateauing near 240 TFLOPs/s.
- **Key Points**:
- At hidden size 4096: ~200 TFLOPs/s
- At hidden size 8192: ~230 TFLOPs/s
- At hidden size 32768: ~240 TFLOPs/s
6. **Brown (h/a=32)**
- **Trend**: Near-maximal performance, plateauing near 245 TFLOPs/s.
- **Key Points**:
- At hidden size 4096: ~220 TFLOPs/s
- At hidden size 8192: ~240 TFLOPs/s
- At hidden size 32768: ~245 TFLOPs/s
7. **Pink (h/a=64)**
- **Trend**: Highest performance, plateauing near 250 TFLOPs/s.
- **Key Points**:
- At hidden size 4096: ~230 TFLOPs/s
- At hidden size 8192: ~245 TFLOPs/s
- At hidden size 32768: ~250 TFLOPs/s
## Spatial Grounding
- **Legend Placement**: Right-aligned, adjacent to the chart.
- **Color Consistency**:
- Blue line matches h/a=1 (legend).
- Pink line matches h/a=64 (legend).
- All other colors align with their respective h/a values.
## Component Isolation
1. **Header**: Chart title centered at the top.
2. **Main Chart**:
- Lines plotted against hidden size (x-axis) and throughput (y-axis).
- Lines converge at higher hidden sizes, indicating diminishing returns.
3. **Footer**: No additional text or components.
## Language and Transcription
- **Primary Language**: English.
- **No Other Languages Detected**.
## Summary
The chart illustrates the relationship between hidden size and throughput (TFLOPs/s) for varying h/a ratios (1, 2, 4, 8, 16, 32, 64). Higher h/a ratios achieve greater throughput but exhibit diminishing returns as hidden size increases. The pink line (h/a=64) achieves the highest throughput (~250 TFLOPs/s), while the blue line (h/a=1) performs the lowest (~95 TFLOPs/s). All lines plateau near their maximum throughput at hidden size 32768.
</details>
Figure 22: Attention key-query score GEMM throughput for 12 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a16.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Chart
## Chart Title
**Attention Key Query Score, a=16**
## Axes
- **X-axis (Horizontal):**
- Label: `Hidden Size`
- Range: `0` to `32768`
- Tick Intervals: `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`
- **Y-axis (Vertical):**
- Label: `Throughput (TFLOPs/s)`
- Range: `0` to `250`
- Tick Intervals: `0`, `50`, `100`, `150`, `200`, `250`
## Legend
- **Location:** Right side of the chart
- **Color-Coded Labels:**
- `h/a = 1` (Blue)
- `h/a = 2` (Orange)
- `h/a = 4` (Green)
- `h/a = 8` (Red)
- `h/a = 16` (Purple)
- `h/a = 32` (Brown)
- `h/a = 64` (Pink)
## Data Series Trends
1. **Blue Line (h/a = 1):**
- Starts near `0` TFLOPs/s at `Hidden Size = 0`.
- Gradually increases, reaching ~`90` TFLOPs/s at `Hidden Size = 32768`.
- Slope: Gentle upward trend with minor fluctuations.
2. **Orange Line (h/a = 2):**
- Starts slightly above blue line.
- Reaches ~`120` TFLOPs/s at `Hidden Size = 32768`.
- Slope: Steeper than blue line, with minor oscillations.
3. **Green Line (h/a = 4):**
- Starts higher than orange line.
- Reaches ~`150` TFLOPs/s at `Hidden Size = 32768`.
- Slope: Moderate upward trend with slight dips.
4. **Red Line (h/a = 8):**
- Starts significantly higher than green line.
- Reaches ~`220` TFLOPs/s at `Hidden Size = 32768`.
- Slope: Sharp initial rise, then stabilizes.
5. **Purple Line (h/a = 16):**
- Starts near red line.
- Reaches ~`230` TFLOPs/s at `Hidden Size = 32768`.
- Slope: Smooth upward trend with minor fluctuations.
6. **Brown Line (h/a = 32):**
- Starts slightly above purple line.
- Reaches ~`240` TFLOPs/s at `Hidden Size = 32768`.
- Slope: Steady increase with minor oscillations.
7. **Pink Line (h/a = 64):**
- Starts highest among all lines.
- Reaches ~`245` TFLOPs/s at `Hidden Size = 32768`.
- Slope: Consistent upward trend with minor dips.
## Key Observations
- **Throughput Correlation:** Higher `h/a` ratios (e.g., `64`) consistently achieve higher throughput than lower ratios (e.g., `1`).
- **Saturation Point:** All lines plateau near `Hidden Size = 32768`, indicating diminishing returns beyond this point.
- **Stability:** Lines for `h/a ≥ 8` exhibit smoother trends compared to lower ratios, which show more variability.
## Spatial Grounding
- **Legend Position:** Right-aligned, outside the chart boundary.
- **Line Placement:**
- Blue (`h/a = 1`) is the lowest line.
- Pink (`h/a = 64`) is the highest line.
- Colors match legend labels exactly.
## Component Isolation
1. **Header:** Chart title centered at the top.
2. **Main Chart:**
- X-axis and Y-axis labeled with numerical ranges.
- Seven data series plotted with distinct colors.
3. **Footer:** No additional text or components.
## Data Table (Reconstructed)
| Hidden Size | h/a = 1 | h/a = 2 | h/a = 4 | h/a = 8 | h/a = 16 | h/a = 32 | h/a = 64 |
|-------------|---------|---------|---------|---------|----------|----------|----------|
| 0 | ~0 | ~0 | ~0 | ~0 | ~0 | ~0 | ~0 |
| 4096 | ~30 | ~40 | ~60 | ~90 | ~110 | ~130 | ~150 |
| 8192 | ~70 | ~90 | ~110 | ~170 | ~190 | ~200 | ~210 |
| 12288 | ~80 | ~100 | ~120 | ~200 | ~215 | ~220 | ~225 |
| 16384 | ~85 | ~105 | ~130 | ~210 | ~225 | ~230 | ~235 |
| 20480 | ~90 | ~110 | ~140 | ~220 | ~230 | ~235 | ~240 |
| 24576 | ~95 | ~115 | ~150 | ~225 | ~235 | ~240 | ~245 |
| 28672 | ~90 | ~110 | ~155 | ~230 | ~240 | ~245 | ~245 |
| 32768 | ~95 | ~120 | ~160 | ~235 | ~245 | ~245 | ~245 |
## Notes
- All numerical values are approximations based on visual inspection of the chart.
- No non-English text or additional data sources are present.
- The chart focuses on the relationship between `Hidden Size` and `Throughput` for varying `h/a` ratios.
</details>
Figure 23: Attention key-query score GEMM throughput for 16 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a20.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key QueryScore Graph
## Title
**Attention Key Query Score, a=20**
## Axes
- **X-axis**: Hidden Size (values: 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768)
- **Y-axis**: Throughput (TFLOPs/s) (values: 0, 50, 100, 150, 200)
## Legend
- **Position**: Right side of the plot (outside the graph area)
- **Labels and Colors**:
- `h/a = 1` (blue)
- `h/a = 2` (orange)
- `h/a = 4` (green)
- `h/a = 8` (red)
- `h/a = 16` (purple)
- `h/a = 32` (brown)
- `h/a = 64` (pink)
## Data Trends
1. **h/a = 1 (blue)**:
- **Trend**: Gentle upward slope, lowest throughput across all hidden sizes.
- **Key Points**: Starts at 0, reaches ~90 TFLOPs/s at hidden size 32768.
2. **h/a = 2 (orange)**:
- **Trend**: Moderate upward slope, surpasses h/a=1 but remains below h/a=4.
- **Key Points**: Starts at 0, reaches ~140 TFLOPs/s at hidden size 32768.
3. **h/a = 4 (green)**:
- **Trend**: Steeper slope than h/a=2, consistent growth.
- **Key Points**: Starts at 0, reaches ~150 TFLOPs/s at hidden size 32768.
4. **h/a = 8 (red)**:
- **Trend**: Sharp upward slope, surpasses h/a=4 significantly.
- **Key Points**: Starts at 0, reaches ~200 TFLOPs/s at hidden size 32768.
5. **h/a = 16 (purple)**:
- **Trend**: Very steep slope, highest throughput until h/a=32.
- **Key Points**: Starts at 0, peaks at ~220 TFLOPs/s near hidden size 24576, then plateaus.
6. **h/a = 32 (brown)**:
- **Trend**: Slightly less steep than h/a=16, but maintains high throughput.
- **Key Points**: Starts at 0, peaks at ~220 TFLOPs/s near hidden size 24576, then plateaus.
7. **h/a = 64 (pink)**:
- **Trend**: Steepest slope, highest throughput overall.
- **Key Points**: Starts at 0, peaks at ~220 TFLOPs/s near hidden size 24576, then plateaus.
## Key Observations
- **Correlation**: Higher `h/a` ratios correspond to higher throughput, especially at larger hidden sizes.
- **Plateaus**: All lines plateau near hidden size 24576–32768, indicating diminishing returns beyond this point.
- **Divergence**: Lines for h/a=16, 32, and 64 overlap significantly in the plateau region, suggesting similar performance limits.
## Spatial Grounding
- **Legend**: Right-aligned, outside the plot area. Colors match line colors exactly (e.g., blue = h/a=1).
- **Data Points**: All lines originate at (0, 0) and increase monotonically with hidden size.
## Conclusion
The graph demonstrates that increasing the `h/a` ratio improves throughput, with diminishing returns observed at larger hidden sizes. The highest performance is achieved with `h/a=64`, though all high-ratio lines converge near the plateau region.
</details>
Figure 24: Attention key-query score GEMM throughput for 20 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a24.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Graph
## Title
**Attention Key Query Score, a=24**
## Axes
- **X-axis (Horizontal):**
- Label: `Hidden Size`
- Range: `0` to `32768`
- Markers: `0`, `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`
- **Y-axis (Vertical):**
- Label: `Throughput (TFLOPs/s)`
- Range: `0` to `200`
- Markers: `0`, `50`, `100`, `150`, `200`
## Legend
- **Location:** Right side of the graph
- **Entries:**
- `h / a = 1` (Blue)
- `h / a = 2` (Orange)
- `h / a = 4` (Green)
- `h / a = 8` (Red)
- `h / a = 16` (Purple)
- `h / a = 32` (Brown)
- `h / a = 64` (Pink)
## Data Series & Trends
1. **Blue Line (`h / a = 1`):**
- **Trend:** Gradual upward slope from `0` to `~90 TFLOPs/s` across the entire `Hidden Size` range.
- **Key Points:**
- At `Hidden Size = 0`: `~0 TFLOPs/s`
- At `Hidden Size = 32768`: `~90 TFLOPs/s`
2. **Orange Line (`h / a = 2`):**
- **Trend:** Steeper initial rise than blue, plateauing at `~150 TFLOPs/s`.
- **Key Points:**
- At `Hidden Size = 0`: `~0 TFLOPs/s`
- At `Hidden Size = 32768`: `~150 TFLOPs/s`
3. **Green Line (`h / a = 4`):**
- **Trend:** Smooth upward trajectory, reaching `~150 TFLOPs/s` by `Hidden Size = 32768`.
- **Key Points:**
- At `Hidden Size = 0`: `~0 TFLOPs/s`
- At `Hidden Size = 32768`: `~150 TFLOPs/s`
4. **Red Line (`h / a = 8`):**
- **Trend:** Sharp initial spike, peaking at `~220 TFLOPs/s` near `Hidden Size = 8192`, then stabilizing.
- **Key Points:**
- At `Hidden Size = 0`: `~0 TFLOPs/s`
- At `Hidden Size = 8192`: `~220 TFLOPs/s`
- At `Hidden Size = 32768`: `~200 TFLOPs/s`
5. **Purple Line (`h / a = 16`):**
- **Trend:** Rapid rise to `~220 TFLOPs/s` by `Hidden Size = 8192`, followed by minor fluctuations.
- **Key Points:**
- At `Hidden Size = 0`: `~0 TFLOPs/s`
- At `Hidden Size = 8192`: `~220 TFLOPs/s`
- At `Hidden Size = 32768`: `~210 TFLOPs/s`
6. **Brown Line (`h / a = 32`):**
- **Trend:** Similar to purple but with sharper fluctuations. Peaks at `~230 TFLOPs/s` near `Hidden Size = 24576`.
- **Key Points:**
- At `Hidden Size = 0`: `~0 TFLOPs/s`
- At `Hidden Size = 24576`: `~230 TFLOPs/s`
- At `Hidden Size = 32768`: `~215 TFLOPs/s`
7. **Pink Line (`h / a = 64`):**
- **Trend:** Highest performance, peaking at `~230 TFLOPs/s` near `Hidden Size = 24576`, then stabilizing.
- **Key Points:**
- At `Hidden Size = 0`: `~0 TFLOPs/s`
- At `Hidden Size = 24576`: `~230 TFLOPs/s`
- At `Hidden Size = 32768`: `~220 TFLOPs/s`
## Observations
- **Performance Correlation:** Higher `h / a` ratios (e.g., `64`) generally achieve higher throughput, especially at larger `Hidden Size` values.
- **Divergence:** Lines for `h / a = 8`, `16`, `32`, and `64` converge at higher `Hidden Size` values, suggesting diminishing returns beyond a certain point.
- **Stability:** All lines stabilize near `Hidden Size = 32768`, indicating saturation of throughput gains.
## Notes
- No non-English text or additional data tables are present.
- All legend colors match their corresponding lines in the graph.
- Spatial grounding confirms the legend is positioned on the right, with no overlap or misalignment.
</details>
Figure 25: Attention key-query score GEMM throughput for 24 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a32.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Analysis
## Chart Overview
The image depicts a line chart titled **"Attention Key Query Score, a=32"**, analyzing the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOP/s)** (y-axis). The chart includes seven data series representing different **h/a** (hidden size to attention head ratio) values, with trends visualized across a logarithmic scale of Hidden Size.
---
### Key Components
1. **Title**:
- *"Attention Key Query Score, a=32"*
- Indicates the analysis focuses on attention mechanisms with a fixed parameter `a=32`.
2. **Axes**:
- **X-axis (Hidden Size)**:
- Logarithmic scale ranging from `0` to `32768`.
- Markers at intervals: `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`.
- **Y-axis (Throughput)**:
- Linear scale from `0` to `200` TFLOP/s.
- Markers at intervals: `0`, `50`, `100`, `150`, `200`.
3. **Legend**:
- Located in the **top-right corner**.
- Colors map to **h/a ratios**:
- `1` (blue), `2` (orange), `4` (green), `8` (red), `16` (purple), `32` (brown), `64` (pink).
- Confirmed spatial grounding: All line colors match legend entries exactly.
---
### Data Series Trends
1. **h/a = 1 (Blue Line)**:
- **Trend**: Gradual upward slope with minor fluctuations.
- **Key Points**:
- At Hidden Size `4096`: ~30 TFLOP/s.
- At Hidden Size `32768`: ~90 TFLOP/s.
2. **h/a = 2 (Orange Line)**:
- **Trend**: Steeper initial rise, then plateaus.
- **Key Points**:
- At Hidden Size `4096`: ~50 TFLOP/s.
- At Hidden Size `32768`: ~140 TFLOP/s.
3. **h/a = 4 (Green Line)**:
- **Trend**: Consistent upward trajectory with minor dips.
- **Key Points**:
- At Hidden Size `4096`: ~70 TFLOP/s.
- At Hidden Size `32768`: ~145 TFLOP/s.
4. **h/a = 8 (Red Line)**:
- **Trend**: Sharp rise, followed by stabilization with oscillations.
- **Key Points**:
- At Hidden Size `4096`: ~90 TFLOP/s.
- At Hidden Size `32768`: ~200 TFLOP/s.
5. **h/a = 16 (Purple Line)**:
- **Trend**: Rapid ascent, then sustained high throughput with minor dips.
- **Key Points**:
- At Hidden Size `4096`: ~120 TFLOP/s.
- At Hidden Size `32768`: ~200 TFLOP/s.
6. **h/a = 32 (Brown Line)**:
- **Trend**: Highest throughput, peaking early and maintaining near-maximum.
- **Key Points**:
- At Hidden Size `4096`: ~150 TFLOP/s.
- At Hidden Size `32768`: ~210 TFLOP/s.
7. **h/a = 64 (Pink Line)**:
- **Trend**: Highest throughput overall, with slight fluctuations.
- **Key Points**:
- At Hidden Size `4096`: ~160 TFLOP/s.
- At Hidden Size `32768`: ~210 TFLOP/s.
---
### Observations
- **Scaling Behavior**: Higher **h/a ratios** (e.g., 32, 64) achieve significantly higher throughput, especially at larger Hidden Sizes.
- **Efficiency**: Lower ratios (e.g., 1, 2) show diminishing returns as Hidden Size increases.
- **Stability**: Lines for h/a ≥ 8 exhibit smoother trends compared to lower ratios, which show more volatility.
---
### Notes
- No non-English text detected.
- No embedded data tables or heatmaps present.
- All textual elements (labels, titles, legend) are in English.
This analysis confirms that throughput scales non-linearly with Hidden Size, with higher h/a ratios achieving optimal performance at larger scales.
</details>
Figure 26: Attention key-query score GEMM throughput for 32 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a40.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Analysis
## Chart Overview
The image is a line chart titled **"Attention Key Query Score, a=40"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOP/s)** (y-axis) for varying **h/a** ratios. The chart includes seven data series, each represented by a distinct color and labeled in the legend.
---
### Axis Labels and Markers
- **X-Axis (Hidden Size)**:
- Range: 0 to 32,768
- Tick Intervals: 0, 4,096, 8,192, 12,288, 16,384, 20,480, 24,576, 28,672, 32,768
- Label: "Hidden Size"
- **Y-Axis (Throughput)**:
- Range: 0 to 200 TFLOP/s
- Tick Intervals: 0, 50, 100, 150, 200
- Label: "Throughput (TFLOP/s)"
---
### Legend
The legend is positioned on the **right side** of the chart. It maps **h/a ratios** to line colors:
| Color | h/a Ratio | Line Style |
|---------|-----------|------------|
| Blue | 1 | Solid |
| Orange | 2 | Solid |
| Green | 4 | Solid |
| Red | 8 | Solid |
| Purple | 16 | Solid |
| Brown | 32 | Solid |
| Pink | 64 | Solid |
---
### Data Series and Trends
1. **h/a = 1 (Blue Line)**
- **Trend**: Gradual, linear increase with minor fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: ~0 TFLOP/s
- At Hidden Size = 32,768: ~90 TFLOP/s
2. **h/a = 2 (Orange Line)**
- **Trend**: Steeper than h/a=1, with oscillations.
- **Key Data Points**:
- At Hidden Size = 0: ~0 TFLOP/s
- At Hidden Size = 32,768: ~140 TFLOP/s
3. **h/a = 4 (Green Line)**
- **Trend**: Moderate growth with periodic dips.
- **Key Data Points**:
- At Hidden Size = 0: ~0 TFLOP/s
- At Hidden Size = 32,768: ~130 TFLOP/s
4. **h/a = 8 (Red Line)**
- **Trend**: Sharp initial rise, plateauing near 180 TFLOP/s.
- **Key Data Points**:
- At Hidden Size = 0: ~0 TFLOP/s
- At Hidden Size = 32,768: ~190 TFLOP/s
5. **h/a = 16 (Purple Line)**
- **Trend**: Rapid ascent, peaking at ~200 TFLOP/s, then slight decline.
- **Key Data Points**:
- At Hidden Size = 0: ~0 TFLOP/s
- At Hidden Size = 32,768: ~200 TFLOP/s
6. **h/a = 32 (Brown Line)**
- **Trend**: Highest throughput, peaking at ~210 TFLOP/s.
- **Key Data Points**:
- At Hidden Size = 0: ~0 TFLOP/s
- At Hidden Size = 32,768: ~210 TFLOP/s
7. **h/a = 64 (Pink Line)**
- **Trend**: Steep rise to ~200 TFLOP/s, then stabilization.
- **Key Data Points**:
- At Hidden Size = 0: ~0 TFLOP/s
- At Hidden Size = 32,768: ~200 TFLOP/s
---
### Observations
- **Performance Scaling**: Higher h/a ratios (e.g., 32, 64) achieve significantly higher throughput than lower ratios (e.g., 1, 2).
- **Optimal Hidden Size**: For h/a ≥ 8, throughput plateaus near 180–210 TFLOP/s, suggesting diminishing returns beyond Hidden Size = 20,480.
- **Efficiency Tradeoff**: h/a=64 achieves near-peak performance with lower Hidden Size compared to h/a=32.
---
### Spatial Grounding
- **Legend Position**: Right-aligned, outside the plot area.
- **Line-Color Consistency**: All lines match their legend labels (e.g., pink = h/a=64).
---
### Conclusion
The chart demonstrates that increasing the **h/a ratio** improves throughput, with diminishing returns at higher Hidden Sizes. The optimal configuration for maximum throughput is **h/a=32** or **h/a=64**, achieving ~200–210 TFLOP/s.
</details>
Figure 27: Attention key-query score GEMM throughput for 40 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a64.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Chart
## Chart Overview
The image is a **line chart** titled **"Attention Key Query Score, a=64"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOP/s)** (y-axis) for varying **h/a** ratios. The chart includes seven data series, each represented by a distinct color and labeled in the legend.
---
## Axis Labels and Markers
- **X-Axis (Hidden Size)**:
- Range: `0` to `32768`
- Tick marks at: `0`, `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`
- Label: **"Hidden Size"**
- **Y-Axis (Throughput)**:
- Range: `0` to `200`
- Label: **"Throughput (TFLOP/s)"**
---
## Legend
- **Location**: Right side of the chart
- **Entries**:
1. **Blue**: `h/a = 1`
2. **Orange**: `h/a = 2`
3. **Green**: `h/a = 4`
4. **Red**: `h/a = 8`
5. **Purple**: `h/a = 16`
6. **Brown**: `h/a = 32`
7. **Pink**: `h/a = 64`
---
## Data Series and Trends
### 1. **h/a = 1 (Blue Line)**
- **Trend**: Starts near `0` at `Hidden Size = 0`, gradually increases to **~80 TFLOP/s** at `Hidden Size = 32768`.
- **Key Data Points**:
- `Hidden Size = 32768`: **80 TFLOP/s**
### 2. **h/a = 2 (Orange Line)**
- **Trend**: Starts slightly above `h/a = 1`, steeper growth, reaching **~130 TFLOP/s** at `Hidden Size = 32768`.
- **Key Data Points**:
- `Hidden Size = 32768`: **130 TFLOP/s**
### 3. **h/a = 4 (Green Line)**
- **Trend**: Starts higher than `h/a = 2`, steeper slope, reaching **~120 TFLOP/s** at `Hidden Size = 32768`.
- **Key Data Points**:
- `Hidden Size = 32768`: **120 TFLOP/s**
### 4. **h/a = 8 (Red Line)**
- **Trend**: Starts significantly higher, steep ascent, peaking at **~190 TFLOP/s** at `Hidden Size = 32768`.
- **Key Data Points**:
- `Hidden Size = 32768`: **190 TFLOP/s**
### 5. **h/a = 16 (Purple Line)**
- **Trend**: Highest starting point, steepest slope, reaching **~195 TFLOP/s** at `Hidden Size = 32768`.
- **Key Data Points**:
- `Hidden Size = 32768`: **195 TFLOP/s**
### 6. **h/a = 32 (Brown Line)**
- **Trend**: Starts highest, steep rise, plateaus near **~200 TFLOP/s** at `Hidden Size = 32768`.
- **Key Data Points**:
- `Hidden Size = 32768`: **200 TFLOP/s**
### 7. **h/a = 64 (Pink Line)**
- **Trend**: Highest starting point, steepest initial growth, ending at **~210 TFLOP/s** at `Hidden Size = 32768`.
- **Key Data Points**:
- `Hidden Size = 32768`: **210 TFLOP/s**
---
## Spatial Grounding and Verification
- **Legend Alignment**:
- Colors in the legend match the line colors in the chart (e.g., blue = `h/a = 1`, pink = `h/a = 64`).
- **Trend Consistency**:
- Lines with higher `h/a` ratios start higher on the y-axis and exhibit steeper slopes, aligning with the described trends.
---
## Summary
The chart demonstrates that **throughput increases with larger hidden sizes** and **higher h/a ratios**. The `h/a = 64` (pink) line achieves the highest throughput (~210 TFLOP/s), while `h/a = 1` (blue) has the lowest (~80 TFLOP/s). All data points and trends are consistent with the legend and axis labels.
</details>
Figure 28: Attention key-query score GEMM throughput for 64 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a80.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Graph
## Image Description
The image is a line graph titled **"Attention Key Query Score, a=80"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOPs/s)** (y-axis). The graph includes seven data series, each represented by a distinct colored line, corresponding to different **h/a** (hidden size to attention head ratio) values. The legend is positioned on the right side of the graph.
---
## Key Components
### Axes
- **X-axis (Hidden Size)**:
- Range: 0 to 32,768
- Tick marks at: 0, 4,096, 8,192, 12,288, 16,384, 20,480, 24,576, 28,672, 32,768
- Label: "Hidden Size"
- **Y-axis (Throughput)**:
- Range: 0 to 150 TFLOPs/s
- Tick marks at: 0, 50, 100, 150
- Label: "Throughput (TFLOPs/s)"
### Legend
- Located on the **right side** of the graph.
- Colors and labels:
- **Blue**: h/a = 1
- **Orange**: h/a = 2
- **Green**: h/a = 4
- **Red**: h/a = 8
- **Purple**: h/a = 16
- **Brown**: h/a = 32
- **Pink**: h/a = 64
---
## Data Series Analysis
### 1. h/a = 1 (Blue Line)
- **Trend**: Starts at 0 TFLOPs/s, rises gradually, and plateaus around **70–80 TFLOPs/s** at Hidden Size = 32,768.
- **Key Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 32,768: ~75 TFLOPs/s
### 2. h/a = 2 (Orange Line)
- **Trend**: Similar to h/a = 1 but slightly higher. Plateaus near **80–90 TFLOPs/s**.
- **Key Points**:
- At Hidden Size = 32,768: ~85 TFLOPs/s
### 3. h/a = 4 (Green Line)
- **Trend**: Steeper ascent than h/a = 2. Plateaus near **100–110 TFLOPs/s**.
- **Key Points**:
- At Hidden Size = 32,768: ~105 TFLOPs/s
### 4. h/a = 8 (Red Line)
- **Trend**: Sharp increase, surpassing h/a = 4. Plateaus near **120–130 TFLOPs/s**.
- **Key Points**:
- At Hidden Size = 32,768: ~125 TFLOPs/s
### 5. h/a = 16 (Purple Line)
- **Trend**: Outperforms h/a = 8. Plateaus near **140–150 TFLOPs/s**.
- **Key Points**:
- At Hidden Size = 32,768: ~145 TFLOPs/s
### 6. h/a = 32 (Brown Line)
- **Trend**: Highest-performing series. Exceeds h/a = 16, plateauing near **160 TFLOPs/s**.
- **Key Points**:
- At Hidden Size = 32,768: ~160 TFLOPs/s
### 7. h/a = 64 (Pink Line)
- **Trend**: Steepest ascent, achieving the highest throughput. Plateaus near **170 TFLOPs/s**.
- **Key Points**:
- At Hidden Size = 32,768: ~170 TFLOPs/s
---
## Cross-Referenced Observations
- **Color Consistency**: All lines match the legend (e.g., pink = h/a = 64, blue = h/a = 1).
- **Spatial Grounding**: The legend is positioned on the right, aligned with the y-axis.
- **Trend Verification**: Higher h/a ratios correlate with higher throughput, especially at larger Hidden Sizes.
---
## Limitations
- The graph does not provide **exact numerical values** for intermediate data points. Approximations are based on visual scaling.
- No textual annotations or additional context (e.g., experimental conditions) are present in the image.
---
## Conclusion
The graph demonstrates a clear trend: **increasing h/a ratios result in higher throughput**, particularly at larger Hidden Sizes. The h/a = 64 (pink) series achieves the highest throughput (~170 TFLOPs/s), while h/a = 1 (blue) is the lowest (~75 TFLOPs/s). This suggests that optimizing h/a ratios can significantly improve computational efficiency in attention-based models.
</details>
Figure 29: Attention key-query score GEMM throughput for 80 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a96.png Details</summary>
### Visual Description
# Technical Document: Analysis of Attention Key Query Score Graph
## Image Description
The image is a line graph titled **"Attention Key Query Score, a=96"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOP/s)** (y-axis). The graph includes seven data series, each represented by a distinct colored line, corresponding to different **h/a** (hidden size to attention head ratio) values. The legend is positioned on the right side of the graph.
---
## Key Components
### 1. **Title**
- **Text**: "Attention Key Query Score, a=96"
- **Purpose**: Indicates the metric being analyzed and the fixed parameter `a=96`.
### 2. **Axes Labels**
- **X-axis**:
- **Label**: "Hidden Size"
- **Values**: 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768
- **Units**: Not explicitly stated, but implied as numerical values.
- **Y-axis**:
- **Label**: "Throughput (TFLOP/s)"
- **Values**: 0, 50, 100, 150
- **Units**: TFLOP/s (Tera Floating Point Operations per second).
### 3. **Legend**
- **Position**: Right side of the graph.
- **Entries**:
- **Blue**: h/a = 1
- **Orange**: h/a = 2
- **Green**: h/a = 4
- **Red**: h/a = 8
- **Purple**: h/a = 16
- **Brown**: h/a = 32
- **Pink**: h/a = 64
- **Color Matching**: Each line color in the graph corresponds exactly to the legend entries.
---
## Data Series and Trends
### 1. **Line Colors and Corresponding h/a Values**
- **Blue (h/a=1)**:
- **Trend**: Starts at (0, 0) and increases gradually, reaching ~70 TFLOP/s at Hidden Size = 32768.
- **Slope**: Gentle upward curve.
- **Orange (h/a=2)**:
- **Trend**: Similar to blue but with a slightly steeper slope. Reaches ~90 TFLOP/s at Hidden Size = 32768.
- **Green (h/a=4)**:
- **Trend**: Steeper than orange. Reaches ~100 TFLOP/s at Hidden Size = 32768.
- **Red (h/a=8)**:
- **Trend**: Steeper than green. Reaches ~120 TFLOP/s at Hidden Size = 32768.
- **Purple (h/a=16)**:
- **Trend**: Steeper than red. Reaches ~130 TFLOP/s at Hidden Size = 32768.
- **Brown (h/a=32)**:
- **Trend**: Steeper than purple. Reaches ~140 TFLOP/s at Hidden Size = 32768.
- **Pink (h/a=64)**:
- **Trend**: Steepest slope. Reaches ~150 TFLOP/s at Hidden Size = 32768.
### 2. **Key Observations**
- **Positive Correlation**: All lines show a **positive linear trend**, indicating that **throughput increases with hidden size**.
- **h/a Ratio Impact**: Higher h/a ratios (e.g., 64) achieve **higher throughput** at the same hidden size compared to lower ratios (e.g., 1).
- **Scalability**: The graph suggests that **larger hidden sizes** (e.g., 32768) are more effective for higher h/a ratios, as the throughput plateaus at lower hidden sizes for smaller h/a values.
---
## Spatial Grounding
- **Legend Position**: Right side of the graph (x = 32768, y = 0–150).
- **Line Placement**:
- Lines are plotted from left to right, with the **pink line (h/a=64)** at the top and the **blue line (h/a=1)** at the bottom.
- All lines originate at (0, 0) and extend to the maximum hidden size (32768).
---
## Component Isolation
### 1. **Header**
- **Title**: "Attention Key Query Score, a=96"
- **Purpose**: Sets the context for the graph.
### 2. **Main Chart**
- **Axes**: X-axis (Hidden Size), Y-axis (Throughput).
- **Data Series**: Seven lines representing h/a ratios.
- **Trend**: All lines show increasing throughput with hidden size, with higher h/a ratios achieving higher throughput.
### 3. **Footer**
- **No additional text or components** in the footer.
---
## Data Table Reconstruction
The graph does not contain an explicit data table, but the following can be inferred from the trends:
| Hidden Size | h/a=1 | h/a=2 | h/a=4 | h/a=8 | h/a=16 | h/a=32 | h/a=64 |
|-------------|-------|-------|-------|-------|--------|--------|--------|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4096 | ~20 | ~30 | ~40 | ~50 | ~60 | ~70 | ~80 |
| 8192 | ~35 | ~50 | ~65 | ~80 | ~95 | ~110 | ~125 |
| 12288 | ~50 | ~70 | ~90 | ~110 | ~130 | ~145 | ~160 |
| 16384 | ~60 | ~85 | ~105 | ~130 | ~150 | ~165 | ~175 |
| 20480 | ~70 | ~95 | ~115 | ~140 | ~160 | ~175 | ~185 |
| 24576 | ~80 | ~110 | ~130 | ~155 | ~170 | ~185 | ~195 |
| 28672 | ~90 | ~125 | ~145 | ~170 | ~185 | ~195 | ~205 |
| 32768 | ~70 | ~90 | ~105 | ~120 | ~130 | ~140 | ~150 |
> **Note**: Values are approximate based on the visual slope of the lines.
---
## Conclusion
The graph demonstrates that **throughput increases with hidden size** and that **higher h/a ratios** (e.g., 64) achieve **higher throughput** at the same hidden size. The data suggests that **larger hidden sizes** are more effective for optimizing throughput, particularly for higher h/a configurations. The legend and color coding are consistent, ensuring accurate interpretation of the data series.
</details>
Figure 30: Attention key-query score GEMM throughput for 96 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a128.png Details</summary>
### Visual Description
# Technical Document Analysis of Line Graph
## 1. **Title and Axes Labels**
- **Title**: "Attention Key Query Score, a=128"
- **Y-Axis**: "Throughput (TFLOP/s)" (ranging from 0 to 150)
- **X-Axis**: "Hidden Size" (ranging from 0 to 32768)
## 2. **Legend and Color Mapping**
- **Legend Location**: Right side of the plot (outside the graph area).
- **Legend Entries**:
- **Blue**: h/a = 1
- **Orange**: h/a = 2
- **Green**: h/a = 4
- **Red**: h/a = 8
- **Purple**: h/a = 16
- **Brown**: h/a = 32
- **Pink**: h/a = 64
## 3. **Key Trends and Data Points**
### **Line Trends**
- All lines exhibit an **upward trend** as hidden size increases.
- **Higher h/a ratios** (e.g., 32, 64) show **steeper slopes** compared to lower ratios (e.g., 1, 2).
- **Lowest throughput**: h/a = 1 (blue line) remains the lowest across all hidden sizes.
- **Highest throughput**: h/a = 64 (pink line) is the steepest and reaches the highest values.
### **Approximate Data Points**
| Hidden Size | h/a = 1 (Blue) | h/a = 2 (Orange) | h/a = 4 (Green) | h/a = 8 (Red) | h/a = 16 (Purple) | h/a = 32 (Brown) | h/a = 64 (Pink) |
|-------------|----------------|------------------|-----------------|---------------|-------------------|------------------|-----------------|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4096 | ~10 | ~15 | ~20 | ~30 | ~40 | ~50 | ~60 |
| 8192 | ~25 | ~35 | ~45 | ~60 | ~75 | ~90 | ~110 |
| 12288 | ~35 | ~50 | ~65 | ~85 | ~105 | ~125 | ~140 |
| 16384 | ~45 | ~65 | ~80 | ~100 | ~120 | ~135 | ~150 |
| 20480 | ~55 | ~75 | ~90 | ~110 | ~130 | ~140 | ~155 |
| 24576 | ~65 | ~85 | ~95 | ~120 | ~140 | ~145 | ~158 |
| 28672 | ~70 | ~95 | ~100 | ~125 | ~145 | ~148 | ~159 |
| 32768 | ~75 | ~105 | ~110 | ~130 | ~150 | ~152 | ~160 |
## 4. **Spatial Grounding**
- **Legend Position**: Right side of the plot (outside the graph area).
- **Line Placement**:
- **Blue (h/a=1)**: Bottommost line.
- **Pink (h/a=64)**: Topmost line.
- All lines are plotted in ascending order of h/a ratio from bottom to top.
## 5. **Trend Verification**
- **h/a = 1 (Blue)**: Gradual, shallow slope. Throughput increases slowly.
- **h/a = 2 (Orange)**: Slightly steeper than h/a = 1.
- **h/a = 4 (Green)**: Moderate slope, surpassing h/a = 2.
- **h/a = 8 (Red)**: Steeper than h/a = 4, with a noticeable acceleration.
- **h/a = 16 (Purple)**: Rapid increase, surpassing h/a = 8.
- **h/a = 32 (Brown)**: Very steep, approaching the maximum y-axis value.
- **h/a = 64 (Pink)**: Steepest line, reaching the highest throughput values.
## 6. **Component Isolation**
- **Header**: Title "Attention Key Query Score, a=128".
- **Main Chart**:
- X-axis (Hidden Size) and Y-axis (Throughput) with labeled scales.
- Seven colored lines representing different h/a ratios.
- **Footer**: No additional text or components.
## 7. **Data Table Reconstruction**
The graph does not contain an explicit data table, but the trends and approximate values are derived from the line placements and axis scales. The table above summarizes the key data points.
## 8. **Language and Text Extraction**
- **Primary Language**: English (all labels, titles, and axis labels are in English).
- **No Other Languages**: No non-English text is present.
## 9. **Critical Notes**
- **Accuracy Check**: All legend colors match the corresponding lines in the graph.
- **No Missing Data**: All h/a ratios (1, 2, 4, 8, 16, 32, 64) are represented.
- **No Ambiguities**: The graph is clear, with no overlapping lines or conflicting trends.
## 10. **Conclusion**
The graph illustrates the relationship between **hidden size** and **throughput (TFLOP/s)** for different **h/a ratios**. Higher h/a ratios consistently yield higher throughput, with the steepest increase observed at h/a = 64. The data suggests a positive correlation between h/a ratio and computational efficiency in this context.
</details>
Figure 31: Attention key-query score GEMM throughput for 128 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a256.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Chart
## Chart Title
**Attention Key Query Score, a=256**
---
## Axis Labels
- **X-Axis (Horizontal):**
Label: `Hidden Size`
Range: `0` to `32768` (logarithmic scale)
Tick Markers: `0`, `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`
- **Y-Axis (Vertical):**
Label: `Throughput (TFLOPs/s)`
Range: `0` to `100`
Tick Markers: `0`, `50`, `100`
---
## Legend
- **Placement:** Right side of the chart
- **Color-Coded Labels:**
- `1` (Blue)
- `2` (Orange)
- `4` (Green)
- `8` (Red)
- `16` (Purple)
- `32` (Brown)
- `64` (Pink)
- `h / a` (Key for ratio interpretation)
---
## Line Trends and Data Points
1. **Blue Line (h/a = 1):**
- **Trend:** Gradual upward slope with minor fluctuations.
- **Data Points:**
- At `Hidden Size = 32768`: `~35 TFLOPs/s`
2. **Orange Line (h/a = 2):**
- **Trend:** Similar to blue line but with sharper fluctuations.
- **Data Points:**
- At `Hidden Size = 32768`: `~70 TFLOPs/s`
3. **Green Line (h/a = 4):**
- **Trend:** Starts flat, sharp rise at `Hidden Size = 20480`, then stabilizes.
- **Data Points:**
- At `Hidden Size = 32768`: `~75 TFLOPs/s`
4. **Red Line (h/a = 8):**
- **Trend:** Steady upward slope.
- **Data Points:**
- At `Hidden Size = 32768`: `~85 TFLOPs/s`
5. **Purple Line (h/a = 16):**
- **Trend:** Steeper slope than red line.
- **Data Points:**
- At `Hidden Size = 32768`: `~95 TFLOPs/s`
6. **Brown Line (h/a = 32):**
- **Trend:** Steepest slope, linear increase.
- **Data Points:**
- At `Hidden Size = 32768`: `~90 TFLOPs/s`
7. **Pink Line (h/a = 64):**
- **Trend:** Highest slope, exceeds y-axis maximum.
- **Data Points:**
- At `Hidden Size = 32768`: `~105 TFLOPs/s`
---
## Key Observations
- **Inverse Relationship:** Higher `h/a` ratios (e.g., 64) achieve significantly higher throughput than lower ratios (e.g., 1).
- **Scalability:** Throughput increases non-linearly with `Hidden Size`, especially for larger `h/a` values.
- **Anomalies:** Green line (`h/a = 4`) shows a sharp spike at `Hidden Size = 20480`, suggesting an optimization point.
---
## Notes
- All textual information extracted from the chart.
- No additional languages or non-textual elements present.
- Data points extrapolated from visual trends; exact values may require interpolation.
</details>
Figure 32: Attention key-query score GEMM throughput for 256 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_a512.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Score Chart
## Chart Title
**Attention Key Query Score, a=512**
## Axes
- **X-axis (Horizontal):**
- Label: **Hidden Size**
- Range: **0 to 32768**
- Tick Marks: **0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768**
- **Y-axis (Vertical):**
- Label: **Throughput (TFLOP/s)**
- Range: **0 to 50**
- Tick Marks: **0, 10, 20, 30, 40, 50**
## Legend
- **Location:** Top-right corner of the chart
- **Color-Coded Labels:**
- **Blue:** h/a = 1
- **Orange:** h/a = 2
- **Green:** h/a = 4
- **Red:** h/a = 8
- **Purple:** h/a = 16
- **Brown:** h/a = 32
- **Pink:** h/a = 64
## Data Series Trends
1. **h/a = 1 (Blue Line):**
- Starts at **0 TFLOP/s** at Hidden Size = 0.
- Gradually increases to **25 TFLOP/s** at Hidden Size = 32768.
- **Trend:** Steady upward slope with minor fluctuations.
2. **h/a = 2 (Orange Line):**
- Starts at **5 TFLOP/s** at Hidden Size = 4096.
- Peaks at **15 TFLOP/s** at Hidden Size = 16384.
- Drops to **20 TFLOP/s** at Hidden Size = 20480.
- **Trend:** Initial rise, peak, then decline.
3. **h/a = 4 (Green Line):**
- Starts at **0 TFLOP/s** at Hidden Size = 0.
- Peaks at **12 TFLOP/s** at Hidden Size = 16384.
- Increases to **18 TFLOP/s** at Hidden Size = 20480.
- **Trend:** Gradual rise with a peak at 16384.
4. **h/a = 8 (Red Line):**
- Starts at **10 TFLOP/s** at Hidden Size = 4096.
- Rises sharply to **50 TFLOP/s** at Hidden Size = 32768.
- **Trend:** Steep upward slope.
5. **h/a = 16 (Purple Line):**
- Starts at **15 TFLOP/s** at Hidden Size = 8192.
- Peaks at **45 TFLOP/s** at Hidden Size = 24576.
- **Trend:** Rapid increase, then plateaus.
6. **h/a = 32 (Brown Line):**
- Starts at **10 TFLOP/s** at Hidden Size = 8192.
- Peaks at **18 TFLOP/s** at Hidden Size = 16384.
- Increases to **22 TFLOP/s** at Hidden Size = 20480.
- **Trend:** Moderate rise with a peak at 16384.
7. **h/a = 64 (Pink Line):**
- Starts at **12 TFLOP/s** at Hidden Size = 8192.
- Peaks at **20 TFLOP/s** at Hidden Size = 16384.
- Increases to **25 TFLOP/s** at Hidden Size = 20480.
- **Trend:** Steady upward slope with a peak at 16384.
## Key Observations
- **h/a = 8 (Red Line)** achieves the highest throughput (50 TFLOP/s) at the largest hidden size (32768).
- **h/a = 16 (Purple Line)** shows the steepest increase, reaching 45 TFLOP/s at 24576.
- **h/a = 2 (Orange Line)** exhibits a non-monotonic trend, peaking at 16384 before declining.
- **h/a = 1 (Blue Line)** has the lowest throughput across all hidden sizes.
## Spatial Grounding
- **Legend Position:** Top-right corner (coordinates: [x=0.85, y=0.95] relative to chart boundaries).
- **Line Colors:** All lines match their legend labels (e.g., red = h/a=8, blue = h/a=1).
## Notes
- No additional text, tables, or diagrams are present.
- All data points and trends are derived from the chart's visual representation.
- No non-English text is detected.
</details>
Figure 33: Attention key-query score GEMM throughput for 512 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_key_query_problem_ha64.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention Key Query Problem
## Chart Title
**Attention Key Query Problem (h/a = 64)**
## Axes
- **X-axis**: Hidden Size (values: 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768)
- **Y-axis**: Throughput (TFLOP/s) (values: 50, 75, 100, 125, 150, 175, 200, 225)
## Legend
- **Location**: Bottom-right corner
- **Color-Coded Labels**:
- Blue: `a:24`
- Red: `a:64`
- Orange: `a:32`
- Purple: `a:80`
- Green: `a:40`
- Brown: `a:96`
## Data Series Analysis
### 1. Blue Line (`a:24`)
- **Trend**: Steep initial increase, then plateaus near 225 TFLOP/s
- **Key Points**:
- [0, 50] → [4096, 100] → [8192, 130] → [12288, 200] → [16384, 225] → [20480, 225] → [24576, 220] → [28672, 220] → [32768, 220]
### 2. Red Line (`a:64`)
- **Trend**: Gradual upward slope, ending near 210 TFLOP/s
- **Key Points**:
- [0, 50] → [4096, 55] → [8192, 105] → [12288, 135] → [16384, 150] → [20480, 170] → [24576, 190] → [28672, 200] → [32768, 210]
### 3. Orange Line (`a:32`)
- **Trend**: Steep rise, plateauing near 215 TFLOP/s
- **Key Points**:
- [0, 50] → [4096, 100] → [8192, 150] → [12288, 190] → [16384, 205] → [20480, 215] → [24576, 210] → [28672, 215] → [32768, 215]
### 4. Purple Line (`a:80`)
- **Trend**: Moderate upward slope, ending near 180 TFLOP/s
- **Key Points**:
- [0, 50] → [4096, 55] → [8192, 110] → [12288, 130] → [16384, 140] → [20480, 150] → [24576, 160] → [28672, 175] → [32768, 180]
### 5. Green Line (`a:40`)
- **Trend**: Steady increase, plateauing near 205 TFLOP/s
- **Key Points**:
- [0, 50] → [4096, 100] → [8192, 120] → [12288, 170] → [16384, 200] → [20480, 205] → [24576, 195] → [28672, 205] → [32768, 205]
### 6. Brown Line (`a:96`)
- **Trend**: Gradual rise, ending near 160 TFLOP/s
- **Key Points**:
- [0, 50] → [4096, 55] → [8192, 65] → [12288, 110] → [16384, 130] → [20480, 140] → [24576, 150] → [28672, 160] → [32768, 160]
## Spatial Grounding
- **Legend Position**: Bottom-right corner (confirmed via visual alignment with data series)
- **Color Consistency**: All legend colors match corresponding data series (e.g., blue = `a:24`, red = `a:64`).
## Critical Observations
1. **Performance Scaling**: Higher `a` values (e.g., `a:24`, `a:32`) achieve higher throughput at larger hidden sizes.
2. **Efficiency Tradeoff**: Larger `a` values (e.g., `a:96`) show diminishing returns despite increased computational cost.
3. **Optimal Range**: Hidden sizes between 16384 and 28672 yield peak performance for most `a` configurations.
## Notes
- The chart explicitly ties performance (TFLOP/s) to hidden size and `a` parameter configurations.
- No additional textual data or tables are present in the image.
</details>
Figure 34: Attention key-query score GEMM throughput assuming fixed ratio of $\frac{h}{a}=64$ .
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a8.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values, a=8
## Chart Overview
- **Title**: Attention over Values, a=8
- **Type**: Line graph
- **Purpose**: Visualizes throughput (TFLOPs/s) across varying hidden sizes for different h/a ratios.
---
## Axis Labels and Markers
- **X-axis (Hidden Size)**:
- Range: 0 to 32,768
- Markers: 0, 4,096, 8,192, 12,288, 16,384, 20,480, 24,576, 28,672, 32,768
- **Y-axis (Throughput (TFLOPs/s))**:
- Range: 0 to 250
- Markers: 0, 50, 100, 150, 200, 250
---
## Legend
- **Location**: Right of the plot area
- **Labels and Colors**:
- `h/a = 1` → Blue
- `h/a = 2` → Orange
- `h/a = 4` → Green
- `h/a = 8` → Red
- `h/a = 16` → Purple
- `h/a = 32` → Brown
- `h/a = 64` → Pink
---
## Key Trends and Data Points
### Line Analysis
1. **Blue (h/a=1)**:
- **Trend**: Gradual increase from 0 to ~100 TFLOPs/s, then plateaus.
- **Key Points**:
- At hidden size 0: 0 TFLOPs/s
- At hidden size 32,768: ~100 TFLOPs/s
2. **Orange (h/a=2)**:
- **Trend**: Similar to blue but with sharper initial rise; plateaus ~150 TFLOPs/s.
- **Key Points**:
- At hidden size 0: 0 TFLOPs/s
- At hidden size 32,768: ~150 TFLOPs/s
3. **Green (h/a=4)**:
- **Trend**: Slightly higher than orange; plateaus ~150 TFLOPs/s.
- **Key Points**:
- At hidden size 0: 0 TFLOPs/s
- At hidden size 32,768: ~150 TFLOPs/s
4. **Red (h/a=8)**:
- **Trend**: Peaks early (~200 TFLOPs/s at hidden size 4,096), then fluctuates downward.
- **Key Points**:
- At hidden size 0: 0 TFLOPs/s
- Peak: ~200 TFLOPs/s at hidden size 4,096
- At hidden size 32,768: ~220 TFLOPs/s
5. **Purple (h/a=16)**:
- **Trend**: Highest initial throughput (~250 TFLOPs/s), stabilizes with minor fluctuations.
- **Key Points**:
- At hidden size 0: 0 TFLOPs/s
- Peak: ~250 TFLOPs/s at hidden size 4,096
- At hidden size 32,768: ~240 TFLOPs/s
6. **Brown (h/a=32)**:
- **Trend**: Matches purple in initial peak (~250 TFLOPs/s), stabilizes.
- **Key Points**:
- At hidden size 0: 0 TFLOPs/s
- Peak: ~250 TFLOPs/s at hidden size 4,096
- At hidden size 32,768: ~240 TFLOPs/s
7. **Pink (h/a=64)**:
- **Trend**: Highest throughput (~250 TFLOPs/s), minimal fluctuations.
- **Key Points**:
- At hidden size 0: 0 TFLOPs/s
- Peak: ~250 TFLOPs/s at hidden size 4,096
- At hidden size 32,768: ~240 TFLOPs/s
---
## Spatial Grounding
- **Legend Position**: Right of the plot area (outside the chart boundaries).
- **Color Consistency**: All lines match their legend labels (e.g., blue = h/a=1, pink = h/a=64).
---
## Component Isolation
1. **Header**:
- Title: "Attention over Values, a=8"
- No additional text.
2. **Main Chart**:
- Axes, gridlines, and seven data series (lines).
3. **Footer**:
- Legend with seven entries (h/a ratios and colors).
---
## Observations
- **Throughput Scaling**: Higher h/a ratios (e.g., 64) achieve higher throughput but plateau earlier.
- **Hidden Size Impact**: Throughput increases with hidden size up to ~4,096, then stabilizes or fluctuates.
- **a=8 Context**: The parameter `a=8` is fixed; variations are driven by h/a ratios.
---
## Notes
- No data tables or embedded text blocks present.
- All textual information extracted from axis labels, legend, and title.
- No non-English text detected.
</details>
Figure 35: Attention over value GEMM throughput for 8 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a12.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values (a=12)
## Chart Overview
- **Title**: Attention over Values, a=12
- **Type**: Line graph
- **Purpose**: Visualizes throughput (TFLOPs/s) across varying hidden sizes for different h/a ratios
## Axes
- **X-axis (Horizontal)**:
- Label: `Hidden Size`
- Range: `0` to `32768`
- Tick Intervals: `0`, `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`
- **Y-axis (Vertical)**:
- Label: `Throughput (TFLOPs/s)`
- Range: `0` to `250`
- Tick Intervals: `0`, `50`, `100`, `150`, `200`, `250`
## Legend
- **Location**: Right side of the chart
- **Labels and Colors**:
- `h/a = 1` → Blue
- `h/a = 2` → Orange
- `h/a = 4` → Green
- `h/a = 8` → Red
- `h/a = 16` → Purple
- `h/a = 32` → Brown
- `h/a = 64` → Pink
## Data Series Analysis
1. **h/a = 1 (Blue)**:
- **Trend**: Gradual upward slope with minor fluctuations.
- **Key Points**:
- At `Hidden Size = 0`: ~0 TFLOPs/s
- At `Hidden Size = 32768`: ~90 TFLOPs/s
2. **h/a = 2 (Orange)**:
- **Trend**: Slightly higher than h/a=1, with similar growth pattern.
- **Key Points**:
- At `Hidden Size = 0`: ~0 TFLOPs/s
- At `Hidden Size = 32768`: ~140 TFLOPs/s
3. **h/a = 4 (Green)**:
- **Trend**: Moderate fluctuations but steeper growth than h/a=2.
- **Key Points**:
- At `Hidden Size = 0`: ~0 TFLOPs/s
- At `Hidden Size = 32768`: ~150 TFLOPs/s
4. **h/a = 8 (Red)**:
- **Trend**: Highest throughput, with sharp initial growth and sustained performance.
- **Key Points**:
- At `Hidden Size = 0`: ~0 TFLOPs/s
- At `Hidden Size = 32768`: ~230 TFLOPs/s
5. **h/a = 16 (Purple)**:
- **Trend**: Overlaps with h/a=32 and h/a=64; minimal fluctuations.
- **Key Points**:
- At `Hidden Size = 0`: ~0 TFLOPs/s
- At `Hidden Size = 32768`: ~240 TFLOPs/s
6. **h/a = 32 (Brown)**:
- **Trend**: Nearly identical to h/a=16 and h/a=64.
- **Key Points**:
- At `Hidden Size = 0`: ~0 TFLOPs/s
- At `Hidden Size = 32768`: ~240 TFLOPs/s
7. **h/a = 64 (Pink)**:
- **Trend**: Overlaps with h/a=16 and h/a=32; highest throughput.
- **Key Points**:
- At `Hidden Size = 0`: ~0 TFLOPs/s
- At `Hidden Size = 32768`: ~240 TFLOPs/s
## Key Observations
- **Diminishing Returns**: Higher h/a ratios (e.g., 16, 32, 64) achieve similar throughput (~240 TFLOPs/s) despite exponential increases in h/a.
- **Scalability**: Throughput increases with hidden size for all h/a ratios, but the rate of growth slows after `Hidden Size = 8192`.
- **Performance Threshold**: h/a=8 (red) achieves ~230 TFLOPs/s, while h/a=16/32/64 plateau near ~240 TFLOPs/s.
## Spatial Grounding
- **Legend Position**: Right-aligned, outside the plot area.
- **Color Consistency**: All lines match their legend labels (e.g., red = h/a=8).
## Language Notes
- **Primary Language**: English
- **No Additional Languages Detected**
## Conclusion
The chart demonstrates that higher h/a ratios correlate with increased throughput, but beyond h/a=8, performance gains plateau. This suggests an optimal h/a ratio for maximizing throughput in this context.
</details>
Figure 36: Attention over value GEMM throughput for 12 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a16.png Details</summary>
### Visual Description
# Technical Document Analysis: Line Graph of "Attention over Values, a=16"
## **1. Image Description**
The image is a **line graph** titled **"Attention over Values, a=16"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOPs/s)** (y-axis). The graph includes **seven distinct data series**, each represented by a colored line corresponding to different **h/a ratios** (1, 2, 4, 8, 16, 32, 64). The legend is positioned on the **right side** of the plot area.
---
## **2. Axis Labels and Markers**
- **X-axis (Hidden Size)**:
- Range: **0 to 32768**
- Tick marks at: **0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768**
- Labels: Numerical values (e.g., "0", "4096", etc.).
- **Y-axis (Throughput (TFLOPs/s))**:
- Range: **0 to 250**
- Tick marks at: **0, 50, 100, 150, 200, 250**
- Labels: Numerical values (e.g., "0", "50", etc.).
---
## **3. Legend and Data Series**
The legend maps **h/a ratios** to **line colors** as follows:
| **h/a Ratio** | **Color** | **Line Description** |
|---------------|-----------|----------------------|
| 1 | Blue | Lowest throughput, gradual increase. |
| 2 | Orange | Moderate throughput, moderate fluctuations. |
| 4 | Green | Higher throughput, moderate fluctuations. |
| 8 | Red | High throughput, sharp peaks and troughs. |
| 16 | Purple | Very high throughput, consistent peaks. |
| 32 | Brown | High throughput, smooth curve. |
| 64 | Pink | Highest throughput, jagged peaks. |
**Spatial Grounding**:
- The legend is located **outside the plot area** on the **right side**.
- Each line’s color **matches the legend entries exactly** (e.g., blue for h/a=1, pink for h/a=64).
---
## **4. Key Trends and Data Points**
### **Line 1 (h/a=1, Blue)**
- **Trend**: Starts at **0 TFLOPs/s**, gradually increases to a **peak of ~100 TFLOPs/s** around **Hidden Size = 8192**, then plateaus.
- **Data Points**:
- At **Hidden Size = 0**: 0 TFLOPs/s.
- At **Hidden Size = 4096**: ~60 TFLOPs/s.
- At **Hidden Size = 8192**: ~100 TFLOPs/s.
- At **Hidden Size = 32768**: ~100 TFLOPs/s.
### **Line 2 (h/a=2, Orange)**
- **Trend**: Starts at **0 TFLOPs/s**, rises to a **peak of ~150 TFLOPs/s** around **Hidden Size = 8192**, then fluctuates.
- **Data Points**:
- At **Hidden Size = 0**: 0 TFLOPs/s.
- At **Hidden Size = 4096**: ~100 TFLOPs/s.
- At **Hidden Size = 8192**: ~150 TFLOPs/s.
- At **Hidden Size = 32768**: ~150 TFLOPs/s.
### **Line 3 (h/a=4, Green)**
- **Trend**: Starts at **0 TFLOPs/s**, rises to a **peak of ~150 TFLOPs/s** around **Hidden Size = 8192**, then fluctuates.
- **Data Points**:
- At **Hidden Size = 0**: 0 TFLOPs/s.
- At **Hidden Size = 4096**: ~120 TFLOPs/s.
- At **Hidden Size = 8192**: ~150 TFLOPs/s.
- At **Hidden Size = 32768**: ~150 TFLOPs/s.
### **Line 4 (h/a=8, Red)**
- **Trend**: Starts at **0 TFLOPs/s**, rises to a **peak of ~200 TFLOPs/s** around **Hidden Size = 8192**, then fluctuates.
- **Data Points**:
- At **Hidden Size = 0**: 0 TFLOPs/s.
- At **Hidden Size = 4096**: ~140 TFLOPs/s.
- At **Hidden Size = 8192**: ~200 TFLOPs/s.
- At **Hidden Size = 32768**: ~200 TFLOPs/s.
### **Line 5 (h/a=16, Purple)**
- **Trend**: Starts at **0 TFLOPs/s**, rises to a **peak of ~230 TFLOPs/s** around **Hidden Size = 8192**, then fluctuates.
- **Data Points**:
- At **Hidden Size = 0**: 0 TFLOPs/s.
- At **Hidden Size = 4096**: ~160 TFLOPs/s.
- At **Hidden Size = 8192**: ~230 TFLOPs/s.
- At **Hidden Size = 32768**: ~230 TFLOPs/s.
### **Line 6 (h/a=32, Brown)**
- **Trend**: Starts at **0 TFLOPs/s**, rises to a **peak of ~230 TFLOPs/s** around **Hidden Size = 8192**, then fluctuates.
- **Data Points**:
- At **Hidden Size = 0**: 0 TFLOPs/s.
- At **Hidden Size = 4096**: ~180 TFLOPs/s.
- At **Hidden Size = 8192**: ~230 TFLOPs/s.
- At **Hidden Size = 32768**: ~230 TFLOPs/s.
### **Line 7 (h/a=64, Pink)**
- **Trend**: Starts at **0 TFLOPs/s**, rises to a **peak of ~240 TFLOPs/s** around **Hidden Size = 8192**, then fluctuates.
- **Data Points**:
- At **Hidden Size = 0**: 0 TFLOPs/s.
- At **Hidden Size = 4096**: ~200 TFLOPs/s.
- At **Hidden Size = 8192**: ~240 TFLOPs/s.
- At **Hidden Size = 32768**: ~240 TFLOPs/s.
---
## **5. Observations**
- **Higher h/a ratios** (e.g., 16, 32, 64) exhibit **higher throughput** and **more consistent performance** compared to lower ratios.
- **Lower h/a ratios** (e.g., 1, 2, 4) show **lower throughput** and **greater variability** in performance.
- The **peak throughput** for all lines occurs near **Hidden Size = 8192**, after which performance stabilizes or declines slightly.
---
## **6. Additional Notes**
- The graph does not include a **data table** or **embedded text** beyond the axis labels and legend.
- The **x-axis** and **y-axis** are clearly labeled, and all numerical values are explicitly marked.
- The **legend** is fully legible and spatially consistent with the data series.
---
## **7. Conclusion**
The graph demonstrates a **positive correlation** between **h/a ratio** and **throughput**, with higher ratios achieving greater computational efficiency. The **Hidden Size** parameter influences performance, with optimal results observed at **Hidden Size = 8192** for most h/a ratios.
</details>
Figure 37: Attention over value GEMM throughput for 16 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a20.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values, a=20
## Chart Overview
- **Title**: Attention over Values, a=20
- **Type**: Line chart
- **Purpose**: Visualizes throughput (TFLOP/s) across varying hidden sizes for different h/a ratios.
## Axes
- **X-axis (Horizontal)**:
- **Label**: Hidden Size
- **Range**: 0 to 32,768
- **Markers**: Incremental ticks at 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768.
- **Y-axis (Vertical)**:
- **Label**: Throughput (TFLOP/s)
- **Range**: 0 to 250
- **Markers**: Incremental ticks at 0, 50, 100, 150, 200, 250.
## Legend
- **Location**: Right side of the chart.
- **Labels and Colors**:
- `h/a = 1` → Blue
- `h/a = 2` → Orange
- `h/a = 4` → Green
- `h/a = 8` → Red
- `h/a = 16` → Purple
- `h/a = 32` → Brown
- `h/a = 64` → Pink
## Data Series Analysis
### 1. h/a = 1 (Blue)
- **Trend**: Gradual upward slope with minimal fluctuations.
- **Key Points**:
- Starts at ~0 TFLOP/s at Hidden Size = 0.
- Reaches ~90 TFLOP/s at Hidden Size = 32,768.
### 2. h/a = 2 (Orange)
- **Trend**: Steeper initial rise, then stabilizes with minor oscillations.
- **Key Points**:
- Peaks at ~140 TFLOP/s near Hidden Size = 8,192.
- Ends at ~140 TFLOP/s at Hidden Size = 32,768.
### 3. h/a = 4 (Green)
- **Trend**: Higher baseline than h/a=2, with moderate fluctuations.
- **Key Points**:
- Peaks at ~150 TFLOP/s near Hidden Size = 16,384.
- Ends at ~145 TFLOP/s at Hidden Size = 32,768.
### 4. h/a = 8 (Red)
- **Trend**: Sharp initial rise, followed by volatility and gradual decline.
- **Key Points**:
- Peaks at ~230 TFLOP/s near Hidden Size = 12,288.
- Ends at ~200 TFLOP/s at Hidden Size = 32,768.
### 5. h/a = 16 (Purple)
- **Trend**: High initial throughput, sustained with oscillations.
- **Key Points**:
- Peaks at ~220 TFLOP/s near Hidden Size = 8,192.
- Ends at ~210 TFLOP/s at Hidden Size = 32,768.
### 6. h/a = 32 (Brown)
- **Trend**: Smooth rise with minor fluctuations.
- **Key Points**:
- Peaks at ~210 TFLOP/s near Hidden Size = 16,384.
- Ends at ~210 TFLOP/s at Hidden Size = 32,768.
### 7. h/a = 64 (Pink)
- **Trend**: Highest initial throughput, with significant volatility.
- **Key Points**:
- Peaks at ~240 TFLOP/s near Hidden Size = 4,096.
- Ends at ~220 TFLOP/s at Hidden Size = 32,768.
## Spatial Grounding
- **Legend Placement**: Right-aligned, outside the chart boundary.
- **Color Consistency**: All line colors match legend labels (e.g., blue = h/a=1, pink = h/a=64).
## Key Observations
1. **Throughput vs. h/a**: Higher h/a ratios generally achieve higher throughput but exhibit greater variability.
2. **Hidden Size Impact**: Throughput increases with hidden size up to a point, after which it plateaus or declines slightly.
3. **Volatility**: Larger h/a values (e.g., 64, 32) show more pronounced oscillations compared to smaller ratios (e.g., 1, 2).
## Data Table Reconstruction
| Hidden Size | h/a=1 | h/a=2 | h/a=4 | h/a=8 | h/a=16 | h/a=32 | h/a=64 |
|-------------|-------|-------|-------|-------|--------|--------|--------|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4096 | ~60 | ~90 | ~110 | ~130 | ~150 | ~170 | ~190 |
| 8192 | ~70 | ~140 | ~135 | ~180 | ~200 | ~195 | ~220 |
| 12288 | ~80 | ~130 | ~140 | ~210 | ~215 | ~205 | ~230 |
| 16384 | ~85 | ~135 | ~145 | ~200 | ~210 | ~205 | ~225 |
| 20480 | ~88 | ~140 | ~142 | ~210 | ~205 | ~208 | ~215 |
| 24576 | ~90 | ~145 | ~148 | ~220 | ~210 | ~212 | ~220 |
| 28672 | ~92 | ~150 | ~150 | ~215 | ~218 | ~215 | ~225 |
| 32768 | ~90 | ~145 | ~145 | ~200 | ~210 | ~210 | ~220 |
## Notes
- All numerical values are approximations based on visual inspection of the chart.
- No additional textual or diagrammatic components are present.
</details>
Figure 38: Attention over value GEMM throughput for 20 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a24.png Details</summary>
### Visual Description
# Technical Document Analysis: Attention over Values, a=24
## Image Description
The image is a line graph titled **"Attention over Values, a=24"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOPs/s)** (y-axis). The graph includes multiple data series, each represented by a distinct colored line, corresponding to different **h/a** ratios. The legend is positioned on the right side of the graph, with color-coded labels for each h/a value.
---
## Key Components
### 1. **Axis Labels and Ranges**
- **X-axis (Hidden Size)**:
- Range: 0 to 32768
- Tick marks at: 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768
- **Y-axis (Throughput (TFLOPs/s))**:
- Range: 0 to 250
- Tick marks at: 0, 50, 100, 150, 200, 250
### 2. **Legend**
- **Position**: Right side of the graph
- **Labels and Colors**:
- **Blue**: h/a = 1
- **Orange**: h/a = 2
- **Green**: h/a = 4
- **Red**: h/a = 8
- **Purple**: h/a = 16
- **Brown**: h/a = 32
- **Pink**: h/a = 64
---
## Data Series Analysis
### 1. **h/a = 1 (Blue Line)**
- **Trend**: Gradual upward slope with minor fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~50 TFLOPs/s
- At Hidden Size = 8192: ~70 TFLOPs/s
- At Hidden Size = 12288: ~80 TFLOPs/s
- At Hidden Size = 16384: ~90 TFLOPs/s
- At Hidden Size = 20480: ~95 TFLOPs/s
- At Hidden Size = 24576: ~98 TFLOPs/s
- At Hidden Size = 28672: ~99 TFLOPs/s
- At Hidden Size = 32768: ~100 TFLOPs/s
### 2. **h/a = 2 (Orange Line)**
- **Trend**: Similar to h/a = 1 but with more pronounced fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~60 TFLOPs/s
- At Hidden Size = 8192: ~80 TFLOPs/s
- At Hidden Size = 12288: ~90 TFLOPs/s
- At Hidden Size = 16384: ~100 TFLOPs/s
- At Hidden Size = 20480: ~105 TFLOPs/s
- At Hidden Size = 24576: ~110 TFLOPs/s
- At Hidden Size = 28672: ~115 TFLOPs/s
- At Hidden Size = 32768: ~120 TFLOPs/s
### 3. **h/a = 4 (Green Line)**
- **Trend**: Steady increase with occasional dips.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~70 TFLOPs/s
- At Hidden Size = 8192: ~90 TFLOPs/s
- At Hidden Size = 12288: ~100 TFLOPs/s
- At Hidden Size = 16384: ~110 TFLOPs/s
- At Hidden Size = 20480: ~115 TFLOPs/s
- At Hidden Size = 24576: ~120 TFLOPs/s
- At Hidden Size = 28672: ~125 TFLOPs/s
- At Hidden Size = 32768: ~130 TFLOPs/s
### 4. **h/a = 8 (Red Line)**
- **Trend**: Sharp peak at Hidden Size = 16384, followed by a decline.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~80 TFLOPs/s
- At Hidden Size = 8192: ~100 TFLOPs/s
- At Hidden Size = 12288: ~120 TFLOPs/s
- At Hidden Size = 16384: ~140 TFLOPs/s
- At Hidden Size = 20480: ~120 TFLOPs/s
- At Hidden Size = 24576: ~110 TFLOPs/s
- At Hidden Size = 28672: ~105 TFLOPs/s
- At Hidden Size = 32768: ~100 TFLOPs/s
### 5. **h/a = 16 (Purple Line)**
- **Trend**: High peak at Hidden Size = 24576, then a decline.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~90 TFLOPs/s
- At Hidden Size = 8192: ~110 TFLOPs/s
- At Hidden Size = 12288: ~130 TFLOPs/s
- At Hidden Size = 16384: ~150 TFLOPs/s
- At Hidden Size = 20480: ~140 TFLOPs/s
- At Hidden Size = 24576: ~160 TFLOPs/s
- At Hidden Size = 28672: ~145 TFLOPs/s
- At Hidden Size = 32768: ~140 TFLOPs/s
### 6. **h/a = 32 (Brown Line)**
- **Trend**: Highest peak at Hidden Size = 28672, then a decline.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~100 TFLOPs/s
- At Hidden Size = 8192: ~120 TFLOPs/s
- At Hidden Size = 12288: ~140 TFLOPs/s
- At Hidden Size = 16384: ~160 TFLOPs/s
- At Hidden Size = 20480: ~150 TFLOPs/s
- At Hidden Size = 24576: ~170 TFLOPs/s
- At Hidden Size = 28672: ~190 TFLOPs/s
- At Hidden Size = 32768: ~170 TFLOPs/s
### 7. **h/a = 64 (Pink Line)**
- **Trend**: Highest throughput overall, peaking at Hidden Size = 32768.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~110 TFLOPs/s
- At Hidden Size = 8192: ~130 TFLOPs/s
- At Hidden Size = 12288: ~150 TFLOPs/s
- At Hidden Size = 16384: ~170 TFLOPs/s
- At Hidden Size = 20480: ~180 TFLOPs/s
- At Hidden Size = 24576: ~200 TFLOPs/s
- At Hidden Size = 28672: ~210 TFLOPs/s
- At Hidden Size = 32768: ~220 TFLOPs/s
---
## Observations
1. **h/a Ratio Impact**: Higher h/a ratios (e.g., 64) generally achieve higher throughput, especially at larger hidden sizes.
2. **Peak Performance**:
- h/a = 8 peaks at Hidden Size = 16384.
- h/a = 16 peaks at Hidden Size = 24576.
- h/a = 32 peaks at Hidden Size = 28672.
- h/a = 64 peaks at Hidden Size = 32768.
3. **Fluctuations**: Lines with higher h/a ratios (e.g., 16, 32, 64) exhibit more pronounced fluctuations, suggesting variability in performance under different conditions.
---
## Notes
- **Language**: All text is in English. No other languages are present.
- **Data Accuracy**: All data points and trends are cross-referenced with the legend and visual trends to ensure consistency.
- **Legend Placement**: The legend is located on the right side of the graph, as specified in the image description.
---
## Conclusion
The graph illustrates how **Throughput (TFLOPs/s)** varies with **Hidden Size** for different **h/a** ratios. Higher h/a values generally correlate with increased throughput, particularly at larger hidden sizes. The data points and trends are consistent with the visual representation, confirming the accuracy of the extracted information.
</details>
Figure 39: Attention over value GEMM throughput for 24 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a32.png Details</summary>
### Visual Description
# Technical Document Extraction: Line Graph Analysis
## Image Description
The image is a line graph titled **"Attention over Values, a=32"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOPs/s)** (y-axis). The graph includes six data series, each represented by a distinct colored line, corresponding to different **h/a** ratios. The legend is positioned on the right side of the graph.
---
## Key Components
### 1. **Title**
- **Text**: "Attention over Values, a=32"
- **Purpose**: Indicates the focus of the analysis (attention mechanisms) and a fixed parameter value (a=32).
### 2. **Axes**
- **X-axis (Hidden Size)**:
- **Label**: "Hidden Size"
- **Range**: 0 to 32768
- **Tick Marks**: 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768
- **Y-axis (Throughput)**:
- **Label**: "Throughput (TFLOPs/s)"
- **Range**: 0 to 200
- **Tick Marks**: 0, 50, 100, 150, 200
### 3. **Legend**
- **Location**: Right side of the graph
- **Entries**:
- **Blue**: h/a = 1
- **Orange**: h/a = 2
- **Green**: h/a = 4
- **Red**: h/a = 8
- **Purple**: h/a = 16
- **Pink**: h/a = 64
---
## Data Series and Trends
### 1. **h/a = 1 (Blue Line)**
- **Trend**: Starts at 0 and increases steadily with minor fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~50 TFLOPs/s
- At Hidden Size = 8192: ~70 TFLOPs/s
- At Hidden Size = 12288: ~80 TFLOPs/s
- At Hidden Size = 16384: ~90 TFLOPs/s
- At Hidden Size = 20480: ~95 TFLOPs/s
- At Hidden Size = 24576: ~100 TFLOPs/s
- At Hidden Size = 28672: ~110 TFLOPs/s
- At Hidden Size = 32768: ~120 TFLOPs/s
### 2. **h/a = 2 (Orange Line)**
- **Trend**: Similar to h/a = 1 but with slightly higher throughput and minor fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~60 TFLOPs/s
- At Hidden Size = 8192: ~90 TFLOPs/s
- At Hidden Size = 12288: ~110 TFLOPs/s
- At Hidden Size = 16384: ~120 TFLOPs/s
- At Hidden Size = 20480: ~130 TFLOPs/s
- At Hidden Size = 24576: ~140 TFLOPs/s
- At Hidden Size = 28672: ~150 TFLOPs/s
- At Hidden Size = 32768: ~160 TFLOPs/s
### 3. **h/a = 4 (Green Line)**
- **Trend**: Higher throughput than h/a = 2, with more pronounced fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~80 TFLOPs/s
- At Hidden Size = 8192: ~120 TFLOPs/s
- At Hidden Size = 12288: ~140 TFLOPs/s
- At Hidden Size = 16384: ~150 TFLOPs/s
- At Hidden Size = 20480: ~160 TFLOPs/s
- At Hidden Size = 24576: ~170 TFLOPs/s
- At Hidden Size = 28672: ~180 TFLOPs/s
- At Hidden Size = 32768: ~190 TFLOPs/s
### 4. **h/a = 8 (Red Line)**
- **Trend**: Higher throughput than h/a = 4, with significant peaks and troughs.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~100 TFLOPs/s
- At Hidden Size = 8192: ~150 TFLOPs/s
- At Hidden Size = 12288: ~180 TFLOPs/s
- At Hidden Size = 16384: ~190 TFLOPs/s
- At Hidden Size = 20480: ~200 TFLOPs/s
- At Hidden Size = 24576: ~210 TFLOPs/s
- At Hidden Size = 28672: ~220 TFLOPs/s
- At Hidden Size = 32768: ~230 TFLOPs/s
### 5. **h/a = 16 (Purple Line)**
- **Trend**: Highest throughput among all series, with sharp peaks and troughs.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~120 TFLOPs/s
- At Hidden Size = 8192: ~170 TFLOPs/s
- At Hidden Size = 12288: ~200 TFLOPs/s
- At Hidden Size = 16384: ~210 TFLOPs/s
- At Hidden Size = 20480: ~220 TFLOPs/s
- At Hidden Size = 24576: ~230 TFLOPs/s
- At Hidden Size = 28672: ~240 TFLOPs/s
- At Hidden Size = 32768: ~250 TFLOPs/s
### 6. **h/a = 64 (Pink Line)**
- **Trend**: Highest throughput with the most pronounced fluctuations.
- **Key Data Points**:
- At Hidden Size = 0: 0 TFLOPs/s
- At Hidden Size = 4096: ~140 TFLOPs/s
- At Hidden Size = 8192: ~200 TFLOPs/s
- At Hidden Size = 12288: ~220 TFLOPs/s
- At Hidden Size = 16384: ~230 TFLOPs/s
- At Hidden Size = 20480: ~240 TFLOPs/s
- At Hidden Size = 24576: ~250 TFLOPs/s
- At Hidden Size = 28672: ~260 TFLOPs/s
- At Hidden Size = 32768: ~270 TFLOPs/s
---
## Observations
1. **Throughput Correlation**: Higher **h/a** ratios (e.g., 64) generally correspond to higher throughput, though with increased variability.
2. **Fluctuations**: Lines with higher **h/a** values (e.g., 16, 64) exhibit more pronounced peaks and troughs compared to lower ratios (e.g., 1, 2).
3. **Consistency**: All lines start at 0 TFLOPs/s and show a general upward trend as Hidden Size increases.
---
## Notes
- **Language**: All text in the image is in English.
- **Data Integrity**: Legend colors and line placements were cross-verified for accuracy.
- **Spatial Grounding**: The legend is positioned on the right side of the graph, ensuring clear association with the data series.
---
## Conclusion
The graph demonstrates that increasing the **h/a** ratio correlates with higher throughput, though with varying degrees of stability. The **h/a = 64** (pink line) achieves the highest throughput but with the most fluctuations, while **h/a = 1** (blue line) shows the most stable but lowest performance.
</details>
Figure 40: Attention over value GEMM throughput for 32 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a40.png Details</summary>
### Visual Description
# Technical Document Analysis: Attention over Values, a=40
## Chart Description
This image is a **line graph** titled **"Attention over Values, a=40"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOPs/s)** (y-axis) for different **h/a** ratios. The graph includes seven data series, each represented by a distinct color and labeled in the legend.
---
## Axis Labels and Markers
- **X-axis (Hidden Size)**:
- Range: `0` to `32768`
- Tick marks: `0`, `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`
- Units: Not explicitly labeled, but implied as numerical values.
- **Y-axis (Throughput (TFLOPs/s))**:
- Range: `0` to `200`
- Tick marks: `0`, `50`, `100`, `150`, `200`
- Units: **TFLOPs/s** (Terabytes per second).
---
## Legend
The legend is located on the **right side** of the graph and maps colors to **h/a** ratios:
| Color | h/a Ratio |
|-------------|-----------|
| Blue | 1 |
| Orange | 2 |
| Green | 4 |
| Red | 8 |
| Purple | 16 |
| Brown | 32 |
| Pink | 64 |
**Spatial Grounding**:
- The legend is positioned **vertically** on the right, with each color aligned to its corresponding h/a ratio.
- Colors match the lines in the graph exactly (e.g., blue line = h/a=1, pink line = h/a=64).
---
## Data Series and Trends
### 1. **h/a=1 (Blue Line)**
- **Trend**: Starts at `0` and increases gradually, reaching approximately `80 TFLOPs/s` at `Hidden Size = 32768`.
- **Key Points**:
- At `Hidden Size = 4096`: ~30 TFLOPs/s
- At `Hidden Size = 8192`: ~50 TFLOPs/s
- At `Hidden Size = 16384`: ~70 TFLOPs/s
- At `Hidden Size = 32768`: ~80 TFLOPs/s
### 2. **h/a=2 (Orange Line)**
- **Trend**: Similar to h/a=1 but with minor fluctuations. Peaks at ~120 TFLOPs/s at `Hidden Size = 32768`.
- **Key Points**:
- At `Hidden Size = 4096`: ~40 TFLOPs/s
- At `Hidden Size = 8192`: ~70 TFLOPs/s
- At `Hidden Size = 16384`: ~100 TFLOPs/s
- At `Hidden Size = 32768`: ~120 TFLOPs/s
### 3. **h/a=4 (Green Line)**
- **Trend**: Steady increase with minor dips. Reaches ~140 TFLOPs/s at `Hidden Size = 32768`.
- **Key Points**:
- At `Hidden Size = 4096`: ~50 TFLOPs/s
- At `Hidden Size = 8192`: ~90 TFLOPs/s
- At `Hidden Size = 16384`: ~120 TFLOPs/s
- At `Hidden Size = 32768`: ~140 TFLOPs/s
### 4. **h/a=8 (Red Line)**
- **Trend**: Sharp increase with peaks. Peaks at ~180 TFLOPs/s at `Hidden Size = 32768`.
- **Key Points**:
- At `Hidden Size = 4096`: ~60 TFLOPs/s
- At `Hidden Size = 8192`: ~110 TFLOPs/s
- At `Hidden Size = 16384`: ~150 TFLOPs/s
- At `Hidden Size = 32768`: ~180 TFLOPs/s
### 5. **h/a=16 (Purple Line)**
- **Trend**: Rapid rise followed by stabilization. Peaks at ~200 TFLOPs/s at `Hidden Size = 32768`.
- **Key Points**:
- At `Hidden Size = 4096`: ~70 TFLOPs/s
- At `Hidden Size = 8192`: ~130 TFLOPs/s
- At `Hidden Size = 16384`: ~180 TFLOPs/s
- At `Hidden Size = 32768`: ~200 TFLOPs/s
### 6. **h/a=32 (Brown Line)**
- **Trend**: Steep increase with fluctuations. Peaks at ~210 TFLOPs/s at `Hidden Size = 32768`.
- **Key Points**:
- At `Hidden Size = 4096`: ~80 TFLOPs/s
- At `Hidden Size = 8192`: ~140 TFLOPs/s
- At `Hidden Size = 16384`: ~190 TFLOPs/s
- At `Hidden Size = 32768`: ~210 TFLOPs/s
### 7. **h/a=64 (Pink Line)**
- **Trend**: Highest peaks, reaching ~220 TFLOPs/s at `Hidden Size = 32768`.
- **Key Points**:
- At `Hidden Size = 4096`: ~90 TFLOPs/s
- At `Hidden Size = 8192`: ~150 TFLOPs/s
- At `Hidden Size = 16384`: ~200 TFLOPs/s
- At `Hidden Size = 32768`: ~220 TFLOPs/s
---
## Observations
1. **Inverse Relationship**: As **h/a** increases, **Throughput (TFLOPs/s)** generally increases, indicating higher computational efficiency for larger h/a ratios.
2. **Fluctuations**: Lines for h/a=8, 16, 32, and 64 show irregular peaks and dips, suggesting variability in performance at specific Hidden Sizes.
3. **Saturation**: The pink line (h/a=64) achieves the highest throughput but shows a slight decline after `Hidden Size = 28672`, possibly indicating diminishing returns.
---
## Notes
- **Language**: All text in the image is in **English**.
- **Data Table**: No explicit data table is present; values are inferred from the graph.
- **Missing Information**: No additional textual annotations or footnotes are visible.
This analysis is based on the visual data and legend provided in the image. Numerical values are approximate and derived from the graph's scale.
</details>
Figure 41: Attention over value GEMM throughput for 40 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a64.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values (a=64)
## Chart Overview
- **Title**: Attention over Values, a=64
- **Type**: Line graph
- **Purpose**: Visualizes throughput (TFLOP/s) across varying hidden sizes for different h/a ratios.
## Axes
- **X-axis (Hidden Size)**:
- Range: 0 to 32768
- Markers: 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768
- **Y-axis (Throughput (TFLOP/s))**:
- Range: 0 to 200
- Markers: 0, 50, 100, 150, 200
## Legend
- **Location**: Right side of the plot (outside the graph area)
- **Labels and Colors**:
- `h/a = 1` (blue)
- `h/a = 2` (orange)
- `h/a = 4` (green)
- `h/a = 8` (red)
- `h/a = 16` (purple)
- `h/a = 32` (brown)
- `h/a = 64` (pink)
## Data Series Analysis
### 1. h/a = 1 (Blue)
- **Trend**: Steady upward slope with minor fluctuations.
- **Key Points**:
- At 0: 0 TFLOP/s
- At 16384: ~100 TFLOP/s
- At 32768: ~100 TFLOP/s
### 2. h/a = 2 (Orange)
- **Trend**: Gradual increase with a dip at 8192 (~90 TFLOP/s).
- **Key Points**:
- At 0: 0 TFLOP/s
- At 8192: ~90 TFLOP/s
- At 32768: ~150 TFLOP/s
### 3. h/a = 4 (Green)
- **Trend**: Sharp rise, peak at 16384 (~150 TFLOP/s), then decline.
- **Key Points**:
- At 0: 0 TFLOP/s
- At 16384: ~150 TFLOP/s
- At 32768: ~150 TFLOP/s
### 4. h/a = 8 (Red)
- **Trend**: Rapid ascent, peak at 16384 (~200 TFLOP/s), then decline.
- **Key Points**:
- At 0: 0 TFLOP/s
- At 16384: ~200 TFLOP/s
- At 32768: ~200 TFLOP/s
### 5. h/a = 16 (Purple)
- **Trend**: Steep rise, peak at 16384 (~200 TFLOP/s), then gradual decline.
- **Key Points**:
- At 0: 0 TFLOP/s
- At 16384: ~200 TFLOP/s
- At 32768: ~200 TFLOP/s
### 6. h/a = 32 (Brown)
- **Trend**: Sharp peak at 16384 (~200 TFLOP/s), then steep decline.
- **Key Points**:
- At 0: 0 TFLOP/s
- At 16384: ~200 TFLOP/s
- At 32768: ~150 TFLOP/s
### 7. h/a = 64 (Pink)
- **Trend**: Steep rise, peak at 16384 (~200 TFLOP/s), then gradual increase.
- **Key Points**:
- At 0: 0 TFLOP/s
- At 16384: ~200 TFLOP/s
- At 32768: ~220 TFLOP/s
## Critical Observations
1. **Peak at 16384**: All lines except `h/a = 1` reach their maximum throughput at 16384 hidden size.
2. **Divergence at 32768**:
- `h/a = 64` (pink) surpasses others, reaching ~220 TFLOP/s.
- `h/a = 32` (brown) declines sharply to ~150 TFLOP/s.
3. **Stability**: `h/a = 1` (blue) shows the most consistent growth without peaks/dips.
## Spatial Grounding
- **Legend Position**: Right-aligned, outside the plot boundary.
- **Color Consistency**: All lines match their legend labels (e.g., blue = `h/a = 1`).
## Conclusion
The graph demonstrates that higher `h/a` ratios (e.g., 64) achieve higher throughput at larger hidden sizes, with `h/a = 64` outperforming others at 32768. Lower ratios (e.g., 1, 2) show more gradual growth. Peaks at 16384 suggest an optimal hidden size for most configurations.
</details>
Figure 42: Attention over value GEMM throughput for 64 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a80.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values, a=80
## Chart Description
This image is a **line chart** titled **"Attention over Values, a=80"**. It visualizes the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOP/s)** (y-axis) for different **h/a** ratios. The chart includes seven data series, each represented by a distinct color and labeled in the legend.
---
### Axis Labels and Ranges
- **X-axis (Hidden Size)**:
- Range: `0` to `32768`
- Tick marks at: `0`, `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`
- **Y-axis (Throughput (TFLOP/s))**:
- Range: `0` to `200`
- Tick marks at: `0`, `50`, `100`, `150`, `200`
---
### Legend
The legend is positioned on the **right side** of the chart. It maps **h/a ratios** to colors:
| Color | h/a Ratio |
|-------------|-----------|
| Blue | 1 |
| Orange | 2 |
| Green | 4 |
| Red | 8 |
| Purple | 16 |
| Brown | 32 |
| Pink | 64 |
---
### Data Series Analysis
#### 1. **h/a = 1 (Blue Line)**
- **Trend**: Gradual upward slope with minor fluctuations.
- **Key Points**:
- At `Hidden Size = 0`: Throughput ≈ `0` TFLOP/s.
- At `Hidden Size = 32768`: Throughput ≈ `75` TFLOP/s.
#### 2. **h/a = 2 (Orange Line)**
- **Trend**: Similar to h/a=1 but with sharper increases and dips.
- **Key Points**:
- At `Hidden Size = 0`: Throughput ≈ `0` TFLOP/s.
- At `Hidden Size = 32768`: Throughput ≈ `125` TFLOP/s.
#### 3. **h/a = 4 (Green Line)**
- **Trend**: Steeper growth with a peak at `Hidden Size = 16384`, followed by a decline.
- **Key Points**:
- At `Hidden Size = 0`: Throughput ≈ `0` TFLOP/s.
- Peak at `Hidden Size = 16384`: Throughput ≈ `150` TFLOP/s.
- At `Hidden Size = 32768`: Throughput ≈ `120` TFLOP/s.
#### 4. **h/a = 8 (Red Line)**
- **Trend**: Sharp peak at `Hidden Size = 20480`, then a decline.
- **Key Points**:
- At `Hidden Size = 0`: Throughput ≈ `0` TFLOP/s.
- Peak at `Hidden Size = 20480`: Throughput ≈ `210` TFLOP/s.
- At `Hidden Size = 32768`: Throughput ≈ `180` TFLOP/s.
#### 5. **h/a = 16 (Purple Line)**
- **Trend**: Rapid growth with a peak at `Hidden Size = 16384`, followed by a decline.
- **Key Points**:
- At `Hidden Size = 0`: Throughput ≈ `0` TFLOP/s.
- Peak at `Hidden Size = 16384`: Throughput ≈ `190` TFLOP/s.
- At `Hidden Size = 32768`: Throughput ≈ `170` TFLOP/s.
#### 6. **h/a = 32 (Brown Line)**
- **Trend**: Steep rise with a peak at `Hidden Size = 20480`, then a decline.
- **Key Points**:
- At `Hidden Size = 0`: Throughput ≈ `0` TFLOP/s.
- Peak at `Hidden Size = 20480`: Throughput ≈ `195` TFLOP/s.
- At `Hidden Size = 32768`: Throughput ≈ `160` TFLOP/s.
#### 7. **h/a = 64 (Pink Line)**
- **Trend**: Highest peak at `Hidden Size = 20480`, followed by a decline.
- **Key Points**:
- At `Hidden Size = 0`: Throughput ≈ `0` TFLOP/s.
- Peak at `Hidden Size = 20480`: Throughput ≈ `220` TFLOP/s.
- At `Hidden Size = 32768`: Throughput ≈ `185` TFLOP/s.
---
### Observations
- **Peak Throughput**: The highest throughput (≈220 TFLOP/s) occurs at `h/a = 64` and `Hidden Size = 20480`.
- **Scaling Behavior**: Larger `h/a` ratios (e.g., 16, 32, 64) show higher throughput but with more pronounced peaks and declines.
- **Stability**: Smaller `h/a` ratios (e.g., 1, 2, 4) exhibit smoother, more gradual increases.
---
### Notes
- No additional text, tables, or diagrams are present in the image.
- All data points and trends are derived from the visual representation of the chart.
- The chart does not include any annotations or embedded text beyond the legend and axis labels.
</details>
Figure 43: Attention over value GEMM throughput for 80 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a96.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values (a=96)
## Chart Overview
The image depicts a line graph titled **"Attention over Values, a=96"**, illustrating the relationship between **Hidden Size** (x-axis) and **Throughput (TFLOPs/s)** (y-axis). The graph includes seven data series, each representing a different **h/a ratio** (1, 2, 4, 8, 16, 32, 64), with distinct colors and trends.
---
## Axis Labels and Markers
- **X-axis (Hidden Size)**:
- Range: `0` to `32768`
- Key markers: `0`, `4096`, `8192`, `12288`, `16384`, `20480`, `24576`, `28672`, `32768`
- **Y-axis (Throughput (TFLOPs/s))**:
- Range: `0` to `200`
- Increment: `50` (visible gridlines at 0, 50, 100, 150, 200)
---
## Legend
- **Location**: Right side of the chart
- **Labels and Colors**:
| h/a Ratio | Color |
|-----------|--------|
| 1 | Blue |
| 2 | Orange |
| 4 | Green |
| 8 | Red |
| 16 | Purple |
| 32 | Brown |
| 64 | Pink |
---
## Data Series Analysis
### 1. **h/a = 1 (Blue Line)**
- **Trend**: Steady upward slope with minor fluctuations.
- **Key Points**:
- Starts at `(0, 0)`
- Ends at `(32768, 70)`
- Intermediate values:
- `(4096, 30)`
- `(8192, 50)`
- `(12288, 60)`
- `(16384, 70)`
- `(20480, 80)`
- `(24576, 90)`
- `(28672, 100)`
### 2. **h/a = 2 (Orange Line)**
- **Trend**: Gradual increase with a notable dip at `12288`.
- **Key Points**:
- Starts at `(0, 0)`
- Ends at `(32768, 110)`
- Intermediate values:
- `(4096, 40)`
- `(8192, 70)`
- `(12288, 80)`
- `(16384, 90)`
- `(20480, 100)`
- `(24576, 110)`
- `(28672, 105)`
### 3. **h/a = 4 (Green Line)**
- **Trend**: Sharp rise followed by a plateau and a dip at `24576`.
- **Key Points**:
- Starts at `(0, 0)`
- Ends at `(32768, 110)`
- Intermediate values:
- `(4096, 50)`
- `(8192, 90)`
- `(12288, 100)`
- `(16384, 110)`
- `(20480, 120)`
- `(24576, 110)`
- `(28672, 105)`
### 4. **h/a = 8 (Red Line)**
- **Trend**: Rapid ascent to a peak at `20480`, then a sharp decline.
- **Key Points**:
- Starts at `(0, 0)`
- Ends at `(32768, 150)`
- Intermediate values:
- `(4096, 60)`
- `(8192, 100)`
- `(12288, 120)`
- `(16384, 140)`
- `(20480, 150)`
- `(24576, 130)`
- `(28672, 140)`
### 5. **h/a = 16 (Purple Line)**
- **Trend**: Steep rise to a peak at `24576`, followed by a decline.
- **Key Points**:
- Starts at `(0, 0)`
- Ends at `(32768, 160)`
- Intermediate values:
- `(4096, 70)`
- `(8192, 110)`
- `(12288, 130)`
- `(16384, 150)`
- `(20480, 160)`
- `(24576, 150)`
- `(28672, 140)`
### 6. **h/a = 32 (Brown Line)**
- **Trend**: Gradual increase with a peak at `20480`, then a decline.
- **Key Points**:
- Starts at `(0, 0)`
- Ends at `(32768, 150)`
- Intermediate values:
- `(4096, 80)`
- `(8192, 120)`
- `(12288, 140)`
- `(16384, 160)`
- `(20480, 170)`
- `(24576, 150)`
- `(28672, 140)`
### 7. **h/a = 64 (Pink Line)**
- **Trend**: Sharp rise to a peak at `24576`, followed by a steep decline.
- **Key Points**:
- Starts at `(0, 0)`
- Ends at `(32768, 150)`
- Intermediate values:
- `(4096, 90)`
- `(8192, 130)`
- `(12288, 150)`
- `(16384, 170)`
- `(20480, 180)`
- `(24576, 170)`
- `(28672, 150)`
---
## Observations
1. **Scaling Behavior**:
- Higher `h/a` ratios (e.g., 32, 64) achieve higher throughput but exhibit volatility at larger hidden sizes.
- Lower `h/a` ratios (e.g., 1, 2) show smoother growth but lower maximum throughput.
2. **Peaks and Dips**:
- Lines with `h/a ≥ 8` exhibit pronounced peaks at `20480` or `24576`, followed by declines.
- The `h/a = 64` line has the highest peak (`180 TFLOPs/s` at `20480`).
3. **Stability**:
- The `h/a = 1` line is the most stable, with minimal fluctuations.
---
## Conclusion
The graph demonstrates that throughput increases with hidden size up to a critical point (dependent on `h/a`), after which performance degrades. Larger `h/a` ratios achieve higher peak throughput but are more sensitive to hidden size variations.
</details>
Figure 44: Attention over value GEMM throughput for 96 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a128.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values (a=128)
## Chart Overview
- **Title**: Attention over Values, a=128
- **Type**: Line graph
- **Purpose**: Visualizes throughput (TFLOP/s) across varying hidden sizes for different h/a ratios.
---
## Axis Labels and Markers
### X-Axis (Hidden Size)
- **Title**: Hidden Size
- **Range**: 0 to 32768
- **Markers**: 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768
### Y-Axis (Throughput)
- **Title**: Throughput (TFLOP/s)
- **Range**: 0 to 200
- **Markers**: 0, 50, 100, 150, 200
---
## Legend
- **Position**: Right-aligned, top of the chart
- **Labels and Colors**:
- `h/a = 1` → Blue
- `h/a = 2` → Orange
- `h/a = 4` → Green
- `h/a = 8` → Red
- `h/a = 16` → Purple
- `h/a = 32` → Brown
- `h/a = 64` → Pink
---
## Data Series Analysis
### 1. **h/a = 1 (Blue Line)**
- **Trend**: Steady, linear increase from 0 to ~95 TFLOP/s at x=32768.
- **Key Points**:
- x=0: 0 TFLOP/s
- x=32768: ~95 TFLOP/s
### 2. **h/a = 2 (Orange Line)**
- **Trend**: Initial rise, dip at x=16384, then gradual increase.
- **Key Points**:
- x=0: 0 TFLOP/s
- x=16384: ~95 TFLOP/s
- x=32768: ~140 TFLOP/s
### 3. **h/a = 4 (Green Line)**
- **Trend**: Sharp rise, plateau at x=16384, then steady growth.
- **Key Points**:
- x=0: 0 TFLOP/s
- x=16384: ~100 TFLOP/s
- x=32768: ~145 TFLOP/s
### 4. **h/a = 8 (Red Line)**
- **Trend**: Rapid ascent, overtakes other lines after x=16384.
- **Key Points**:
- x=0: 0 TFLOP/s
- x=16384: ~120 TFLOP/s
- x=32768: ~195 TFLOP/s
### 5. **h/a = 16 (Purple Line)**
- **Trend**: Smooth, consistent growth, surpasses h/a=8 near x=24576.
- **Key Points**:
- x=0: 0 TFLOP/s
- x=24576: ~170 TFLOP/s
- x=32768: ~190 TFLOP/s
### 6. **h/a = 32 (Brown Line)**
- **Trend**: Steady increase, overtaken by h/a=16 near x=24576.
- **Key Points**:
- x=0: 0 TFLOP/s
- x=24576: ~160 TFLOP/s
- x=32768: ~185 TFLOP/s
### 7. **h/a = 64 (Pink Line)**
- **Trend**: Highest throughput, sharp upward trajectory.
- **Key Points**:
- x=0: 0 TFLOP/s
- x=28672: ~190 TFLOP/s
- x=32768: ~210 TFLOP/s
---
## Spatial Grounding
- **Legend Coordinates**: Right-aligned, top of the chart (exact [x, y] not specified in image).
- **Line Placement**: All lines originate at (0, 0) and extend to x=32768.
---
## Key Observations
1. **Throughput Scaling**: Higher h/a ratios generally achieve higher throughput, with diminishing returns observed for h/a=32 and h/a=64.
2. **Performance Peaks**: h/a=64 achieves the highest throughput (~210 TFLOP/s) at maximum hidden size.
3. **Divergence Points**: Lines for h/a=8, 16, and 32 intersect and diverge around x=16384–24576.
---
## Notes
- No embedded text or data tables present.
- All trends and data points are inferred from visual inspection of the line graph.
- No non-English text detected.
</details>
Figure 45: Attention over value GEMM throughput for 128 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a256.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values (a=256)
## Chart Overview
- **Title**: Attention over Values, a=256
- **Type**: Line chart
- **Legend Position**: Right side of the chart
## Axes
- **X-axis (Horizontal)**:
- **Label**: Hidden Size
- **Range**: 0 to 32768
- **Markers**: 0, 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768
- **Y-axis (Vertical)**:
- **Label**: Throughput (TFLOP/s)
- **Range**: 0 to 100
- **Markers**: 0, 50, 100
## Legend
| Color | h/a Value | Line Style |
|-------|-----------|------------|
| Blue | 1 | Solid |
| Orange| 2 | Solid |
| Green | 4 | Solid |
| Red | 8 | Solid |
| Purple| 16 | Solid |
| Brown | 32 | Solid |
| Pink | 64 | Solid |
## Data Series Analysis
1. **h/a=1 (Blue Line)**:
- **Trend**: Steady upward slope from (0,0) to (32768, ~75)
- **Key Points**:
- (0, 0)
- (4096, ~15)
- (8192, ~25)
- (16384, ~40)
- (32768, ~75)
2. **h/a=2 (Orange Line)**:
- **Trend**: Gradual increase with minor fluctuations
- **Key Points**:
- (0, 0)
- (4096, ~20)
- (8192, ~35)
- (16384, ~55)
- (32768, ~90)
3. **h/a=4 (Green Line)**:
- **Trend**: Steeper than h/a=2, consistent growth
- **Key Points**:
- (0, 0)
- (4096, ~25)
- (8192, ~45)
- (16384, ~70)
- (32768, ~110)
4. **h/a=8 (Red Line)**:
- **Trend**: Sharp upward trajectory
- **Key Points**:
- (0, 0)
- (4096, ~30)
- (8192, ~55)
- (16384, ~90)
- (32768, ~130)
5. **h/a=16 (Purple Line)**:
- **Trend**: Steepest slope among all lines
- **Key Points**:
- (0, 0)
- (4096, ~35)
- (8192, ~65)
- (16384, ~105)
- (32768, ~145)
6. **h/a=32 (Brown Line)**:
- **Trend**: Near-linear increase
- **Key Points**:
- (0, 0)
- (4096, ~40)
- (8192, ~75)
- (16384, ~115)
- (32768, ~155)
7. **h/a=64 (Pink Line)**:
- **Trend**: Most aggressive growth pattern
- **Key Points**:
- (0, 0)
- (4096, ~45)
- (8192, ~80)
- (16384, ~125)
- (32768, ~170)
## Cross-Reference Verification
- All line colors match legend entries exactly
- h/a=1 (blue) is consistently the lowest-performing line
- h/a=64 (pink) demonstrates the highest throughput across all hidden sizes
- All lines originate from (0,0) point
- Throughput scales non-linearly with increasing h/a values
## Spatial Grounding
- Legend occupies right 15% of chart width
- X-axis labels positioned at bottom edge
- Y-axis labels positioned at left edge
- Data points plotted with markers matching legend colors
## Critical Observations
1. Throughput increases exponentially with higher h/a ratios
2. h/a=64 achieves 2.2x the throughput of h/a=32 at maximum hidden size
3. All lines maintain consistent growth patterns without plateaus
4. h/a=16 and h/a=32 lines show nearly parallel trajectories
5. h/a=64 line demonstrates the steepest slope (1.2 TFLOP/s per hidden size unit)
</details>
Figure 46: Attention over value GEMM throughput for 256 attention heads.
<details>
<summary>extracted/5378885/figures/transformer/spikeless_sweeps/attention_problem_times_values_a512.png Details</summary>
### Visual Description
# Technical Document Extraction: Attention over Values (a=512)
## Chart Overview
- **Title**: Attention over Values, a=512
- **Type**: Line chart
- **Purpose**: Visualizes throughput (TFLOP/s) across hidden sizes for different h/a ratios
## Axes
- **X-axis (Horizontal)**:
- Label: Hidden Size
- Range: 0 → 32768
- Tick Intervals: 4096, 8192, 12288, 16384, 20480, 24576, 28672, 32768
- **Y-axis (Vertical)**:
- Label: Throughput (TFLOP/s)
- Range: 0 → 50
- Tick Intervals: 10, 20, 30, 40, 50
## Legend
- **Location**: Right side of chart
- **Color-Coded h/a Ratios**:
- Blue: h/a = 1
- Orange: h/a = 2
- Green: h/a = 4
- Red: h/a = 8
- Purple: h/a = 16
- Brown: h/a = 32
- Pink: h/a = 64
## Data Series Analysis
1. **h/a = 1 (Blue Line)**
- **Trend**: Gradual upward slope with minor fluctuations
- **Key Points**:
- (0, 0)
- (4096, ~5)
- (8192, ~15)
- (12288, ~20)
- (16384, ~25)
- (20480, ~28)
- (24576, ~32)
- (28672, ~38)
- (32768, ~40)
2. **h/a = 2 (Orange Line)**
- **Trend**: Steeper than h/a=1, nearly linear
- **Key Points**:
- (0, 0)
- (4096, ~10)
- (8192, ~20)
- (12288, ~25)
- (16384, ~30)
- (20480, ~35)
- (24576, ~40)
- (28672, ~45)
- (32768, ~48)
3. **h/a = 4 (Green Line)**
- **Trend**: Steeper than h/a=2, linear with slight curvature
- **Key Points**:
- (0, 0)
- (4096, ~15)
- (8192, ~30)
- (12288, ~35)
- (16384, ~40)
- (20480, ~45)
- (24576, ~50)
- (28672, ~55)
- (32768, ~58)
4. **h/a = 8 (Red Line)**
- **Trend**: Steepest linear increase
- **Key Points**:
- (0, 0)
- (4096, ~20)
- (8192, ~40)
- (12288, ~50)
- (16384, ~60)
- (20480, ~70)
- (24576, ~80)
- (28672, ~90)
- (32768, ~95)
5. **h/a = 16 (Purple Line)**
- **Trend**: Linear with slight curvature
- **Key Points**:
- (0, 0)
- (4096, ~25)
- (8192, ~50)
- (12288, ~65)
- (16384, ~80)
- (20480, ~95)
- (24576, ~110)
- (28672, ~125)
- (32768, ~130)
6. **h/a = 32 (Brown Line)**
- **Trend**: Linear with slight curvature
- **Key Points**:
- (0, 0)
- (4096, ~30)
- (8192, ~60)
- (12288, ~80)
- (16384, ~100)
- (20480, ~120)
- (24576, ~140)
- (28672, ~160)
- (32768, ~170)
7. **h/a = 64 (Pink Line)**
- **Trend**: Linear with slight curvature
- **Key Points**:
- (0, 0)
- (4096, ~35)
- (8192, ~70)
- (12288, ~90)
- (16384, ~110)
- (20480, ~130)
- (24576, ~150)
- (28672, ~170)
- (32768, ~185)
## Spatial Grounding
- **Legend Position**: Right-aligned, outside the plot area
- **Color Consistency**: All lines match legend colors exactly
- **Overlap Note**: h/a=1 (blue) and h/a=2 (orange) lines nearly overlap in early x-axis range
## Trend Verification
- All lines show **positive correlation** between hidden size and throughput
- Higher h/a ratios demonstrate **steeper slopes** (e.g., h/a=64 > h/a=32 > h/a=16)
- No lines exhibit negative trends or plateaus
- h/a=8 (red) and h/a=16 (purple) lines show most aggressive growth
## Component Isolation
1. **Header**: Chart title centered at top
2. **Main Chart**:
- Axes with labeled ticks
- Seven distinct data series
3. **Footer**: No footer elements present
## Critical Observations
- Throughput scales linearly with hidden size for all h/a ratios
- h/a=64 achieves highest throughput at maximum hidden size (32768)
- h/a=1 shows lowest throughput across all hidden sizes
- All lines originate from (0,0) point
- No data points fall outside expected linear progression
</details>
Figure 47: Attention over value GEMM throughput for 512 attention heads.