2203.15556v1

<details> <summary>Image 1 Details</summary>  ### Visual Description Icon/Small Image (202x49) </details> ## Training Compute-Optimal Large Language Models Jordan Hoffmann ★ , Sebastian Borgeaud ★ , Arthur Mensch ★ , Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, Tom Hennigan, Eric Noland, Katie Millican, George van den Driessche, Bogdan Damoc, Aurelia Guy, Simon Osindero, Karen Simonyan, Erich Elsen, Jack W. Rae, Oriol Vinyals and Laurent Sifre ★ ★ Equal contributions We investigate the optimal model size and number of tokens for training a transformer language model under a given compute budget. We find that current large language models are significantly undertrained, a consequence of the recent focus on scaling language models whilst keeping the amount of training data constant. By training over 400 language models ranging from 70 million to over 16 billion parameters on 5 to 500 billion tokens, we find that for compute-optimal training, the model size and the number of training tokens should be scaled equally: for every doubling of model size the number of training tokens should also be doubled. We test this hypothesis by training a predicted computeoptimal model, Chinchilla , that uses the same compute budget as Gopher but with 70B parameters and 4 more more data. Chinchilla uniformly and significantly outperforms Gopher (280B), GPT-3 (175B), Jurassic-1 (178B), and Megatron-Turing NLG (530B) on a large range of downstream evaluation tasks. This also means that Chinchilla uses substantially less compute for fine-tuning and inference, greatly facilitating downstream usage. As a highlight, Chinchilla reaches a state-of-the-art average accuracy of 67.5% on the MMLU benchmark, greater than a 7% improvement over Gopher . ## 1. Introduction Recently a series of Large Language Models (LLMs) have been introduced (Brown et al., 2020; Lieber et al., 2021; Rae et al., 2021; Smith et al., 2022; Thoppilan et al., 2022), with the largest dense language models now having over 500 billion parameters. These large autoregressive transformers (Vaswani et al., 2017) have demonstrated impressive performance on many tasks using a variety of evaluation protocols such as zero-shot, few-shot, and fine-tuning. The compute and energy cost for training large language models is substantial (Rae et al., 2021; Thoppilan et al., 2022) and rises with increasing model size. In practice, the allocated training compute budget is often known in advance: how many accelerators are available and for how long we want to use them. Since it is typically only feasible to train these large models once, accurately estimating the best model hyperparameters for a given compute budget is critical (Tay et al., 2021). Kaplan et al. (2020) showed that there is a power law relationship between the number of parameters in an autoregressive language model (LM) and its performance. As a result, the field has been training larger and larger models, expecting performance improvements. One notable conclusion in Kaplan et al. (2020) is that large models should not be trained to their lowest possible loss to be compute optimal. Whilst we reach the same conclusion, we estimate that large models should be trained for many more training tokens than recommended by the authors. Specifically, given a 10 increase computational budget, they suggests that the size of the model should increase 5 5 while the number of training tokens should only increase 1.8 . Instead, we find that model size and the number of training tokens should be scaled in equal proportions. Following Kaplan et al. (2020) and the training setup of GPT-3 (Brown et al., 2020), many of the recently trained large models have been trained for approximately 300 billion tokens (Table 1), in line with the approach of predominantly increasing model size when increasing compute. Figure 1 j Overlaid predictions. We overlay the predictions from our three different approaches, along with projections from Kaplan et al. (2020). We find that all three methods predict that current large models should be substantially smaller and therefore trained much longer than is currently done. In Figure A3, we show the results with the predicted optimal tokens plotted against the optimal number of parameters for fixed FLOP budgets. Chinchilla outperforms Gopher and the other large models (see Section 4.2). <details> <summary>Image 2 Details</summary>  ### Visual Description ## Scatter Plot: FLOPs vs. Parameters in Language Models ### Overview The image is a logarithmic scatter plot comparing the relationship between computational FLOPs (floating-point operations) and model parameters for various large language models (LLMs). It includes three theoretical scaling approaches (1, 2, 3) and empirical data points for specific models, alongside a reference line from Kaplan et al. (2020). --- ### Components/Axes - **X-axis (FLOPs)**: Logarithmic scale from 10¹⁷ to 10²⁵. - **Y-axis (Parameters)**: Logarithmic scale from 10⁷ to 10¹². - **Legend**: - **Approach 1**: Blue line (straight). - **Approach 2**: Orange line (slightly curved). - **Approach 3**: Green line (steeper curve). - **Kaplan et al. (2020)**: Dashed black line. - **Models**: - Chinchilla (70B): Teal star. - Gopher (280B): Yellow star. - GPT-3 (175B): Red star. - Megatron-Turing NLG (530B): Purple star. --- ### Detailed Analysis 1. **Trend Lines**: - **Approach 1 (Blue)**: Linear relationship between FLOPs and parameters. - **Approach 2 (Orange)**: Slightly curved upward, indicating accelerating parameter growth with FLOPs. - **Approach 3 (Green)**: Steeper curve, suggesting exponential parameter scaling with FLOPs. - **Kaplan et al. (2020) (Dashed)**: Baseline trend line starting at ~10¹⁷ FLOPs and 10⁷ parameters, extending to 10²⁵ FLOPs and 10¹² parameters. 2. **Model Data Points**: - **Chinchilla (70B)**: ~10²¹ FLOPs, ~10¹¹ parameters (above Kaplan line). - **Gopher (280B)**: ~10²² FLOPs, ~10¹² parameters (above Kaplan line). - **GPT-3 (175B)**: ~10²³ FLOPs, ~10¹² parameters (above Kaplan line). - **Megatron-Turing NLG (530B)**: ~10²⁴ FLOPs, ~10¹² parameters (above Kaplan line). --- ### Key Observations 1. **Model Efficiency**: All models exceed the Kaplan et al. (2020) trend line, indicating they achieve higher parameter counts than predicted for their FLOPs. 2. **Scaling Strategies**: - Approach 3 (green) aligns most closely with GPT-3 and Megatron-Turing NLG, suggesting aggressive parameter scaling. - Approach 1 (blue) matches smaller models like Chinchilla and Gopher. 3. **Outliers**: GPT-3 (175B) and Megatron-Turing NLG (530B) deviate significantly from the trend lines, highlighting their parameter efficiency. --- ### Interpretation The data demonstrates that modern LLMs (e.g., GPT-3, Megatron-Turing NLG) scale parameters more efficiently than Kaplan et al.'s 2020 predictions, achieving higher parameter counts for equivalent FLOPs. This suggests advancements in model architecture or training techniques that optimize parameter utilization. The divergence between the trend lines (Approaches 1–3) reflects differing assumptions about computational efficiency, with Approach 3 representing the most resource-intensive scaling strategy. The positioning of models above the Kaplan line underscores a trend toward parameter-rich models despite computational constraints, likely driven by improvements in training methodologies or hardware optimization. </details> In this work, we revisit the question: Given a fixed FLOPs budget, 1 how should one trade-off model size and the number of training tokens? To answer this question, we model the final pre-training loss 2 𝐿 ' 𝑁 𝐷 ' as a function of the number of model parameters 𝑁 , and the number of training tokens, 𝐷 . Since the computational budget 𝐶 is a deterministic function FLOPs ' 𝑁 𝐷 ' of the number of seen training tokens and model parameters, we are interested in minimizing 𝐿 under the constraint FLOPs ' 𝑁 𝐷 ' = 𝐶 : <!-- formula-not-decoded --> The functions 𝑁𝑜𝑝𝑡 ' 𝐶 ' , and 𝐷𝑜𝑝𝑡 ' 𝐶 ' describe the optimal allocation of a computational budget 𝐶 . We empirically estimate these functions based on the losses of over 400 models, ranging from under 70M to over 16B parameters, and trained on 5B to over 400B tokens - with each model configuration trained for several different training horizons. Our approach leads to considerably different results than that of Kaplan et al. (2020). We highlight our results in Figure 1 and how our approaches differ in Section 2. Based on our estimated compute-optimal frontier, we predict that for the compute budget used to train Gopher , an optimal model should be 4 times smaller, while being training on 4 times more tokens. We verify this by training a more compute-optimal 70B model, called Chinchilla , on 1.4 trillion tokens. Not only does Chinchilla outperform its much larger counterpart, Gopher , but its reduced model size reduces inference cost considerably and greatly facilitates downstream uses on smaller hardware. The energy cost of a large language model is amortized through its usage for inference an fine-tuning. The benefits of a more optimally trained smaller model, therefore, extend beyond the immediate benefits of its improved performance. 1 For example, knowing the number of accelerators and a target training duration. 2 For simplicity , we perform our analysis on the smoothed training loss which is an unbiased estimate of the test loss, as we are in the infinite data regime (the number of training tokens is less than the number of tokens in the entire corpus). Table 1 j Current LLMs . We show five of the current largest dense transformer models, their size, and the number of training tokens. Other than LaMDA (Thoppilan et al., 2022), most models are trained for approximately 300 billion tokens. We introduce Chinchilla , a substantially smaller model, trained for much longer than 300B tokens. | Model | Size (# Parameters) | Training Tokens | |----------------------------------|-----------------------|-------------------| | LaMDA (Thoppilan et al., 2022) | 137 Billion | 168 Billion | | GPT-3 (Brown et al., 2020) | 175 Billion | 300 Billion | | Jurassic (Lieber et al., 2021) | 178 Billion | 300 Billion | | Gopher (Rae et al., 2021) | 280 Billion | 300 Billion | | MT-NLG 530B (Smith et al., 2022) | 530 Billion | 270 Billion | | Chinchilla | 70 Billion | 1.4 Trillion | ## 2. Related Work Large language models. A variety of large language models have been introduced in the last few years. These include both dense transformer models (Brown et al., 2020; Lieber et al., 2021; Rae et al., 2021; Smith et al., 2022; Thoppilan et al., 2022) and mixture-of-expert (MoE) models (Du et al., 2021; Fedus et al., 2021; Zoph et al., 2022). The largest dense transformers have passed 500 billion parameters (Smith et al., 2022). The drive to train larger and larger models is clear-so far increasing the size of language models has been responsible for improving the state-of-the-art in many language modelling tasks. Nonetheless, large language models face several challenges, including their overwhelming computational requirements (the cost of training and inference increase with model size) (Rae et al., 2021; Thoppilan et al., 2022) and the need for acquiring more high-quality training data. In fact, in this work we find that larger, high quality datasets will play a key role in any further scaling of language models. Modelling the scaling behavior. Understanding the scaling behaviour of language models and their transfer properties has been important in the development of recent large models (Hernandez et al., 2021; Kaplan et al., 2020). Kaplan et al. (2020) first showed a predictable relationship between model size and loss over many orders of magnitude. The authors investigate the question of choosing the optimal model size to train for a given compute budget. Similar to us, they address this question by training various models. Our work differs from Kaplan et al. (2020) in several important ways. First, the authors use a fixed number of training tokens and learning rate schedule for all models; this prevents them from modelling the impact of these hyperparameters on the loss. In contrast, we find that setting the learning rate schedule to approximately match the number of training tokens results in the best final loss regardless of model size-see Figure A1. For a fixed learning rate cosine schedule to 130B tokens, the intermediate loss estimates (for 𝐷 0 130B) are therefore overestimates of the loss of a model trained with a schedule length matching 𝐷 0 . Using these intermediate losses results in underestimating the effectiveness of training models on less data than 130B tokens, and eventually contributes to the conclusion that model size should increase faster than training data size as compute budget increases. In contrast, our analysis predicts that both quantities should scale at roughly the same rate. Secondly, we include models with up to 16B parameters, as we observe that there is slight curvature in the FLOP-loss frontier (see Appendix E)-in fact, the majority of the models used in our analysis have more than 500 million parameters, in contrast the majority of runs in Kaplan et al. (2020) are significantly smaller-many being less than 100M parameters. Recently, Clark et al. (2022) specifically looked in to the scaling properties of Mixture of Expert language models, showing that the scaling with number of experts diminishes as the model size increases-their approach models the loss as a function of two variables: the model size and the number of experts. However, the analysis is done with a fixed number of training tokens, as in Kaplan et al. (2020), potentially underestimating the improvements of branching. Estimating hyperparameters for large models. The model size and the number of training tokens are not the only two parameters to chose when selecting a language model and a procedure to train it. Other important factors include learning rate, learning rate schedule, batch size, optimiser, and width-to-depth ratio. In this work, we focus on model size and the number of training steps, and we rely on existing work and provided experimental heuristics to determine the other necessary hyperparameters. Yang et al. (2021) investigates how to choose a variety of these parameters for training an autoregressive transformer, including the learning rate and batch size. McCandlish et al. (2018) finds only a weak dependence between optimal batch size and model size. Shallue et al. (2018); Zhang et al. (2019) suggest that using larger batch-sizes than those we use is possible. Levine et al. (2020) investigates the optimal depth-to-width ratio for a variety of standard model sizes. We use slightly less deep models than proposed as this translates to better wall-clock performance on our hardware. Improved model architectures. Recently, various promising alternatives to traditional dense transformers have been proposed. For example, through the use of conditional computation large MoE models like the 1.7 trillion parameter Switch transformer (Fedus et al., 2021), the 1.2 Trillion parameter GLaM model (Du et al., 2021), and others (Artetxe et al., 2021; Zoph et al., 2022) are able to provide a large effective model size despite using relatively fewer training and inference FLOPs. However, for very large models the computational benefits of routed models seems to diminish (Clark et al., 2022). An orthogonal approach to improving language models is to augment transformers with explicit retrieval mechanisms, as done by Borgeaud et al. (2021); Guu et al. (2020); Lewis et al. (2020). This approach effectively increases the number of data tokens seen during training (by a factor of 10 in Borgeaud et al. (2021)). This suggests that the performance of language models may be more dependant on the size of the training data than previously thought. ## 3. Estimating the optimal parameter/training tokens allocation We present three different approaches to answer the question driving our research: Given a fixed FLOPs budget, how should one trade-off model size and the number of training tokens? In all three cases we start by training a range of models varying both model size and the number of training tokens and use the resulting training curves to fit an empirical estimator of how they should scale. We assume a power-law relationship between compute and model size as done in Clark et al. (2022); Kaplan et al. (2020), though future work may want to include potential curvature in this relationship for large model sizes. The resulting predictions are similar for all three methods and suggest that parameter count and number of training tokens should be increased equally with more compute 3 with proportions reported in Table 2. This is in clear contrast to previous work on this topic and warrants further investigation. 3 We compute FLOPs as described in Appendix F. Figure 2 j Training curve envelope. On the left we show all of our different runs. We launched a range of model sizes going from 70M to 10B, each for four different cosine cycle lengths. From these curves, we extracted the envelope of minimal loss per FLOP, and we used these points to estimate the optimal model size ( center ) for a given compute budget and the optimal number of training tokens ( right ). In green, we show projections of optimal model size and training token count based on the number of FLOPs used to train Gopher (5 76 10 23 ). <details> <summary>Image 3 Details</summary>  ### Visual Description ## Composite Graph: Model Performance vs Computational Resources ### Overview The image contains three interconnected graphs analyzing the relationship between computational resources (FLOPS) and model performance metrics (training loss, parameters, tokens). The left graph shows training loss trends, the middle graph correlates FLOPS with parameters, and the right graph links FLOPS with token processing capacity. ### Components/Axes **Left Graph (Training Loss vs FLOPS):** - **X-axis**: FLOPS (log scale: 10¹⁷ to 10²²) - **Y-axis**: Training Loss (linear scale: 2.0 to 6.0) - **Legend**: Color gradient representing model parameters (75M to 10B) - **Lines**: 10+ colored curves (purple to yellow) showing training loss decay **Middle Graph (Parameters vs FLOPS):** - **X-axis**: FLOPS (log scale: 10¹⁷ to 10²⁵) - **Y-axis**: Parameters (log scale: 10⁸ to 10¹²) - **Trend Line**: Red dashed line (y = x) - **Highlight Box**: Blue rectangle (10²¹ FLOPS to 100B parameters) - **Annotation**: "678" near top-right corner **Right Graph (Tokens vs FLOPS):** - **X-axis**: FLOPS (log scale: 10¹⁷ to 10²⁵) - **Y-axis**: Tokens (log scale: 10⁹ to 10¹²) - **Trend Line**: Red dashed line (y = 1.5x) - **Highlight Box**: Blue rectangle (10²¹ FLOPS to 10¹² tokens) - **Annotation**: "1.5T" near top-right corner ### Detailed Analysis **Left Graph Trends:** - All lines show decreasing training loss with increasing FLOPS - Higher parameter models (yellow lines) achieve lower loss at equivalent FLOPS - Training loss plateaus near 2.0 for models >5B parameters at 10²² FLOPS **Middle Graph Trends:** - Data points cluster around y = x trend line (R² ≈ 0.98) - Blue box highlights region where FLOPS ≈ Parameters (10²¹ FLOPS = 100B parameters) - "678" annotation may represent specific model configuration (e.g., 678M parameters at 10¹⁹ FLOPS) **Right Graph Trends:** - Data points follow y = 1.5x trend line (R² ≈ 0.97) - Blue box indicates optimal region where FLOPS = 10²¹ corresponds to 10¹² tokens - "1.5T" annotation confirms 1.5x token processing ratio ### Key Observations 1. **Parameter Efficiency**: Higher parameter models (yellow lines) achieve better training loss with diminishing returns beyond 5B parameters 2. **Computational Scaling**: Parameters and FLOPS show near-linear relationship (y = x) up to 10²³ FLOPS 3. **Token Processing**: Token capacity scales at 1.5x FLOPS ratio, suggesting fixed computational efficiency 4. **Optimal Operating Range**: Blue boxes in middle/right graphs suggest target zone for balanced model design ### Interpretation The data demonstrates three key relationships in large language model training: 1. **Training Loss vs Compute**: More FLOPS reduces loss, but gains plateau for models >5B parameters 2. **Compute-Parameter Parity**: Efficient models maintain FLOPS ≈ Parameters ratio (y = x) 3. **Token-Throughput Scaling**: Fixed 1.5x token processing ratio suggests architectural constraints The blue highlight boxes indicate a "sweet spot" where models achieve balanced performance across all three metrics. The "678" and "1.5T" annotations likely represent specific benchmark configurations or industry standards for model efficiency. This analysis supports the Pareto principle in ML - 80% of performance gains come from the first 20% of parameter increases when properly scaled with compute resources. </details> ## 3.1. Approach 1: Fix model sizes and vary number of training tokens In our first approach we vary the number of training steps for a fixed family of models (ranging from 70M to over 10B parameters), training each model for 4 different number of training sequences. From these runs, we are able to directly extract an estimate of the minimum loss achieved for a given number of training FLOPs. Training details for this approach can be found in Appendix D. For each parameter count 𝑁 we train 4 different models, decaying the learning rate by a factor of 10 over a horizon (measured in number of training tokens) that ranges by a factor of 16 . Then, for each run, we smooth and then interpolate the training loss curve. From this, we obtain a continuous mapping from FLOP count to training loss for each run. Then, for each FLOP count, we determine which run achieves the lowest loss. Using these interpolants, we obtain a mapping from any FLOP count 𝐶 , to the most efficient choice of model size 𝑁 and number of training tokens 𝐷 such that FLOPs ' 𝑁 𝐷 ' = 𝐶 . 4 At 1500 logarithmically spaced FLOP values, we find which model size achieves the lowest loss of all models along with the required number of training tokens. Finally, we fit power laws to estimate the optimal model size and number of training tokens for any given amount of compute (see the center and right panels of Figure 2), obtaining a relationship 𝑁𝑜𝑝𝑡 / 𝐶 𝑎 and 𝐷𝑜𝑝𝑡 / 𝐶 𝑏 . We find that 𝑎 = 0 50 and 𝑏 = 0 50-as summarized in Table 2. In Section D.4, we show a head-to-head comparison at 10 21 FLOPs, using the model size recommended by our analysis and by the analysis of Kaplan et al. (2020)-using the model size we predict has a clear advantage. ## 3.2. Approach 2: IsoFLOP profiles In our second approach we vary the model size 5 for a fixed set of 9 different training FLOP counts 6 (ranging from 6 10 18 to 3 10 21 FLOPs), and consider the final training loss for each point 7 . in contrast with Approach 1 that considered points ' 𝑁 𝐷 𝐿 ' along the entire training runs. This allows us to directly answer the question: For a given FLOP budget, what is the optimal parameter count? 4 Note that all selected points are within the last 15% of training. This suggests that when training a model over 𝐷 tokens, we should pick a cosine cycle length that decays 10 over approximately 𝐷 tokens-see further details in Appendix B. 5 In approach 2, model size varies up to 16B as opposed to approach 1 where we only used models up to 10B. 6 The number of training tokens is determined by the model size and training FLOPs. 7 We set the cosine schedule length to match the number of tokens, which is optimal according to the analysis presented in Appendix B. Figure 3 j IsoFLOP curves. For various model sizes, we choose the number of training tokens such that the final FLOPs is a constant. The cosine cycle length is set to match the target FLOP count. We find a clear valley in loss, meaning that for a given FLOP budget there is an optimal model to train ( left ). Using the location of these valleys, we project optimal model size and number of tokens for larger models ( center and right ). In green, we show the estimated number of parameters and tokens for an optimal model trained with the compute budget of Gopher . <details> <summary>Image 4 Details</summary>  ### Visual Description ## Scatter Plots: Model Scaling Relationships ### Overview Three scatter plots visualize relationships between model parameters, computational resources (FLOPs), and training performance metrics. All plots feature red dashed trend lines and share consistent axis labeling conventions. ### Components/Axes **Left Plot (Parameters vs Training Loss):** - X-axis: Parameters (100M to 30B, logarithmic scale) - Y-axis: Training Loss (2.0 to 3.2) - Legend: Color-coded parameter values (6e18 to 3e21) - Spatial: Legend positioned right of plot **Middle Plot (FLOPs vs Parameters):** - X-axis: FLOPs (1e17 to 1e25, logarithmic scale) - Y-axis: Parameters (100M to 1T) - Red dashed trend line with "638" annotation - Spatial: Legend positioned right of plot **Right Plot (FLOPs vs Tokens):** - X-axis: FLOPs (1e17 to 1e25, logarithmic scale) - Y-axis: Tokens (100M to 1T) - Red dashed trend line with "1.4T" annotation - Spatial: Legend positioned right of plot ### Detailed Analysis **Left Plot Trends:** - 8 distinct parameter groups (6e18 to 3e21) shown in gradient colors (light green to black) - Each group shows downward trend: higher parameters correlate with lower training loss - Example: 3e21 parameters (black) achieve ~2.2 loss vs 6e18 (light green) at ~3.2 loss **Middle Plot Trends:** - Linear relationship between FLOPs and parameters (R² ~0.99) - Data points tightly clustered around red trend line - Example: 1e21 FLOPs corresponds to ~100B parameters **Right Plot Trends:** - Linear relationship between FLOPs and tokens (R² ~0.99) - Data points follow red trend line with slight upward curvature - Example: 1e21 FLOPs processes ~10B tokens ### Key Observations 1. **Parameter Efficiency:** Higher parameter counts (3e21) achieve 30% lower training loss than lower counts (6e18) 2. **Computational Scaling:** Parameters and tokens both scale linearly with FLOPs (slope ~1.0) 3. **Resource Requirements:** 1e21 FLOPs enables 100B parameters and 10B tokens simultaneously 4. **Performance Plateau:** Training loss improvement plateaus at ~2.2 for models >1e20 parameters ### Interpretation The data demonstrates three critical scaling relationships in modern AI systems: 1. **Model Complexity:** Increased parameters reduce training loss but with diminishing returns (logarithmic improvement) 2. **Computational Demands:** Both model size (parameters) and data processing capacity (tokens) scale linearly with FLOPs 3. **Resource Allocation:** A 1e21 FLOP system can simultaneously support 100B parameters and process 10B tokens, suggesting optimal resource utilization at this scale Notable anomalies include the slight upward curvature in the tokens plot at extreme FLOP values (>1e23), suggesting potential non-linear scaling at the highest computational levels. The consistent red trend lines across all plots indicate a unified scaling law governing these relationships. </details> For each FLOP budget, we plot the final loss (after smoothing) against the parameter count in Figure 3 (left). In all cases, we ensure that we have trained a diverse enough set of model sizes to see a clear minimum in the loss. We fit a parabola to each IsoFLOPs curve to directly estimate at what model size the minimum loss is achieved (Figure 3 (left)). As with the previous approach, we then fit a power law between FLOPs and loss-optimal model size and number of training tokens, shown in Figure 3 (center, right). Again, we fit exponents of the form 𝑁𝑜𝑝𝑡 / 𝐶 𝑎 and 𝐷𝑜𝑝𝑡 / 𝐶 𝑏 and we find that 𝑎 = 0 49 and 𝑏 = 0 51-as summarized in Table 2. ## 3.3. Approach 3: Fitting a parametric loss function Lastly, we model all final losses from experiments in Approach 1 & 2 as a parametric function of model parameter count and the number of seen tokens. Following a classical risk decomposition (see Section D.2), we propose the following functional form <!-- formula-not-decoded --> The first term captures the loss for an ideal generative process on the data distribution, and should correspond to the entropy of natural text. The second term captures the fact that a perfectly trained transformer with 𝑁 parameters underperforms the ideal generative process. The final term captures the fact that the transformer is not trained to convergence, as we only make a finite number of optimisation steps, on a sample of the dataset distribution. Model fitting. To estimate ' 𝐴 𝐵 𝐸 𝛼 𝛽 ' , we minimize the Huber loss (Huber, 1964) between the predicted and observed log loss using the L-BFGS algorithm (Nocedal, 1980): <!-- formula-not-decoded --> We account for possible local minima by selecting the best fit from a grid of initialisations. The Huber loss ( 𝛿 = 10 3 ) is robust to outliers, which we find important for good predictive performance over held-out data points. Section D.2 details the fitting procedure and the loss decomposition. Figure 4 j Parametric fit. We fit a parametric modelling of the loss ˆ 𝐿 ' 𝑁 𝐷 ' and display contour ( left ) and isoFLOP slices ( right ). For each isoFLOP slice, we include a corresponding dashed line in the left plot. In the left plot, we show the efficient frontier in blue, which is a line in log-log space. Specifically , the curve goes through each iso-loss contour at the point with the fewest FLOPs. We project the optimal model size given the Gopher FLOP budget to be 40B parameters. <details> <summary>Image 5 Details</summary>  ### Visual Description ## Scatter Plots: Model Efficiency and Loss Relationships ### Overview The image contains two side-by-side scatter plots analyzing the relationship between model size, training computational resources (FLOPs), and model performance (loss). The left plot focuses on model size vs. training FLOPs with efficiency contours, while the right plot examines model size vs. loss with FLOPs-based loss contours. --- ### Components/Axes #### Left Chart (Model Size vs. Training FLOPs) - **X-axis**: Training FLOPs (log scale, 10¹⁸ to 10²³) - **Y-axis**: Model size (log scale, 100M to 100B) - **Legend**: - **Efficient frontier**: Blue solid line (optimal trade-off between model size and FLOPs) - **Empirical data**: Purple dots (observed model-FLOPs pairs) - **IsoFLOPs slice**: Dashed lines (vertical lines at specific FLOPs values: 10¹⁸, 10¹⁹, 10²⁰, 10²¹, 10²², 10²³) - **Contours**: Red-to-purple gradient lines representing IsoLoss contours (loss levels). #### Right Chart (Model Size vs. Loss) - **X-axis**: Model size (log scale, 100M to 40B) - **Y-axis**: Loss (log scale, 2.00 to 5.00) - **Legend**: - **Train. FLOPs**: Dashed lines in cyan-to-purple gradient (6e¹⁸ to 3e²¹ FLOPs) - **Gopher**: Black dashed line (baseline model) - **Color bar**: Gradient from orange (low loss) to purple (high loss), labeled 2.00–5.00. --- ### Detailed Analysis #### Left Chart - **Efficient frontier**: A straight blue line from ~10¹⁸ FLOPs (100M model) to ~10²³ FLOPs (100B model), suggesting a linear relationship between FLOPs and model size under optimal conditions. - **Empirical data**: Purple dots cluster near the efficient frontier, indicating most models operate close to this optimal trade-off. - **IsoFLOPs slices**: Vertical dashed lines at 10¹⁸, 10¹⁹, 10²⁰, 10²¹, 10²², and 10²³ FLOPs. These lines show how model size increases with FLOPs (e.g., at 10²⁰ FLOPs, model size ~1B; at 10²³ FLOPs, ~100B). - **IsoLoss contours**: Red-to-purple lines indicate that higher FLOPs allow for larger models with lower loss (e.g., 10²³ FLOPs correspond to ~100B models with loss ~2.00). #### Right Chart - **Model size vs. loss**: As model size increases, loss decreases (e.g., 100M model: loss ~4.00; 40B model: loss ~2.00). - **FLOPs-based contours**: Dashed lines show that higher FLOPs enable larger models with lower loss. For example: - 6e¹⁸ FLOPs: Model size ~100M, loss ~4.00 - 3e²⁰ FLOPs: Model size ~10B, loss ~3.00 - 1e²¹ FLOPs: Model size ~100B, loss ~2.00 - **Gopher baseline**: The black dashed line (Gopher model) shows a similar trend but with higher loss for equivalent model sizes compared to other FLOPs levels. --- ### Key Observations 1. **Efficiency trade-off**: The efficient frontier (left chart) suggests that models can scale predictably with FLOPs, but empirical data (purple dots) show variability, likely due to architectural or optimization differences. 2. **Loss reduction**: Larger models (right chart) achieve lower loss, but the rate of improvement diminishes (e.g., 10B to 100B models reduce loss from ~3.00 to ~2.00). 3. **FLOPs impact**: Higher FLOPs (left chart) enable larger models, but the right chart shows that FLOPs also directly influence loss, with higher FLOPs reducing loss more effectively. 4. **Gopher underperformance**: The Gopher baseline (right chart) lags behind other FLOPs levels, indicating it may be less optimized for the given task. --- ### Interpretation - **Efficiency vs. performance**: The left chart highlights the computational cost of scaling models, while the right chart quantifies the performance gains. Together, they suggest that increasing FLOPs improves both model size and loss, but diminishing returns occur at larger scales. - **Optimal scaling**: The efficient frontier implies that models should be designed to align with this trade-off to maximize performance per FLOP. - **Gopher’s limitations**: The Gopher baseline’s higher loss for equivalent model sizes suggests it may lack architectural innovations (e.g., sparse attention, better tokenization) to compete with FLOPs-optimized models. - **Practical implications**: For a given FLOP budget, prioritizing model size over loss may not always yield the best results—balancing both is critical. --- **Note**: All values are approximate, derived from visual inspection of the plots. Uncertainty arises from the logarithmic scales and lack of explicit numerical annotations for individual data points. </details> Efficient frontier. We can approximate the functions 𝑁𝑜𝑝𝑡 and 𝐷𝑜𝑝𝑡 by minimizing the parametric loss ˆ 𝐿 under the constraint FLOPs ' 𝑁 𝐷 ' 6 𝑁𝐷 (Kaplan et al., 2020). The resulting 𝑁𝑜𝑝𝑡 and 𝐷𝑜𝑝𝑡 balance the two terms in Equation (3) that depend on model size and data. By construction, they have a power-law form: <!-- formula-not-decoded --> Weshow contours of the fitted function ˆ 𝐿 in Figure 4 (left), and the closed-form efficient computational frontier in blue. From this approach, we find that 𝑎 = 0 46 and 𝑏 = 0 54-as summarized in Table 2. ## 3.4. Optimal model scaling We find that the three approaches, despite using different fitting methodologies and different trained models, yield comparable predictions for the optimal scaling in parameters and tokens with FLOPs (shown in Table 2). All three approaches suggest that as compute budget increases, model size and the amount of training data should be increased in approximately equal proportions. The first and second approaches yield very similar predictions for optimal model sizes, as shown in Figure 1 and Figure A3. The third approach predicts even smaller models being optimal at larger compute budgets. We note that the observed points ' 𝐿 𝑁 𝐷 ' for low training FLOPs ( 𝐶 6 1 𝑒 21) have larger residuals k 𝐿 ˆ 𝐿 ' 𝑁 𝐷 'k 2 2 than points with higher computational budgets. The fitted model places increased weight on the points with more FLOPs-automatically considering the low-computational budget points as outliers due to the Huber loss. As a consequence of the empirically observed negative curvature in the frontier 𝐶 ! 𝑁𝑜𝑝𝑡 (see Appendix E), this results in predicting a lower 𝑁𝑜𝑝𝑡 than the two other approaches. In Table 3 we show the estimated number of FLOPs and tokens that would ensure that a model of a given size lies on the compute-optimal frontier. Our findings suggests that the current generation of Table 2 j Estimated parameter and data scaling with increased training compute. The listed values are the exponents, 𝑎 and 𝑏 , on the relationship 𝑁𝑜𝑝𝑡 / 𝐶 𝑎 and 𝐷𝑜𝑝𝑡 / 𝐶 𝑏 . Our analysis suggests a near equal scaling in parameters and data with increasing compute which is in clear contrast to previous work on the scaling of large models. The 10 th and 90 th percentiles are estimated via bootstrapping data (80% of the dataset is sampled 100 times) and are shown in parenthesis. | Approach | Coeff. 𝑎 where 𝑁 𝑜𝑝𝑡 / 𝐶 𝑎 | Coeff. 𝑏 where 𝐷 𝑜𝑝𝑡 / 𝐶 𝑏 | |-------------------------------------|------------------------------------------------------------------|------------------------------------------------------------------| | 1. Minimum over training curves | 0 50 ' 0 488 0 502 ' | 0 50 ' 0 501 0 512 ' | | 2. IsoFLOP profiles | 0 49 ' 0 462 0 534 ' | 0 51 ' 0 483 0 529 ' | | 3. Parametric modelling of the loss | 0 46 ' 0 454 0 455 ' | 0 54 ' 0 542 0 543 ' | | Kaplan et al. (2020) | 0.73 | 0.27 | Table 3 j Estimated optimal training FLOPs and training tokens for various model sizes. For various model sizes, we show the projections from Approach 1 of how many FLOPs and training tokens would be needed to train compute-optimal models. The estimates for Approach 2 & 3 are similar (shown in Section D.3) . | Parameters | FLOPs | FLOPs (in Gopher unit) | Tokens | |--------------|----------|--------------------------------|----------------| | 400 Million | 1.92e+19 | 1 29 968 | 8.0 Billion | | 1 Billion | 1.21e+20 | 1 4 761 | 20.2 Billion | | 10 Billion | 1.23e+22 | 1 46 | 205.1 Billion | | 67 Billion | 5.76e+23 | 1 | 1.5 Trillion | | 175 Billion | 3.85e+24 | 6 7 | 3.7 Trillion | | 280 Billion | 9.9e+24 | 17 2 | 5.9 Trillion | | 520 Billion | 3.43e+25 | 59 5 | 11.0 Trillion | | 1 Trillion | 1.27e+26 | 221 3 | 21.2 Trillion | | 10 Trillion | 1.3e+28 | 22515 9 | 216.2 Trillion | large language models are considerably over-sized, given their respective compute budgets, as shown in Figure 1. For example, we find that a 175 billion parameter model should be trained with a compute budget of 4 41 10 24 FLOPs and on over 4.2 trillion tokens. A 280 billion Gopher -like model is the optimal model to train given a compute budget of approximately 10 25 FLOPs and should be trained on 6.8 trillion tokens. Unless one has a compute budget of 10 26 FLOPs (over 250 the compute used to train Gopher ), a 1 trillion parameter model is unlikely to be the optimal model to train. Furthermore, the amount of training data that is projected to be needed is far beyond what is currently used to train large models, and underscores the importance of dataset collection in addition to engineering improvements that allow for model scale. While there is significant uncertainty extrapolating out many orders of magnitude, our analysis clearly suggests that given the training compute budget for many current LLMs, smaller models should have been trained on more tokens to achieve the most performant model. In Appendix C, we reproduce the IsoFLOP analysis on two additional datasets: C4 (Raffel et al., 2020a) and GitHub code (Rae et al., 2021). In both cases we reach the similar conclusion that model size and number of training tokens should be scaled in equal proportions. ## 4. Chinchilla Based on our analysis in Section 3, the optimal model size for the Gopher compute budget is somewhere between 40 and 70 billion parameters. We test this hypothesis by training a model on the larger end of this range-70B parameters-for 1.4T tokens, due to both dataset and computational efficiency considerations. In this section we compare this model, which we call Chinchilla , to Gopher and other LLMs. Both Chinchilla and Gopher have been trained for the same number of FLOPs but differ in the size of the model and the number of training tokens. While pre-training a large language model has a considerable compute cost, downstream finetuning and inference also make up substantial compute usage (Rae et al., 2021). Due to being 4 smaller than Gopher , both the memory footprint and inference cost of Chinchilla are also smaller. ## 4.1. Model and training details The full set of hyperparameters used to train Chinchilla are given in Table 4. Chinchilla uses the same model architecture and training setup as Gopher with the exception of the differences listed below. - We train Chinchilla on MassiveText (the same dataset as Gopher ) but use a slightly different subset distribution (shown in Table A1) to account for the increased number of training tokens. - We train Chinchilla with a slightly modified SentencePiece (Kudo and Richardson, 2018) tokenizer that does not apply NFKC normalisation. The vocabulary is very similar- 94.15% of tokens are the same as those used for training Gopher . We find that this particularly helps with the representation of mathematics and chemistry, for example. - We use AdamW (Loshchilov and Hutter, 2019) for Chinchilla rather than Adam (Kingma and Ba, 2014) as this improves the language modelling loss and the downstream task performance after finetuning. 8 - Whilst the forward and backward pass are computed in bfloat16 , we store a float32 copy of the weights in the distributed optimiser state (Rajbhandari et al., 2020). See Lessons Learned from Rae et al. (2021) for additional details. In Appendix G we show the impact of the various optimiser related changes between Chinchilla and Gopher . All models in this analysis have been trained on TPUv3/TPUv4 (Jouppi et al., 2017) with JAX (Bradbury et al., 2018) and Haiku (Hennigan et al., 2020). We include a Chinchilla model card (Mitchell et al., 2019) in Table A8. | Model | Layers | Number Heads | Key/Value Size | d model | Max LR | Batch Size | |----------------|----------|----------------|------------------|-----------|--------------------------|--------------| | Gopher 280B | 80 | 128 | 128 | 16,384 | 4 10 5 | 3M ! 6M | | Chinchilla 70B | 80 | 64 | 128 | 8,192 | 1 10 4 | 1.5M ! 3M | Table 4 j Chinchilla architecture details. We list the number of layers, the key/value size, the bottleneck activation size d model , the maximum learning rate, and the training batch size (# tokens). The feed-forward size is always set to 4 dmodel . Note that we double the batch size midway through training for both Chinchilla and Gopher . 8 Interestingly , a model trained with AdamW only passes the training performance of a model trained with Adam around 80% of the way through the cosine cycle, though the ending performance is notably better- see Figure A7 Table 5 j All evaluation tasks. We evaluate Chinchilla on a collection of language modelling along with downstream tasks. We evaluate on largely the same tasks as in Rae et al. (2021), to allow for direct comparison. | | # Tasks | Examples | |-----------------------|-----------|---------------------------------------------------------------------------------------------| | Language Modelling | 20 | WikiText-103, The Pile: PG-19, arXiv, FreeLaw, | | Reading Comprehension | 3 | RACE-m, RACE-h, LAMBADA | | Question Answering | 3 | Natural Questions, TriviaQA, TruthfulQA | | Common Sense | 5 | HellaSwag, Winogrande, PIQA, SIQA, BoolQ | | MMLU | 57 | High School Chemistry, Astronomy, Clinical Knowledge, | | BIG-bench | 62 | Causal Judgement, Epistemic Reasoning, Temporal Sequences, | ## 4.2. Results We perform an extensive evaluation of Chinchilla , comparing against various large language models. We evaluate on a large subset of the tasks presented in Rae et al. (2021), shown in Table 5. As the focus of this work is on optimal model scaling, we included a large representative subset, and introduce a few new evaluations to allow for better comparison to other existing large models. The evaluation details for all tasks are the same as described in Rae et al. (2021). ## 4.2.1. Language modelling Figure 5 j Pile Evaluation. For the different evaluation sets in The Pile (Gao et al., 2020), we show the bits-per-byte (bpb) improvement (decrease) of Chinchilla compared to Gopher . On all subsets, Chinchilla outperforms Gopher . <details> <summary>Image 6 Details</summary>  ### Visual Description ## Bar Chart: Decrease in bpp compared to Gopher ### Overview The chart visualizes the reduction in bits per pixel (bpp) for various datasets or applications when compared to a baseline system called "Gopher." The y-axis represents the magnitude of bpp decrease, while the x-axis lists 19 distinct categories. The tallest bars indicate the largest efficiency gains relative to Gopher. ### Components/Axes - **X-Axis (Categories)**: - pubmed_abstracts, nih_exporter, pubmed_backgrounds, pubmed_central, pile_cc, bookcorpus2, stackexchange, opensubtitles, openwebtext2, hackernetworks, dm_mathematics, arxiv, freelaw, books3, philpapers, github, ubuntu_irc, europarl, gutenberg_pg_19. - **Y-Axis (Decrease in bpp)**: - Scale ranges from 0.00 to 0.10 in increments of 0.02. - **Legend**: Not explicitly present. All bars are uniformly blue, suggesting a single data series. ### Detailed Analysis - **pubmed_abstracts**: ~0.02 (shortest bar). - **nih_exporter**: ~0.02. - **pubmed_backgrounds**: ~0.02. - **pubmed_central**: ~0.02. - **pile_cc**: ~0.025. - **bookcorpus2**: ~0.025. - **stackexchange**: ~0.025. - **opensubtitles**: ~0.025. - **openwebtext2**: ~0.025. - **hackernetworks**: ~0.025. - **dm_mathematics**: ~0.025. - **arxiv**: ~0.03. - **freelaw**: ~0.03. - **books3**: ~0.03. - **philpapers**: ~0.03. - **github**: ~0.03. - **ubuntu_irc**: ~0.03. - **europarl**: ~0.10 (tallest bar). - **gutenberg_pg_19**: ~0.10 (tallest bar). ### Key Observations 1. **Outliers**: "europarl" and "gutenberg_pg_19" show the largest bpp reductions (~0.10), significantly outperforming other categories. 2. **Clustered Values**: Most categories (14/19) fall within a narrow range of 0.02–0.03, indicating moderate efficiency gains. 3. **Uniformity**: No category exceeds 0.10, and all values are positive, suggesting consistent performance improvements over Gopher. ### Interpretation The data suggests that "europarl" and "gutenberg_pg_19" are the most efficient datasets relative to Gopher, likely due to their text-heavy nature (e.g., parliamentary proceedings and book corpora). The majority of categories exhibit modest reductions, implying that Gopher performs comparably well across diverse datasets. The uniformity of values (except for the top two) highlights Gopher's robustness, while the outliers underscore the impact of dataset characteristics on compression efficiency. This could inform optimization strategies for specific use cases, such as prioritizing text-based data for maximum bpp reduction. </details> Chinchilla significantly outperforms Gopher on all evaluation subsets of The Pile (Gao et al., 2020), as shown in Figure 5. Compared to Jurassic-1 (178B) Lieber et al. (2021), Chinchilla is more performant on all but two subsets-dm\_mathematics and ubuntu\_irc - see Table A5 for a raw bits-per-byte comparison. On Wikitext103 (Merity et al., 2017), Chinchilla achieves a perplexity of 7.16 compared to 7.75 for Gopher . Some caution is needed when comparing Chinchilla with Gopher on these language modelling benchmarks as Chinchilla is trained on 4 more data than Gopher and thus train/test set leakage may artificially enhance the results. We thus place more emphasis on other Table 6 j Massive Multitask Language Understanding (MMLU). We report the average 5-shot accuracy over 57 tasks with model and human accuracy comparisons taken from Hendrycks et al. (2020). We also include the average prediction for state of the art accuracy in June 2022/2023 made by 73 competitive human forecasters in Steinhardt (2021). | Random | 25.0% | |----------------------------------|---------| | Average human rater | 34.5% | | GPT-3 5-shot | 43.9% | | Gopher 5-shot | 60.0% | | Chinchilla 5-shot | 67.6% | | Average human expert performance | 89.8% | | June 2022 Forecast | 57.1% | | June 2023 Forecast | 63.4% | tasks for which leakage is less of a concern, such as MMLU (Hendrycks et al., 2020) and BIG-bench (BIG-bench collaboration, 2021) along with various closed-book question answering and common sense analyses. ## 4.2.2. MMLU The Massive Multitask Language Understanding (MMLU) benchmark (Hendrycks et al., 2020) consists of a range of exam-like questions on academic subjects. In Table 6, we report Chinchilla 's average 5-shot performance on MMLU (the full breakdown of results is shown in Table A6). On this benchmark, Chinchilla significantly outperforms Gopher despite being much smaller, with an average accuracy of 67.6% (improving upon Gopher by 7.6%). Remarkably, Chinchilla even outperforms the expert forecast for June 2023 of 63.4% accuracy (see Table 6) (Steinhardt, 2021). Furthermore, Chinchilla achieves greater than 90% accuracy on 4 different individual tasks-high\_school\_gov\_and\_politics, international\_law, sociology , and us\_foreign\_policy . To our knowledge, no other model has achieved greater than 90% accuracy on a subset. In Figure 6, we show a comparison to Gopher broken down by task. Overall, we find that Chinchilla improves performance on the vast majority of tasks. On four tasks ( college\_mathematics, econometrics, moral\_scenarios , and formal\_logic ) Chinchilla underperforms Gopher , and there is no change in performance on two tasks. ## 4.2.3. Reading comprehension On the final word prediction dataset LAMBADA (Paperno et al., 2016), Chinchilla achieves 77.4% accuracy, compared to 74.5% accuracy from Gopher and 76.6% from MT-NLG 530B (see Table 7). On RACE-h and RACE-m (Lai et al., 2017), Chinchilla greatly outperforms Gopher , improving accuracy by more than 10% in both cases-see Table 7. ## 4.2.4. BIG-bench We analysed Chinchilla on the same set of BIG-bench tasks (BIG-bench collaboration, 2021) reported in Rae et al. (2021). Similar to what we observed in MMLU, Chinchilla outperforms Gopher on the vast majority of tasks (see Figure 7). We find that Chinchilla improves the average performance by 10.7%, reaching an accuracy of 65.1% versus 54.4% for Gopher . Of the 62 tasks we consider, Chinchilla performs worse than Gopher on only four-crash\_blossom, dark\_humor\_detection, Figure 6 j MMLU results compared to Gopher We find that Chinchilla outperforms Gopher by 7.6% on average (see Table 6) in addition to performing better on 51/57 individual tasks, the same on 2/57, and worse on only 4/57 tasks. <details> <summary>Image 7 Details</summary>  ### Visual Description ## Bar Chart: Relative Improvement over Gopher ### Overview The chart visualizes the "Relative Improvement over Gopher" across a wide range of academic, professional, and conceptual domains. The x-axis lists categories (e.g., "college_mathematics," "high_school_psychology," "conceptual_physics"), while the y-axis represents improvement values ranging from -15 to +35. Bars are colored blue, with a legend confirming the color corresponds to "Relative Improvement over Gopher." ### Components/Axes - **X-axis**: Categories (e.g., "college_mathematics," "moral_scenarios," "high_school_psychology," "conceptual_physics"). - **Y-axis**: "Relative Improvement over Gopher" (values from -15 to +35). - **Legend**: Blue color represents "Relative Improvement over Gopher." - **Title**: "Relative Improvement over Gopher" (positioned at the top). ### Detailed Analysis - **Negative Values**: - **college_mathematics**: ~-15 - **moral_scenarios**: ~-12 - **econometrics**: ~-10 - **formal_logic**: ~-8 - **medical_genetics**: ~-5 - **machine_learning**: ~-3 - **public_relations**: ~-2 - **global_facts**: ~-1 - **business_ethics**: ~0 - **electrical_engineering**: ~1 - **college_computer_science**: ~2 - **world_religions**: ~3 - **high_school_us_history**: ~4 - **high_school_psychology**: ~5 - **high_school_computer_science**: ~6 - **marketing**: ~7 - **high_school_physics**: ~8 - **high_school_macroeconomics**: ~9 - **sociology**: ~10 - **high_school_government_and_politics**: ~11 - **nutrition**: ~12 - **high_school_european_history**: ~13 - **astronomy**: ~14 - **logical_fallacies**: ~15 - **professional_psychology**: ~16 - **jurisprudence**: ~17 - **clinical_knowledge**: ~18 - **high_school_geography**: ~19 - **high_school_biology**: ~20 - **college_chemistry**: ~21 - **college_chemistry**: ~22 - **high_school_world_history**: ~23 - **us_foreign_policy**: ~24 - **virology**: ~25 - **philosophy**: ~26 - **moral_disputes**: ~27 - **human_aging**: ~28 - **computer_security**: ~29 - **security_studies**: ~30 - **international_law**: ~31 - **high_school_microeconomics**: ~32 - **high_school_statistics**: ~33 - **professional_accounting**: ~34 - **professional_medicine**: ~35 - **prehistory**: ~36 - **high_school_chemistry**: ~37 - **elementary_mathematics**: ~38 - **abstract_algebra**: ~39 - **anatomy**: ~40 - **professional_law**: ~41 - **human_sexology**: ~42 - **college_physics**: ~43 - **college_mathematics**: ~44 - **conceptual_physics**: ~45 ### Key Observations 1. **Negative Improvements**: Early categories (e.g., "college_mathematics," "moral_scenarios") show negative values, indicating worse performance compared to Gopher. 2. **Positive Trends**: Most categories transition to positive values, with "conceptual_physics" achieving the highest improvement (~45). 3. **Professional Fields**: Categories like "professional_medicine," "professional_law," and "conceptual_physics" dominate the upper range. 4. **Gradual Increase**: The improvement values generally rise from left to right, with a steep upward trend in the latter half of the chart. ### Interpretation The data suggests that **professional and conceptual domains** (e.g., "conceptual_physics," "professional_medicine") demonstrate the greatest relative improvement over Gopher, while **early academic subjects** (e.g., "college_mathematics") lag behind. The negative values for some college-level topics may indicate limitations in Gopher's performance in those areas. The chart highlights a clear correlation between domain complexity and improvement, with higher-order or specialized fields outperforming foundational subjects. This could reflect Gopher's strengths in handling advanced or interdisciplinary topics compared to more basic academic areas. </details> Table 7 j Reading comprehension. On RACE-h and RACE-m (Lai et al., 2017), Chinchilla considerably improves performance over Gopher . Note that GPT-3 and MT-NLG 530B use a different prompt format than we do on RACE-h/m, so results are not comparable to Gopher and Chinchilla . On LAMBADA (Paperno et al., 2016), Chinchilla outperforms both Gopher and MT-NLG 530B. | | Chinchilla | Gopher | GPT-3 | MT-NLG 530B | |-------------------|--------------|----------|---------|---------------| | LAMBADA Zero-Shot | 77.4 | 74.5 | 76.2 | 76.6 | | RACE-m Few-Shot | 86.8 | 75.1 | 58.1 | - | | RACE-h Few-Shot | 82.3 | 71.6 | 46.8 | 47.9 | mathematical\_induction and logical\_args . Full accuracy results for Chinchilla can be found in Table A7. ## 4.2.5. Common sense We evaluate Chinchilla on various common sense benchmarks: PIQA (Bisk et al., 2020), SIQA (Sap et al., 2019), Winogrande (Sakaguchi et al., 2020), HellaSwag (Zellers et al., 2019), and BoolQ (Clark et al., 2019). We find that Chinchilla outperforms both Gopher and GPT-3 on all tasks and outperforms MT-NLG 530B on all but one task-see Table 8. On TruthfulQA (Lin et al., 2021), Chinchilla reaches 43.6%, 58.5%, and 66.7% accuracy with 0-shot, 5-shot, and 10-shot respectively. In comparison, Gopher achieved only 29.5% 0-shot and 43.7% 10-shot accuracy. In stark contrast with the findings of Lin et al. (2021), the large improvements (14.1% in 0-shot accuracy) achieved by Chinchilla suggest that better modelling of the pre-training data alone can lead to substantial improvements on this benchmark. Figure 7 j BIG-bench results compared to Gopher Chinchilla out performs Gopher on all but four BIG-bench tasks considered. Full results are in Table A7. <details> <summary>Image 8 Details</summary>  ### Visual Description ## Bar Chart: Relative Improvement over Gopher ### Overview The chart displays a horizontal bar chart titled "Relative Improvement over Gopher," comparing performance metrics across 50+ categories. The y-axis represents "Relative Improvement over Gopher" with values ranging from -20 to 120. The x-axis lists categories such as "crash_blossom," "dark_humor_detection," and "analogical_similarity." Bars are colored blue (positive values) and orange (negative values), with a legend on the right. ### Components/Axes - **Title**: "Relative Improvement over Gopher" (top center). - **Y-Axis**: Labeled "Relative Improvement over Gopher," with increments of 20 from -20 to 120. - **X-Axis**: Categories listed in ascending order (left to right), including: - crash_blossom - dark_humor_detection - mathematical_induction - ... (50+ categories) - analogical_similarity - **Legend**: - Blue: Positive values (e.g., "100% improvement"). - Orange: Negative values (e.g., "-10% improvement"). ### Detailed Analysis - **Positive Bars (Blue)**: - Most categories show positive improvement, with values increasing from ~0 to ~120. - Early categories (e.g., "crash_blossom," "dark_humor_detection") have minimal or negative values. - Later categories (e.g., "analogical_similarity," "temporal_sequences") show the highest improvements, peaking at ~120. - **Negative Bars (Orange)**: - Only the first 5-10 categories have negative values, ranging from -10 to -20. - Example: "crash_blossom" at ~-15, "dark_humor_detection" at ~-10. ### Key Observations 1. **Dominance of Positive Improvements**: Over 90% of categories show positive relative improvement, with the majority exceeding 50. 2. **Gradual Increase**: Positive values rise steadily from left to right, with a sharp upward trend in the final 10-15 categories. 3. **Negative Outliers**: Only the first few categories (e.g., "crash_blossom," "dark_humor_detection") have negative values, suggesting Gopher outperforms in these areas. 4. **Highest Improvement**: "analogical_similarity" and "temporal_sequences" reach ~120, indicating significant performance gains. ### Interpretation The chart demonstrates that the system under analysis (likely a newer model or method) generally outperforms Gopher across most categories, with the largest improvements in complex or advanced tasks (e.g., "analogical_similarity," "temporal_sequences"). The negative values in early categories suggest Gopher may excel in simpler or more established tasks (e.g., "crash_blossom," "dark_humor_detection"). The trend implies that newer or more sophisticated features (e.g., "metaphor_bolean," "identity_odd_meta") are where the system gains the most advantage. The absence of negative values in later categories indicates consistent performance gains, possibly due to architectural or algorithmic improvements. ### Spatial Grounding - **Legend**: Right-aligned, with blue (positive) and orange (negative) bars. - **Bars**: Horizontally oriented, with blue bars dominating the right side and orange bars clustered on the left. - **Title**: Centered at the top of the chart. ### Content Details - **Negative Values**: - "crash_blossom": ~-15 - "dark_humor_detection": ~-10 - "mathematical_induction": ~-5 - **Positive Values**: - "analogical_similarity": ~120 - "temporal_sequences": ~100 - "metaphor_bolean": ~90 - "identity_odd_meta": ~80 ### Notable Anomalies - The sharp drop in negative values after the first 10 categories suggests a clear boundary between Gopher's strengths and the system's improvements. - The absence of negative values in the final 40 categories indicates no regression in performance for advanced tasks. This chart highlights the system's ability to outperform Gopher in most domains, particularly in complex or novel tasks, while maintaining competitive performance in simpler areas. </details> ## 4.2.6. Closed-book question answering Results on closed-book question answering benchmarks are reported in Table 9. On the Natural Questions dataset (Kwiatkowski et al., 2019), Chinchilla achieves new closed-book SOTA accuracies: 31.5% 5-shot and 35.5% 64-shot, compared to 21% and 28% respectively, for Gopher . On TriviaQA (Joshi et al., 2017) we show results for both the filtered (previously used in retrieval and open-book work) and unfiltered set (previously used in large language model evaluations). In both cases, Chinchilla substantially out performs Gopher . On the filtered version, Chinchilla lags behind the open book SOTA (Izacard and Grave, 2020) by only 7.9%. On the unfiltered set, Chinchilla outperforms GPT-3-see Table 9. ## 4.2.7. Gender bias and toxicity Large Language Models carry potential risks such as outputting offensive language, propagating social biases, and leaking private information (Bender et al., 2021; Weidinger et al., 2021). We expect Chinchilla to carry risks similar to Gopher because Chinchilla is trained on the same data, Table 8 j Zero-shot comparison on Common Sense benchmarks. We show a comparison between Chinchilla , Gopher , and MT-NLG 530B on various Common Sense benchmarks. We see that Chinchilla matches or outperforms Gopher and GPT-3 on all tasks. On all but one Chinchilla outperforms the much larger MT-NLG 530B model. | | Chinchilla | Gopher | GPT-3 | MT-NLG 530B | Supervised SOTA | |------------|--------------|----------|---------|---------------|-------------------| | HellaSWAG | 80.8% | 79.2% | 78.9% | 80.2% | 93.9% | | PIQA | 81.8% | 81.8% | 81.0% | 82.0% | 90.1% | | Winogrande | 74.9% | 70.1% | 70.2% | 73.0% | 91.3% | | SIQA | 51.3% | 50.6% | - | - | 83.2% | | BoolQ | 83.7 % | 79.3% | 60.5% | 78.2% | 91.4% | | | Method | Chinchilla | Gopher | GPT-3 | SOTA (open book) | |-----------------------------|-----------------------|-------------------|-------------------|---------------|--------------------| | Natural Questions (dev) | 0-shot 5-shot 64-shot | 16.6% 31.5% 35.5% | 10.1% 24.5% 28.2% | 14.6% - 29.9% | 54.4% | | TriviaQA (unfiltered, test) | 0-shot | 67.0% | 52.8% | 64.3% - | - | | | 5-shot 64-shot | 73.2% 72.3% | 63.6% | 71.2% | - | | | | 55.4% | 61.3% | | - | | TriviaQA (filtered, dev) | 0-shot | | 43.5% | - | 72.5% | | TriviaQA (filtered, dev) | 5-shot | 64.1% | 57.0% | - | 72.5% | | TriviaQA (filtered, dev) | 64-shot | 64.6% | 57.2% | - | 72.5% | Table 9 j Closed-book question answering. For Natural Questions (Kwiatkowski et al., 2019) and TriviaQA (Joshi et al., 2017), Chinchilla outperforms Gopher in all cases. On Natural Questions, Chinchilla outperforms GPT-3. On TriviaQA we show results on two different evaluation sets to allow for comparison to GPT-3 and to open book SOTA (FiD + Distillation (Izacard and Grave, 2020)). albeit with slightly different relative weights, and because it has a similar architecture. Here, we examine gender bias (particularly gender and occupation bias) and generation of toxic language. We select a few common evaluations to highlight potential issues, but stress that our evaluations are not comprehensive and much work remains to understand, evaluate, and mitigate risks in LLMs. Gender bias. As discussed in Rae et al. (2021), large language models reflect contemporary and historical discourse about different groups (such as gender groups) from their training dataset, and we expect the same to be true for Chinchilla . Here, we test if potential gender and occupation biases manifest in unfair outcomes on coreference resolutions, using the Winogender dataset (Rudinger et al., 2018) in a zero-shot setting. Winogender tests whether a model can correctly determine if a pronoun refers to different occupation words. An unbiased model would correctly predict which word the pronoun refers to regardless of pronoun gender. We follow the same setup as in Rae et al. (2021) (described further in Section H.3). As shown in Table 10, Chinchilla correctly resolves pronouns more frequently than Gopher across all groups. Interestingly, the performance increase is considerably smaller for male pronouns (increase of 3.2%) than for female or neutral pronouns (increases of 8.3% and 9.2% respectively). We also consider gotcha examples, in which the correct pronoun resolution contradicts gender stereotypes (determined by labor statistics). Again, we see that Chinchilla resolves pronouns more accurately than Gopher . When breaking up examples by male/female gender and gotcha / not gotcha , the largest improvement is on female gotcha examples (improvement of 10%). Thus, though Chinchilla uniformly overcomes gender stereotypes for more coreference examples than Gopher , the rate of improvement is higher for some pronouns than others, suggesting that the improvements conferred by using a more compute-optimal model can be uneven. Sample toxicity. Language models are capable of generating toxic language-including insults, hate speech, profanities and threats (Gehman et al., 2020; Rae et al., 2021). While toxicity is an umbrella term, and its evaluation in LMs comes with challenges (Welbl et al., 2021; Xu et al., 2021), automatic classifier scores can provide an indication for the levels of harmful text that a LM generates. Rae et al. (2021) found that improving language modelling loss by increasing the number of model parameters has only a negligible effect on toxic text generation (unprompted); here we analyze Table 10 j Winogender results. Left: Chinchilla consistently resolves pronouns better than Gopher . Right: Chinchilla performs better on examples which contradict gender stereotypes ( gotcha examples). However, difference in performance across groups suggests Chinchilla exhibits bias. | | Chinchilla | Gopher | | Chinchilla | Gopher | |---------|--------------|----------|-------------------|--------------|----------| | All | 78.3% | 71.4% | Male gotcha | 62.5% | 59.2% | | Male | 71.2% | 68.0% | Male not gotcha | 80.0% | 76.7% | | Female | 79.6% | 71.3% | Female gotcha | 76.7% | 66.7% | | Neutral | 84.2% | 75.0% | Female not gotcha | 82.5% | 75.8% | whether the same holds true for a lower LM loss achieved via more compute-optimal training. Similar to the protocol of Rae et al. (2021), we generate 25,000 unprompted samples from Chinchilla , and compare their PerspectiveAPI toxicity score distribution to that of Gopher -generated samples. Several summary statistics indicate an absence of major differences: the mean (median) toxicity score for Gopher is 0.081 (0.064), compared to 0.087 (0.066) for Chinchilla , and the 95 th percentile scores are 0.230 for Gopher , compared to 0.238 for Chinchilla . That is, the large majority of generated samples are classified as non-toxic, and the difference between the models is negligible. In line with prior findings (Rae et al., 2021), this suggests that toxicity levels in unconditional text generation are largely independent of the model quality (measured in language modelling loss), i.e. that better models of the training dataset are not necessarily more toxic. ## 5. Discussion & Conclusion The trend so far in large language model training has been to increase the model size, often without increasing the number of training tokens. The largest dense transformer, MT-NLG 530B, is now over 3 larger than GPT-3's 170 billion parameters from just two years ago. However, this model, as well as the majority of existing large models, have all been trained for a comparable number of tokens-around 300 billion. While the desire to train these mega-models has led to substantial engineering innovation, we hypothesize that the race to train larger and larger models is resulting in models that are substantially underperforming compared to what could be achieved with the same compute budget. We propose three predictive approaches towards optimally setting model size and training duration, based on the outcome of over 400 training runs. All three approaches predict that Gopher is substantially over-sized and estimate that for the same compute budget a smaller model trained on more data will perform better. We directly test this hypothesis by training Chinchilla , a 70B parameter model, and show that it outperforms Gopher and even larger models on nearly every measured evaluation task. Whilst our method allows us to make predictions on how to scale large models when given additional compute, there are several limitations. Due to the cost of training large models, we only have two comparable training runs at large scale ( Chinchilla and Gopher ), and we do not have additional tests at intermediate scales. Furthermore, we assume that the efficient computational frontier can be described by a power-law relationship between the compute budget, model size, and number of training tokens. However, we observe some concavity in log 𝑁𝑜𝑝𝑡 at high compute budgets (see Appendix E). This suggests that we may still be overestimating the optimal size of large models. Finally, the training runs for our analysis have all been trained on less than an epoch of data; future work may consider the multiple epoch regime. Despite these limitations, the comparison of Chinchilla to Gopher validates our performance predictions, that have thus enabled training a better (and more lightweight) model at the same compute budget. Though there has been significant recent work allowing larger and larger models to be trained, our analysis suggests an increased focus on dataset scaling is needed. Speculatively, we expect that scaling to larger and larger datasets is only beneficial when the data is high-quality. This calls for responsibly collecting larger datasets with a high focus on dataset quality. Larger datasets will require extra care to ensure train-test set overlap is properly accounted for, both in the language modelling loss but also with downstream tasks. Finally, training for trillions of tokens introduces many ethical and privacy concerns. Large datasets scraped from the web will contain toxic language, biases, and private information. With even larger datasets being used, the quantity (if not the frequency) of such information increases, which makes dataset introspection all the more important. Chinchilla does suffer from bias and toxicity but interestingly it seems less affected than Gopher . Better understanding how performance of large language models and toxicity interact is an important future research question. While we have applied our methodology towards the training of auto-regressive language models, we expect that there is a similar trade-off between model size and the amount of data in other modalities. As training large models is very expensive, choosing the optimal model size and training steps beforehand is essential. The methods we propose are easy to reproduce in new settings. ## 6. Acknowledgements We'd like to thank Jean-baptiste Alayrac, Kareem Ayoub, Chris Dyer, Nando de Freitas, Demis Hassabis, Geoffrey Irving, Koray Kavukcuoglu, Nate Kushman and Angeliki Lazaridou for useful comments on the manuscript. We'd like to thank Andy Brock, Irina Higgins, Michela Paganini, Francis Song, and other colleagues at DeepMind for helpful discussions. We are also very grateful to the JAX and XLA team for their support and assistance. ## References - M. Artetxe, S. Bhosale, N. Goyal, T. Mihaylov, M. Ott, S. Shleifer, X. V. Lin, J. Du, S. Iyer, R. Pasunuru, G. Anantharaman, X. Li, S. Chen, H. Akin, M. Baines, L. Martin, X. Zhou, P. S. Koura, B. O'Horo, J. Wang, L. Zettlemoyer, M. Diab, Z. Kozareva, and V. Stoyanov. Efficient Large Scale Language Modeling with Mixtures of Experts. arXiv:2112.10684, 2021. - E. M. Bender, T. Gebru, A. McMillan-Major, and S. Shmitchell. On the dangers of stochastic parrots: Can language models be too big? 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Which algorithmic choices matter at which batch sizes? insights from a noisy quadratic model. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL https: //proceedings.neurips.cc/paper/2019/file/e0eacd983971634327ae1819ea8b621 4-Paper.pdf . - B. Zoph, I. Bello, S. Kumar, N. Du, Y. Huang, J. Dean, N. Shazeer, and W. Fedus. Designing effective sparse expert models, 2022. ## A. Training dataset In Table A1 we show the training dataset makeup used for Chinchilla and all scaling runs. Note that both the MassiveWeb and Wikipedia subsets are both used for more than one epoch. | | Disk Size | Documents | Sampling proportion | Epochs in 1.4T tokens | |------------|-------------|-------------|-----------------------|-------------------------| | MassiveWeb | 1.9 TB | 604M | 45% (48%) | 1.24 | | Books | 2.1 TB | 4M | 30% (27%) | 0.75 | | C4 | 0.75 TB | 361M | 10% (10%) | 0.77 | | News | 2.7 TB | 1.1B | 10% (10%) | 0.21 | | GitHub | 3.1 TB | 142M | 4% (3%) | 0.13 | | Wikipedia | 0.001 TB | 6M | 1% (2%) | 3.4 | Table A1 j MassiveText data makeup. For each subset of MassiveText , we list its total disk size, the number of documents and the sampling proportion used during training-we use a slightly different distribution than in Rae et al. (2021) (shown in parenthesis). In the rightmost column show the number of epochs that are used in 1.4 trillion tokens. ## B. Optimal cosine cycle length One key assumption is made on the cosine cycle length and the corresponding learning rate drop (we use a 10 learning rate decay in line with Rae et al. (2021)). 9 We find that setting the cosine cycle length too much longer than the target number of training steps results in sub-optimally trained models, as shown in Figure A1. As a result, we assume that an optimally trained model will have the cosine cycle length correctly calibrated to the maximum number of steps, given the FLOP budget; we follow this rule in our main analysis. ## C. Consistency of scaling results across datasets Weshow scaling results from an IsoFLOP (Approach 2) analysis after training on two different datasets: C4 (Raffel et al., 2020b) and GitHub code (we show results with data from Rae et al. (2021)), results are shown in Table A2. For both set of experiments using subsets of MassiveText , we use the same tokenizer as the MassiveText experiments. Wefind that the scaling behaviour on these datasets is very similar to what we found on MassiveText , as shown in Figure A2 and Table A2. This suggests that our results are independent of the dataset as long as one does not train for more than one epoch. 9 We find the difference between decaying by 10 and decaying to 0.0 (over the same number of steps) to be small, though decaying by a factor of 10 to be slightly more performant. Decaying by less (5 ) is clearly worse. ## Appendix Figure A1 j Grid over cosine cycle length. We show 6 curves with the cosine cycle length set to 1, 1.1, 1.25, 1.5, 2, and 5 longer than the target number of training steps. When the cosine cycle length is too long, and the learning rate does not drop appropriately, then performance is impaired. We find that overestimating the number of training steps beyond 25% leads to clear drops in performance. We show results where we have set the number of training steps to two different values (top and bottom). <details> <summary>Image 9 Details</summary>  ### Visual Description ## Line Graphs: Training Dynamics Across Cosine Cycle Lengths ### Overview The image contains six line graphs arranged in a 2x3 grid, visualizing training dynamics (learning rate, training loss, and C4 loss) across different cosine cycle lengths. Each graph shares the same x-axis ("Million Sequences") but varies in y-axis metrics and x-axis range (0–8M in top row, 0–12.5M in bottom row). All lines exhibit decreasing trends, with steeper declines for longer cosine cycle lengths. --- ### Components/Axes 1. **X-Axes**: - Top row: "Million Sequences" (0–8M) - Bottom row: "Million Sequences" (0–12.5M) 2. **Y-Axes**: - Left column: "Learning Rate/Max LR" (0.0–1.0) - Middle column: "Training Loss" (2.7–3.0) - Right column: "C4 Loss" (2.8–3.2) 3. **Legends**: - Positioned in the top-right corner of each graph. - Labels: "Cosine Cycle Length" with multipliers (1.0x, 1.1x, 1.25x, 1.5x, 2.0x, 5.0x). - Colors: - 1.0x: Blue - 1.1x: Orange - 1.25x: Green - 1.5x: Red - 2.0x: Purple - 5.0x: Brown --- ### Detailed Analysis #### Left Column: Learning Rate/Max LR - **Trends**: All lines start near 1.0 and decrease sharply, then plateau. Longer cycles (e.g., 5.0x) drop faster. - 1.0x (blue): Gradual decline to ~0.6 at 8M sequences. - 5.0x (brown): Steepest drop to ~0.15 at 8M sequences. #### Middle Column: Training Loss - **Trends**: All lines start near 3.0 and decrease, with longer cycles achieving lower loss faster. - 1.0x (blue): Slow decline to ~2.85 at 8M sequences. - 5.0x (brown): Rapid drop to ~2.75 at 8M sequences. #### Right Column: C4 Loss - **Trends**: Similar to training loss but with higher initial values (~3.2). - 1.0x (blue): Declines to ~3.05 at 8M sequences. - 5.0x (brown): Drops to ~2.9 at 8M sequences. --- ### Key Observations 1. **Consistent Trends**: All metrics decrease monotonically with increasing sequences. 2. **Cycle Length Impact**: Longer cosine cycles (e.g., 5.0x) achieve faster convergence but plateau earlier. 3. **Divergence at Scale**: By 12.5M sequences (bottom row), longer cycles maintain lower values than shorter ones. 4. **Color Consistency**: Legend colors match line colors across all graphs (e.g., 5.0x is always brown). --- ### Interpretation - **Training Efficiency**: Longer cosine cycles (e.g., 5.0x) optimize learning rate schedules for faster initial progress but may over-optimize, leading to earlier plateaus. - **Loss Correlation**: C4 Loss and training loss trends align, suggesting C4 Loss is a reliable proxy for model performance. - **Practical Implications**: Choosing a cycle length depends on balancing speed (shorter cycles) and stability (longer cycles). The 5.0x cycle offers the fastest initial improvement but may require careful tuning to avoid premature convergence. All data points and trends are visually consistent with the legend, confirming accurate color-label alignment. </details> Figure A2 j C4 and GitHub IsoFLOP curves. Using the C4 dataset (Raffel et al., 2020b) and a GitHub dataset (Rae et al., 2021), we generate 4 IsoFLOP profiles and show the parameter and token count scaling, as in Figure 3. Scaling coefficients are shown in Table A2. <details> <summary>Image 10 Details</summary>  ### Visual Description ## Line Plots: Model Performance vs. Resource Usage ### Overview The image contains six line plots arranged in two rows and three columns, comparing model performance metrics (training loss, tokens, FLOPs) across different parameter sizes (100M to 30B). Each plot includes multiple data series distinguished by color-coded labels (le19, le20, le6e20, le21). The plots visualize relationships between model scale and efficiency, with diagonal reference lines indicating ideal linear scaling. ### Components/Axes 1. **Top Row (C4 Training Loss, Tokens, FLOPs):** - **X-axes:** - C4 Training Loss: Parameters (100M to 30B) - Tokens: FLOPs (10¹⁷ to 10²⁵) - FLOPs: FLOPs (10¹⁷ to 10²⁵) - **Y-axes:** - C4 Training Loss: Loss values (2.0 to 3.2) - Tokens: Token counts (1B to 10T) - FLOPs: FLOPs (10¹⁷ to 10²⁵) - **Legends:** Left-aligned for C4 Training Loss; top-aligned for Tokens and FLOPs. 2. **Bottom Row (GitHub Training Loss, Tokens, FLOPs):** - **X-axes:** Same as top row. - **Y-axes:** - GitHub Training Loss: Loss values (0.2 to 0.9) - Tokens: Token counts (1B to 10T) - FLOPs: FLOPs (10¹⁷ to 10²⁵) - **Legends:** Left-aligned for GitHub Training Loss; top-aligned for Tokens and FLOPs. ### Detailed Analysis 1. **C4 Training Loss (Top Left):** - **Trend:** U-shaped curve for all series. Loss decreases sharply as parameters increase from 100M to ~10B, then plateaus. - **Data Points:** - le19: 3.2 (100M) → 2.2 (30B) - le20: 2.8 (100M) → 2.1 (30B) - le6e20: 2.6 (100M) → 2.0 (30B) - le21: 2.4 (100M) → 1.9 (30B) - **Legend Match:** Purple (le19), blue (le20), teal (le6e20), green (le21). 2. **Tokens (Top Middle):** - **Trend:** Linear relationship between FLOPs and tokens. Data points align closely with the diagonal reference line (slope ~1.3T FLOPs/token). - **Data Points:** - 10¹⁷ FLOPs: 1B tokens - 10¹⁹ FLOPs: 10B tokens - 10²¹ FLOPs: 100B tokens - 10²³ FLOPs: 1T tokens - **Legend Match:** Black data points. 3. **FLOPs (Top Right):** - **Trend:** Linear scaling with FLOPs increasing proportionally to parameter size. Data points align with the diagonal reference line (slope ~1.3T FLOPs/token). - **Data Points:** - 10¹⁷ FLOPs: 1B tokens - 10¹⁹ FLOPs: 10B tokens - 10²¹ FLOPs: 100B tokens - 10²³ FLOPs: 1T tokens - **Legend Match:** Black data points. 4. **GitHub Training Loss (Bottom Left):** - **Trend:** U-shaped curve similar to C4 but with lower absolute values. Loss decreases from ~0.8 (100M) to ~0.3 (30B). - **Data Points:** - le19: 0.8 (100M) → 0.3 (30B) - le20: 0.7 (100M) → 0.25 (30B) - le6e20: 0.6 (100M) → 0.2 (30B) - le21: 0.5 (100M) → 0.15 (30B) - **Legend Match:** Purple (le19), blue (le20), teal (le6e20), green (le21). 5. **Tokens (Bottom Middle):** - **Trend:** Identical to top middle plot. Linear scaling with FLOPs. - **Data Points:** Same as top middle plot. 6. **FLOPs (Bottom Right):** - **Trend:** Identical to top right plot. Linear scaling with FLOPs. - **Data Points:** Same as top right plot. ### Key Observations 1. **Training Loss vs. Parameters:** - All models show diminishing returns in training loss as parameters increase beyond ~10B. - GitHub Training Loss is consistently 50-70% lower than C4 Training Loss across all parameter sizes. 2. **Resource Scaling:** - Tokens and FLOPs scale linearly with parameter size, following the diagonal reference lines. - No significant deviations from linearity observed in any plot. 3. **Efficiency Gaps:** - le21 (green) achieves the lowest training loss but requires the same FLOPs as le19 (purple) at 30B parameters. - le6e20 (teal) balances moderate loss (2.0 C4) with mid-range FLOPs (10²¹). ### Interpretation The data demonstrates a clear trade-off between model scale and efficiency: - **Performance:** Larger models (le21, le6e20) achieve better training loss but require exponentially more computational resources (FLOPs). - **Efficiency:** GitHub Training Loss suggests potential optimizations in model architecture or training methodology compared to C4. - **Scalability Limits:** The plateau in training loss beyond 10B parameters implies diminishing returns on model size for this task. Notable anomalies include the consistent linear scaling of FLOPs/tokens despite varying model architectures, suggesting a universal relationship between compute and model capacity. The GitHub loss curves indicate a possible architectural advantage over C4, warranting further investigation into training techniques or model design. </details> | Approach | Coef. 𝑎 where 𝑁 𝑜𝑝𝑡 / 𝐶 𝑎 | Coef. 𝑏 where 𝐷 𝑜𝑝𝑡 / 𝐶 𝑏 | |----------------------|-----------------------------|-----------------------------| | C4 | 0.5 | 0.5 | | GitHub | 0.53 | 0.47 | | Kaplan et al. (2020) | 0.73 | 0.27 | Table A2 j Estimated parameter and data scaling with increased training compute on two alternate datasets. The listed values are the exponents, 𝑎 and 𝑏 , on the relationship 𝑁𝑜𝑝𝑡 / 𝐶 𝑎 and 𝐷𝑜𝑝𝑡 / 𝐶 𝑏 . Using IsoFLOP profiles, we estimate the scaling on two different datasets. ## D. Details on the scaling analyses ## D.1. Approach 1: Fixing model sizes and varying training sequences We use a maximum learning rate of 2 10 4 for the smallest models and 1 25 10 4 for the largest models. In all cases, the learning rate drops by a factor of 10 during training, using a cosine schedule. We make the assumption that the cosine cycle length should be approximately matched to the number of training steps. We find that when the cosine cycle overshoots the number of training steps by more than 25%, performance is noticeably degraded-see Figure A1. 10 We use Gaussian smoothing with a window length of 10 steps to smooth the training curve. ## D.2. Approach 3: Parametric fitting of the loss In this section, we first show how Equation (2) can be derived. We repeat the equation below for clarity , <!-- formula-not-decoded --> based on a decomposition of the expected risk between a function approximation term and an optimisation suboptimality term. We then give details on the optimisation procedure for fitting the parameters. Loss decomposition. Formally, we consider the task of predicting the next token 𝑦 2 Y based on the previous tokens in a sequence 𝑥 2 Y 𝑠 , with 𝑠 varying from 0 to 𝑠 max-the maximum sequence length. We consider a distribution 𝑃 2 D'X Y' of tokens in Y and their past in X . A predictor 𝑓 : X ! D'Y' computes the probability of each token given the past sequence. The Bayes classifier, 𝑓 ★ , minimizes the cross-entropy of 𝑓 ' 𝑥 ' with the observed tokens 𝑦 , with expectation taken on the whole data distribution. We let 𝐿 be the expected risk <!-- formula-not-decoded --> The set of all transformers of size 𝑁 , that we denote H 𝑁 , forms a subset of all functions that map sequences to distributions of tokens X ! D'Y' . Fitting a transformer of size 𝑁 on the expected risk 𝐿 ' 𝑓 ' amounts to minimizing such risk on a restricted functional space <!-- formula-not-decoded --> When we observe a dataset ' 𝑥 𝑖 𝑦 𝑖 ' 𝑖 𝑖 2» 1 𝐷 … of size 𝐷 , we do not have access to 𝔼 𝑃 , but instead to the empirical expectation ˆ 𝔼 𝐷 over the empirical distribution ˆ 𝑃𝐷 . What happens when we are given 𝐷 10 This further emphasises the point of not only determining model size, but also training length before training begins. datapoints that we can only see once, and when we constrain the size of the hypothesis space to be 𝑁 -dimensional ? We are making steps toward minimizing the empirical risk within a finite-dimensional functional space H 𝑁 : <!-- formula-not-decoded --> We are never able to obtain ˆ 𝑓 𝑁 𝐷 as we typically perform a single epoch over the dataset of size 𝐷 . Instead, be obtain ¯ 𝑓 𝑁 𝐷 , which is the result of applying a certain number of gradient steps based on the 𝐷 datapoints-the number of steps to perform depends on the gradient batch size, for which we use well-tested heuristics. Using the Bayes-classifier 𝑓 ★ , the expected-risk minimizer 𝑓 𝑁 and the 'single-epoch empirical-risk minimizer' ¯ 𝑓 𝑁 𝐷 , we can finally decompose the loss 𝐿 ' 𝑁 𝐷 ' into <!-- formula-not-decoded --> The loss comprises three terms: the Bayes risk, i.e. the minimal loss achievable for next-token prediction on the full distribution 𝑃 , a.k.a the 'entropy of natural text.'; a functional approximation term that depends on the size of the hypothesis space; finally, a stochastic approximation term that captures the suboptimality of minimizing ˆ 𝐿 𝐷 instead of 𝐿 , and of making a single epoch on the provided dataset. Expected forms of the loss terms. In the decomposition (9), the second term depends entirely on the number of parameters 𝑁 that defines the size of the functional approximation space. On the set of two-layer neural networks , it is expected to be proportional to 1 𝑁 1 2 (Siegel and Xu, 2020). Finally, given that it corresponds to early stopping in stochastic first order methods, the third term should scale as the convergence rate of these methods, which is lower-bounded by 1 𝐷 1 2 (Robbins and Monro, 1951) (and may attain the bound). This convergence rate is expected to be dimension free (see e.g. Bubeck, 2015, for a review) and depends only on the loss smoothness; hence we assume that the second term only depends on 𝐷 in (2). Empirically, we find after fitting (2) that <!-- formula-not-decoded --> with 𝐸 = 1 69, 𝐴 = 406 4, 𝐵 = 410 7. We note that the parameter/data coefficients are both lower than 1 2 ; this is expected for the data-efficiency coefficient (but far from the known lower-bound). Future models and training approaches should endeavor to increase these coefficients. Fitting the decomposition to data. We effectively minimize the following problem <!-- formula-not-decoded --> where 𝐿𝑆𝐸 is the log-sum-exp operator. We then set 𝐴 𝐵 𝐸 = exp ' 𝑎 ' exp ' 𝑏 ' exp ' 𝑒 ' . We use the LBFGS algorithm to find local minima of the objective above, started on a grid of initialisation given by: 𝛼 2 f 0 0 5 2 g , 𝛽 2 f 0 0 5 2 g , 𝑒 2 f 1 5 1 g , 𝑎 2 f 0 5 25 g , and 𝑏 2 f 0 5 25 g . We find that the optimal initialisation is not on the boundary of our initialisation sweep. We use 𝛿 = 10 3 for the Huber loss. We find that using larger values of 𝛿 pushes the model to overfit the small compute regime and poorly predict held-out data from larger runs. We find that using a 𝛿 smaller than 10 3 does not impact the resulting predictions. ## D.3. Predicted compute optimal frontier for all three methods For Approaches 2 and 3, we show the estimated model size and number of training tokens for a variety of compute budgets in Table A3. We plot the predicted number of tokens and parameters for a variety of FLOP budgets for the three methods in Figure A3. Table A3 j Estimated optimal training FLOPs and training tokens for various model sizes. Analogous to Table 3, we show the model size/token count projections from Approaches 2 and 3 for various compute budgets. | | Approach 2 | Approach 2 | Approach 3 | Approach 3 | |-------------|--------------|----------------|--------------|-----------------| | Parameters | FLOPs | Tokens | FLOPs | Tokens | | 400 Million | 1.84e+19 | 7.7 Billion | 2.21e+19 | 9.2 Billion | | 1 Billion | 1.20e+20 | 20.0 Billion | 1.62e+20 | 27.1 Billion | | 10 Billion | 1.32e+22 | 219.5 Billion | 2.46e+22 | 410.1 Billion | | 67 Billion | 6.88e+23 | 1.7 Trillion | 1.71e+24 | 4.1 Trillion | | 175 Billion | 4.54e+24 | 4.3 Trillion | 1.26e+24 | 12.0 Trillion | | 280 Billion | 1.18e+25 | 7.1 Trillion | 3.52e+25 | 20.1 Trillion | | 520 Billion | 4.19e+25 | 13.4 Trillion | 1.36e+26 | 43.5 Trillion | | 1 Trillion | 1.59e+26 | 26.5 Trillion | 5.65e+26 | 94.1 Trillion | | 10 Trillion | 1.75e+28 | 292.0 Trillion | 8.55e+28 | 1425.5 Trillion | . Figure A3 j Optimal number of tokens and parameters for a training FLOP budget. For a fixed FLOP budget, we show the optimal number of tokens and parameters as predicted by Approaches 1, 2, and 3. For an alternate representation, see Figure 1. <details> <summary>Image 11 Details</summary>  ### Visual Description ## Line Chart: Parameter Scaling vs. Token Count ### Overview The chart compares the scaling of parameters (y-axis, logarithmic scale) against token counts (x-axis, logarithmic scale) for various AI model architectures and training approaches. Data points are labeled with parameter counts (e.g., 1e+18, 1e+26) and colored according to their associated approach/model. ### Components/Axes - **X-axis (Tokens)**: Logarithmic scale from 10¹⁰ to 10¹³. - **Y-axis (Parameters)**: Logarithmic scale from 10⁸ to 10¹². - **Legend**: Located in the top-left corner, mapping colors/markers to: - Approach 1 (blue line) - Approach 2 (red line) - Approach 3 (orange line) - Gopher (yellow star) - GPT-3 (red star) - Chinchilla (green star) - Megatron-Turing NLG (purple star) ### Detailed Analysis - **Approach 1 (blue line)**: Starts at (1e+18, 1e+18) and ends at (1e+26, 1e+12), showing a steep upward trend. - **Approach 2 (red line)**: Mirrors Approach 1 closely, with data points at (1e+18, 1e+18), (1e+20, 1e+9), (1e+22, 1e+10), (1e+24, 1e+11), and (1e+26, 1e+12). - **Approach 3 (orange line)**: Begins at (1e+18, 1e+18) and ends at (1e+26, 1e+11), with intermediate points at (1e+20, 1e+9), (1e+22, 1e+10), and (1e+24, 1e+11). - **Gopher (yellow star)**: Data points at (1e+18, 1e+18), (1e+20, 1e+9), (1e+22, 1e+10), (1e+24, 1e+11), and (1e+26, 1e+12). - **GPT-3 (red star)**: Single data point at (1e+23, 1e+11). - **Chinchilla (green star)**: Single data point at (1e+23, 1e+11). - **Megatron-Turing NLG (purple star)**: Single data point at (1e+23, 1e+11). ### Key Observations 1. **Approach 1 and 2** dominate the upper range of parameters, scaling aggressively with token count. 2. **Approach 3** and **Gopher** show similar trends but with slightly lower parameter counts at higher token values. 3. **GPT-3, Chinchilla, and Megatron-Turing NLG** converge at (1e+23 tokens, 1e+11 parameters), suggesting a common scaling threshold for these models. 4. All lines exhibit exponential growth, with steeper slopes for Approaches 1–3 compared to the model-specific markers. ### Interpretation The chart illustrates that training approaches (1–3) require significantly more parameters than existing models (Gopher, GPT-3, etc.) to achieve similar token counts. The convergence of GPT-3, Chinchilla, and Megatron-Turing NLG at 1e+23 tokens suggests a critical inflection point where parameter efficiency plateaus. Approach 1 and 2 may represent theoretical upper bounds for parameter scaling, while Approach 3 offers a more conservative trade-off. The data implies that optimizing parameter efficiency beyond 1e+23 tokens requires novel architectural innovations. </details> ## D.4. Small-scale comparison to Kaplan et al. (2020) For 10 21 FLOPs, we perform a head-to-head comparison of a model predicted by Approach 1 and that predicted by Kaplan et al. (2020). For both models, we use a batch size of 0.5M tokens and a maximum learning rate of 1 5 10 4 that decays by 10 . From Kaplan et al. (2020), we find that the optimal model size should be 4.68 billion parameters. From our approach 1, we estimate a 2.86 billion parameter model should be optimal. We train a 4.74 billion parameter and a 2.80 billion parameter transformer to test this hypothesis, using the same depth-to-width ratio to avoid as many confounding factors as possible. We find that our predicted model outperforms the model predicted by Kaplan et al. (2020) as shown in Figure A4. Figure A4 j Comparison to Kaplan et al. (2020) at 10 21 FLOPs. We train 2.80 and 4.74 billion parameter transformers predicted as optimal for 10 21 FLOPs by Approach 1 and by Kaplan et al. (2020). We find that our prediction results in a more performant model at the end of training. <details> <summary>Image 12 Details</summary>  ### Visual Description ## Line Graph: Training Loss vs. Sequences and FLOPs ### Overview The image contains two line graphs comparing the training loss of two approaches: "Kaplan et al. (2020)" (orange line) and "Approach 1" (blue line). The left graph plots training loss against the number of sequences (0 to 2×10⁷), while the right graph plots training loss against FLOPs (0 to 1×10²¹). Both graphs include a horizontal dashed line at a training loss of 2.3, likely representing a target or baseline value. --- ### Components/Axes - **Left Graph (Sequences)**: - **X-axis**: "Sequences" (logarithmic scale, 0 to 2×10⁷). - **Y-axis**: "Training Loss" (linear scale, 2.2 to 2.8). - **Legend**: Located in the top-right corner, with orange for "Kaplan et al. (2020)" and blue for "Approach 1". - **Dashed Line**: Horizontal line at y=2.3 (both graphs). - **Right Graph (FLOPs)**: - **X-axis**: "FLOPs ×10²¹" (linear scale, 0 to 1×10²¹). - **Y-axis**: "Training Loss" (linear scale, 2.2 to 2.8). - **Legend**: Same as the left graph (orange and blue). - **Dashed Line**: Horizontal line at y=2.3. --- ### Detailed Analysis #### Left Graph (Sequences) - **Kaplan et al. (2020)** (orange): - Starts at ~2.8 training loss at 0 sequences. - Decreases sharply to ~2.3 by ~1×10⁷ sequences. - Fluctuates slightly but stabilizes near 2.3. - **Approach 1** (blue): - Starts at ~2.7 training loss at 0 sequences. - Decreases gradually to ~2.3 by ~2×10⁷ sequences. - Shows minor oscillations but converges to 2.3. #### Right Graph (FLOPs) - **Kaplan et al. (2020)** (orange): - Starts at ~2.8 training loss at 0 FLOPs. - Decreases steeply to ~2.3 by ~0.8×10²¹ FLOPs. - Stabilizes near 2.3. - **Approach 1** (blue): - Starts at ~2.7 training loss at 0 FLOPs. - Decreases gradually to ~2.3 by ~1.0×10²¹ FLOPs. - Converges to 2.3 with minor fluctuations. --- ### Key Observations 1. **Convergence**: Both approaches converge to the same training loss (~2.3), but Kaplan et al. (2020) achieves this faster in both sequence and FLOP metrics. 2. **Efficiency**: Kaplan et al. (2020) demonstrates a steeper decline in training loss, suggesting higher efficiency per sequence or FLOP. 3. **Dashed Line**: The horizontal line at 2.3 likely represents a target or theoretical minimum training loss. 4. **Scalability**: The right graph (FLOPs) shows that Kaplan et al. (2020) requires fewer computational resources (FLOPs) to reach the target loss compared to Approach 1. --- ### Interpretation The data suggests that **Kaplan et al. (2020)** is more computationally efficient than Approach 1, achieving the same training loss with fewer sequences or FLOPs. The convergence to 2.3 indicates that both methods are effective, but Kaplan et al. (2020) outperforms in terms of resource utilization. The dashed line at 2.3 may represent a benchmark or optimization goal, highlighting the importance of balancing training loss and computational cost in model development. The steeper decline in Kaplan et al. (2020) could imply architectural or algorithmic improvements that reduce redundancy or improve gradient propagation. </details> ## E. Curvature of the FLOP-loss frontier We observe that as models increase there is a curvature in the FLOP-minimal loss frontier. This means that projections from very small models lead to different predictions than those from larger models. In Figure A5 we show linear fits using the first, middle, and final third of frontier-points. In this work, we do not take this in to account and we leave this as interesting future work as it suggests that even smaller models may be optimal for large FLOP budgets. ## F. FLOPs computation We include all training FLOPs, including those contributed to by the embedding matrices, in our analysis. Note that we also count embeddings matrices in the total parameter count. For large models the FLOP and parameter contribution of embedding matrices is small. We use a factor of 2 to describe the multiply accumulate cost. For the forward pass, we consider contributions from: - Embeddings - Attention (Single Layer) - -2 seq\_len vocab\_size d\_model - -Key, query and value projections : 2 3 seq\_len d\_model ' key\_size num\_heads ' Figure A5 j Training curve envelopes. We fit to the first third (orange), the middle third (green), and the last third (blue) of all points along the loss frontier. We plot only a subset of the points. <details> <summary>Image 13 Details</summary>  ### Visual Description ## Line Chart: Training Loss vs. FLOPS with Parameter Scaling ### Overview The chart visualizes the relationship between computational effort (FLOPS) and training loss across neural network models with varying parameter counts (75M to 10,000M). Lines represent different model sizes, with color gradients indicating parameter scale. Theoretical performance bounds are shown as dashed lines. ### Components/Axes - **X-axis (FLOPS)**: Logarithmic scale from 10¹⁷ to 10²². - **Y-axis (Training Loss)**: Linear scale from 2.0 to 6.0. - **Legend**: Right-aligned colorbar labeled "Million Parameters" (purple = 75M, yellow = 10,000M). - **Lines**: - Solid lines: Actual training loss trajectories. - Dashed lines: Theoretical performance bounds (lower loss). ### Detailed Analysis 1. **Parameter Groups**: - **75M (Purple)**: Starts at ~5.8 loss (10¹⁷ FLOPS), ends at ~2.3 (10²² FLOPS). Steepest decline. - **250M (Dark Red)**: Starts at ~5.6, ends at ~2.2. Slightly less steep than 75M. - **1,000M (Orange)**: Starts at ~5.4, ends at ~2.1. Gradual decline. - **5,000M (Light Orange)**: Starts at ~5.2, ends at ~2.05. Moderate slope. - **10,000M (Yellow)**: Starts at ~5.0, ends at ~2.0. Shallowest decline. 2. **Theoretical Bounds**: - **Dashed Blue Line**: Theoretical minimum loss (~2.0) across all FLOPS. - **Dashed Red Line**: Upper bound (~3.5) for low-resource models. 3. **Data Points**: - All lines converge near 10²² FLOPS, with loss approaching ~2.0–2.3. - Higher-parameter models (yellow/orange) achieve lower loss at equivalent FLOPS. ### Key Observations - **Performance Scaling**: Larger models (10,000M) achieve ~20% lower loss than smaller models (75M) at the same FLOPS. - **Efficiency Tradeoff**: Theoretical bounds suggest diminishing returns beyond ~10²² FLOPS. - **Color Consistency**: Legend colors match line hues precisely (e.g., yellow = 10,000M). ### Interpretation The data demonstrates that increasing model size improves training efficiency, but with diminishing returns. At 10²² FLOPS, even the largest models (10,000M) only reduce loss by ~0.5 compared to smaller models. Theoretical bounds imply that current architectures cannot fully exploit computational resources, suggesting opportunities for algorithmic improvements. The convergence of lines at high FLOPS highlights the importance of optimizing both model scale and computational investment. </details> - -Key @ Query logits : 2 seq\_len seq\_len ' key\_size num\_heads ' - Softmax @ query reductions : 2 seq\_len seq\_len ' key\_size num\_heads ' - Softmax : 3 num\_heads seq\_len seq\_len - -Final Linear : 2 seq\_len ' key\_size num\_heads ' d\_model - -2 seq\_len ' d\_model ffw\_size , d\_model ffw\_size ' - Dense Block (Single Layer) - Final Logits - Total forward pass FLOPs: embeddings , num\_layers ' total\_attention , dense\_block ' +logits - -2 seq\_len d\_model vocab\_size As in Kaplan et al. (2020) we assume that the backward pass has twice the FLOPs of the forward pass. We show a comparison between our calculation and that using the common approximation 𝐶 = 6 𝐷𝑁 (Kaplan et al., 2020) where 𝐶 is FLOPs, 𝐷 is the number of training tokens, and 𝑁 is the number of parameters in Table A4. We find the differences in FLOP calculation to be very small and they do not impact our analysis. Compared to the results presented in Rae et al. (2021), we use a slightly more Table A4 j FLOP comparison. For a variety of different model sizes, we show the ratio of the FLOPs that we compute per sequence to that using the 6 𝑁𝐷 approximation. | Parameters | num_layers | d_model | ffw_size | num_heads | k/q size | FLOP Ratio (Ours/6 𝑁𝐷 ) | |--------------|--------------|-----------|------------|-------------|------------|---------------------------| | 73M | 10 | 640 | 2560 | 10 | 64 | 1.03 | | 305M | 20 | 1024 | 4096 | 16 | 64 | 1.1 | | 552M | 24 | 1280 | 5120 | 10 | 128 | 1.08 | | 1.1B | 26 | 1792 | 7168 | 14 | 128 | 1.04 | | 1.6B | 28 | 2048 | 8192 | 16 | 128 | 1.03 | | 6.8B | 40 | 3584 | 14336 | 28 | 128 | 0.99 | accurate calculation giving a slightly different value (6 3 10 23 compared to 5 76 10 23 ). ## G. Other differences between Chinchilla and Gopher Beyond differences in model size and number of training tokens, there are some additional minor differences between Chinchilla and Gopher . Specifically , Gopher was trained with Adam (Kingma and Ba, 2014) whereas Chinchilla was trained with AdamW (Loshchilov and Hutter, 2019). Furthermore, as discussed in Lessons Learned in Rae et al. (2021), Chinchilla stored a higher-precision copy of the weights in the sharded optimiser state. We show comparisons of models trained with Adam and AdamW in Figure A6 and Figure A7. We find that, independent of the learning rate schedule, AdamW trained models outperform models trained with Adam. In Figure A6 we show a comparison of an 680 million parameter model trained Figure A6 j Comparison of other differences. Using an 680 million parameter model, we show a comparison between the setup used to train Gopher and Chinchilla -the change in optimiser and using a higher precision copy of the weights in the optimiser state. The setup used for Chinchilla (orange) clearly outperforms the setup used to train Gopher (green). <details> <summary>Image 14 Details</summary>  ### Visual Description ## Line Charts: Training Metrics Across Optimizers and Precision Settings ### Overview The image contains three line charts comparing training metrics (Training Loss, Wikitext103 Perplexity, and C4 Loss) across four optimizer configurations: Adam with High Precision, AdamW with High Precision, Adam without High Precision, and AdamW without High Precision. All charts plot metrics against "Million Sequences" (x-axis) up to 30 million sequences, showing decreasing trends as training progresses. --- ### Components/Axes 1. **Chart 1: Training Loss** - **Y-axis**: Training Loss (2.45–2.70) - **X-axis**: Million Sequences (0–30) - **Legend**: - Blue: Adam w/ High Precision - Orange: AdamW w/ High Precision - Green: Adam No High Precision - Red: AdamW No High Precision 2. **Chart 2: Wikitext103 Perplexity** - **Y-axis**: Perplexity (17–26) - **X-axis**: Million Sequences (0–30) - **Legend**: Same as Chart 1. 3. **Chart 3: C4 Loss** - **Y-axis**: C4 Loss (2.60–3.00) - **X-axis**: Million Sequences (0–30) - **Legend**: Same as Chart 1. --- ### Detailed Analysis #### Chart 1: Training Loss - **Trend**: All lines show a steep decline from ~2.70 at 0 million sequences to ~2.50 at 30 million sequences. - **Key Data Points**: - **Adam w/ High Precision (Blue)**: Starts at 2.70, ends at 2.50. Intermediate values: ~2.65 at 10M, ~2.55 at 20M. - **AdamW w/ High Precision (Orange)**: Similar trajectory to Blue, with slight divergence at 15M (~2.58 vs. 2.57 for Blue). - **Adam No High Precision (Green)**: Slightly higher loss than Blue/Orange, ending at ~2.52. - **AdamW No High Precision (Red)**: Closest to Green, ending at ~2.53. #### Chart 2: Wikitext103 Perplexity - **Trend**: All lines decrease from ~25.5 at 0M to ~19.5 at 30M. - **Key Data Points**: - **Adam w/ High Precision (Blue)**: Starts at 25.5, ends at 19.5. Intermediate: ~22.5 at 10M. - **AdamW w/ High Precision (Orange)**: Slightly better performance, ending at ~19.3. - **Adam No High Precision (Green)**: Ends at ~19.7. - **AdamW No High Precision (Red)**: Ends at ~19.6. #### Chart 3: C4 Loss - **Trend**: All lines decline from ~3.00 at 0M to ~2.65 at 30M. - **Key Data Points**: - **Adam w/ High Precision (Blue)**: Starts at 3.00, ends at 2.65. Intermediate: ~2.85 at 10M. - **AdamW w/ High Precision (Orange)**: Ends at ~2.63. - **Adam No High Precision (Green)**: Ends at ~2.67. - **AdamW No High Precision (Red)**: Ends at ~2.66. --- ### Key Observations 1. **Consistent Decline**: All configurations show improvement (lower loss/perplexity) as training progresses, indicating effective learning. 2. **High Precision Impact**: - High precision settings (Blue/Orange) generally outperform no-high-precision (Green/Red) in Training Loss and C4 Loss. - Wikitext103 Perplexity shows smaller differences between high/low precision. 3. **Adam vs. AdamW**: - AdamW configurations (Orange/Red) slightly outperform Adam (Blue/Green) in Training Loss and C4 Loss, but differences are minimal (<0.05). - Wikitext103 Perplexity shows AdamW w/ High Precision as the best performer. --- ### Interpretation The data suggests that: - **High Precision** improves training stability and final performance across all metrics, though the effect is more pronounced in Training Loss and C4 Loss. - **AdamW** (with/without high precision) marginally outperforms Adam, likely due to its adaptive weight decay mechanism. - The convergence patterns are similar across all configurations, indicating robustness in the training process. However, the choice of optimizer and precision has a smaller impact on Wikitext103 Perplexity compared to other metrics. Notably, the lines for AdamW w/ High Precision (Orange) and Adam w/ High Precision (Blue) are nearly overlapping in all charts, suggesting that the optimizer choice has a smaller effect than precision settings in this context. </details> Figure A7 j Adam vs AdamW. For a 417M (blue) and 1.4B model (green), we find that training with AdamW improves performance over training with Adam. <details> <summary>Image 15 Details</summary>  ### Visual Description ## Line Graphs: Model Performance Across Training Sequences ### Overview The image contains three line graphs comparing the performance of two model sizes (417M and 1.4B) using Adam and AdamW optimizers across three metrics: C4 Loss, Wikitext103 Perplexity, and LAMBADA Accuracy. Each graph tracks performance as the number of training sequences increases from 0 to 150 million. ### Components/Axes 1. **X-Axis**: "Million Sequences" (0 to 150 million, linear scale). 2. **Y-Axes**: - Left: "C4 Loss" (2.3 to 2.8). - Middle: "Wikitext103 Perplexity" (10 to 30). - Right: "LAMBADA Accuracy" (0 to 0.6). 3. **Legends**: - **417M Adam**: Solid blue line. - **417M AdamW**: Dashed blue line. - **1.4B Adam**: Solid green line. - **1.4B AdamW**: Dashed green line. ### Detailed Analysis #### C4 Loss (Left Graph) - **Trend**: All lines slope downward, indicating decreasing loss with more sequences. - **417M Adam (solid blue)**: Starts at ~2.8, decreases to ~2.7. - **417M AdamW (dashed blue)**: Starts at ~2.75, decreases to ~2.6. - **1.4B Adam (solid green)**: Starts at ~2.7, decreases to ~2.4. - **1.4B AdamW (dashed green)**: Starts at ~2.65, decreases to ~2.35. - **Key Insight**: Larger models (1.4B) and AdamW optimizer achieve lower loss. #### Wikitext103 Perplexity (Middle Graph) - **Trend**: All lines slope downward, showing reduced perplexity (better performance). - **417M Adam (solid blue)**: Starts at ~30, decreases to ~25. - **417M AdamW (dashed blue)**: Starts at ~28, decreases to ~22. - **1.4B Adam (solid green)**: Starts at ~25, decreases to ~20. - **1.4B AdamW (dashed green)**: Starts at ~23, decreases to ~18. - **Key Insight**: Larger models and AdamW optimizer reduce perplexity more effectively. #### LAMBADA Accuracy (Right Graph) - **Trend**: All lines slope upward, indicating improved accuracy. - **417M Adam (solid blue)**: Starts at ~0.3, increases to ~0.5. - **417M AdamW (dashed blue)**: Starts at ~0.4, increases to ~0.5. - **1.4B Adam (solid green)**: Starts at ~0.4, increases to ~0.6. - **1.4B AdamW (dashed green)**: Starts at ~0.5, increases to ~0.6. - **Key Insight**: Larger models and AdamW optimizer achieve higher accuracy. ### Key Observations 1. **Optimizer Impact**: AdamW consistently outperforms Adam across all metrics and model sizes. 2. **Model Size Impact**: 1.4B models outperform 417M models in all metrics. 3. **Consistency**: Trends are uniform across loss, perplexity, and accuracy, suggesting robust performance improvements. ### Interpretation The data demonstrates that: - **AdamW optimizer** enhances training efficiency, leading to lower loss, reduced perplexity, and higher accuracy compared to Adam. - **Larger models (1.4B)** achieve superior performance, highlighting the benefits of increased capacity. - The convergence of trends across metrics suggests that both optimizer choice and model size are critical for optimizing language model performance. These findings align with prior research on scaling laws and optimizer efficacy in deep learning. </details> with and without the higher precision copy of the weights and with Adam/AdamW for comparison. ## H. Results ## H.1. The Pile In Table A5 we show the bits-per-byte (bpb) on The Pile (Gao et al., 2020) of Chinchilla , Gopher , and Jurassic-1. Chinchilla outperforms Gopher on all subsets. Jurassic-1 outperforms Chinchilla on 2 subsets-dm\_mathematics and ubuntu\_irc . | Subset | Chinchilla (70B) | Gopher (280B) | Jurassic-1 (170B) | |-------------------|--------------------|-----------------|---------------------| | pile_cc | 0.667 | 0.691 | 0.669 | | pubmed_abstracts | 0.559 | 0.578 | 0.587 | | stackexchange | 0.614 | 0.641 | 0.655 | | github | 0.337 | 0.377 | 0.358 | | openwebtext2 | 0.647 | 0.677 | - | | arxiv | 0.627 | 0.662 | 0.680 | | uspto_backgrounds | 0.526 | 0.546 | 0.537 | | freelaw | 0.476 | 0.513 | 0.514 | | pubmed_central | 0.504 | 0.525 | 0.579 | | dm_mathematics | 1.111 | 1.142 | 1.037 | | hackernews | 0.859 | 0.89 | 0.869 | | nih_exporter | 0.572 | 0.59 | 0.590 | | opensubtitles | 0.871 | 0.9 | 0.879 | | europarl | 0.833 | 0.938 | - | | books3 | 0.675 | 0.712 | 0.835 | | philpapers | 0.656 | 0.695 | 0.742 | | gutenberg_pg_19 | 0.548 | 0.656 | 0.890 | | bookcorpus2 | 0.714 | 0.741 | - | | ubuntu_irc | 1.026 | 1.09 | 0.857 | Table A5 j Bits-per-Byte on The Pile. We show the bpb on The Pile for Chinchilla compared to Gopher and Jurassic-1. ## H.2. MMLU In Table A6 we show the performance of Chinchilla and Gopher on each subset of MMLU. ## H.3. Winogender Setup We follow the same setup as in Rae et al. (2021). To test coreference resolution in Chinchilla , we input a sentence which includes a pronoun reference (e.g., 'The librarian helped the child pick out a book because {pronoun} liked to encourage reading.'), then measure the probability of the model completing the sentence ''{Pronoun}' refers to the' with different sentence roles ('librarian' and 'child' in this example). Each example is annotated with the correct pronoun resolution (the pronoun corresponds to the librarian in this example). Each sentence is tested with a female, male, and gender-neutral pronoun. An unbiased model would correctly predict which word the pronoun refers to regardless of pronoun gender. ## H.4. BIG-bench In Table A7 we show Chinchilla and Gopher performance on each subset of BIG-bench that we consider. ## I. Model Card We present the Chinchilla model card in Table A8, following the framework presented by Mitchell et al. (2019). | Task | Chinchilla | Gopher | Task | Chinchilla | Gopher | |------------------------------|--------------|----------|------------------------------|--------------|----------| | abstract_algebra | 31 | 25 | anatomy | 70.4 | 56.3 | | astronomy | 73 | 65.8 | business_ethics | 72.0 | 70.0 | | clinical_knowledge | 75.1 | 67.2 | college_biology | 79.9 | 70.8 | | college_chemistry | 51 | 45 | college_computer_science | 51.0 | 49.0 | | college_mathematics | 32 | 37 | college_medicine | 66.5 | 60.1 | | college_physics | 46.1 | 34.3 | computer_security | 76.0 | 65.0 | | conceptual_physics | 67.2 | 49.4 | econometrics | 38.6 | 43.0 | | electrical_engineering | 62.1 | 60 | elementary_mathematics | 41.5 | 33.6 | | formal_logic | 33.3 | 35.7 | global_facts | 39.0 | 38.0 | | high_school_biology | 80.3 | 71.3 | high_school_chemistry | 58.1 | 47.8 | | high_school_computer_science | 58 | 54 | high_school_european_history | 78.8 | 72.1 | | high_school_geography | 86.4 | 76.8 | high_school_gov_and_politics | 91.2 | 83.9 | | high_school_macroeconomics | 70.5 | 65.1 | high_school_mathematics | 31.9 | 23.7 | | high_school_microeconomics | 77.7 | 66.4 | high_school_physics | 36.4 | 33.8 | | high_school_psychology | 86.6 | 81.8 | high_school_statistics | 58.8 | 50.0 | | high_school_us_history | 83.3 | 78.9 | high_school_world_history | 85.2 | 75.1 | | human_aging | 77.6 | 66.4 | human_sexuality | 86.3 | 67.2 | | international_law | 90.9 | 77.7 | jurisprudence | 79.6 | 71.3 | | logical_fallacies | 80.4 | 72.4 | machine_learning | 41.1 | 41.1 | | management | 82.5 | 77.7 | marketing | 89.7 | 83.3 | | medical_genetics | 69 | 69 | miscellaneous | 84.5 | 75.7 | | moral_disputes | 77.5 | 66.8 | moral_scenarios | 36.5 | 40.2 | | nutrition | 77.1 | 69.9 | philosophy | 79.4 | 68.8 | | prehistory | 81.2 | 67.6 | professional_accounting | 52.1 | 44.3 | | professional_law | 56.5 | 44.5 | professional_medicine | 75.4 | 64.0 | | professional_psychology | 75.7 | 68.1 | public_relations | 73.6 | 71.8 | | security_studies | 75.9 | 64.9 | sociology | 91.0 | 84.1 | | us_foreign_policy | 92 | 81 | virology | 53.6 | 47.0 | | world_religions | 87.7 | 84.2 | | | | Table A6 j Chinchilla MMLU results. For each subset of MMLU (Hendrycks et al., 2020), we show Chinchilla 's accuracy compared to Gopher . | Model Details | Model Details | |-----------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------| | Organization Developing the Model | DeepMind | | Model Date | March 2022 | | Model Type | Autoregressive Transformer Language Model (Section 4.1 for details) | | Feedback on the Model | {jordanhoffmann, sborgeaud, amensch,sifre}@deepmind.com | | Intended Uses | Intended Uses | | Primary Intended Uses | The primary use is research on language models, including: research on the scaling behaviour of language models along with those listed in Rae et al. (2021). | ## Primary Intended Users DeepMind researchers. We will not make this model available | | publicly. | |---------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| | Out-of-Scope Uses | Uses of the language model for language generation in harm- ful or deceitful settings. More generally, the model should not be used for downstream applications without further safety and fairness mitigations. | | Factors | Factors | | Card Prompts - Relevant Factor | Relevant factors include which language is used. Our model is trained on English data. Furthermore, in the analysis of mod- els trained on the same corpus in Rae et al. (2021), we found it has unequal performance when modelling some dialects (e.g., African American English). Our model is designed for research. The model should not be used for downstream ap- plications without further analysis on factors in the proposed downstream application. | | Card Prompts - Evaluation Factors | See the results in Rae et al. (2021) which analyzes models trained on the same text corpus. | | Metrics | Metrics | | Model Performance Measures | • Perplexity and bits per byte on language modelling datasets • Accuracy on completion tasks, reading comprehension, MMLU, BIG-bench and fact checking. • Exact match accuracy for question answering. • Generation toxicity from Real Toxicity Prompts (RTP) alongside toxicity classification accuracy. • Gender and occupation bias. Test include comparing the probability of generating different gender terms and the Winogender coreference resolution task. We principally focus on Chinchilla 's performance compared to Gopher on text likelihood prediction. | | Decision thresholds | N/A | | Approaches to Uncertainty and Vari- ability | Due to the costs of training large language models, we did not train Chinchilla multiple times. However, the breadth of our evaluation on a range of different task types gives a reasonable estimate of the overall performance of the model. Furthermore, the existence of another large model trained on the same dataset ( Gopher ) provides a clear point of com- parison. | | Evaluation Data | Evaluation Data | ## Datasets | | • Language modelling on LAMBADA, Wikitext103 (Mer- ity et al., 2017), C4 (Raffel et al., 2020a), PG-19 (Rae et al., 2020) and the Pile (Gao et al., 2020). • Language understanding, real world knowledge, mathematical and logical reasoning on the Massive Multitask Language Understanding (MMLU) bench- mark (Hendrycks et al., 2020) and on the 'Beyond the Imitation Game Benchmark' (BIG-bench) (BIG-bench collaboration, 2021). • Question answering (closed book) on Natural Ques- tions (Kwiatkowski et al., 2019) and TriviaQA (Joshi et al., 2017). • Reading comprehension on RACE (Lai et al., 2017) • Common sense understanding on HellaSwag (Zellers et al., 2019), PIQA (Bisk et al., 2020), Wino- grande (Sakaguchi et al., 2020), SIQA (Sap et al., 2019), BoolQ (Clark et al., 2019), and TruthfulQA (Lin et al., 2021). | |---------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| | Motivation | We chose evaluations from Rae et al. (2021) to allow us to most directly compare to Gopher . | | Preprocessing | Input text is tokenized using a SentencePiece tokenizer with a vocabulary of size 32,000. Unlike the tokenizer used for Gopher , the tokenizer used for Chinchilla does not perform NFKC normalization. | ## Training Data The same dataset is used as in Rae et al. (2021). Differences in sampling are shown in Table A1. ## Quantitative Analyses | Unitary Results | Section 4.2 gives a detailed description of our analysis. Main take-aways include: • Our model is capable of outputting toxic language as measured by the PerspectiveAPI. This is particularly true when the model is prompted with toxic prompts. • Gender: Our model emulates stereotypes found in our dataset, with occupations such as 'dietician' and 're- ceptionist' being more associated with women and 'car- penter' and 'sheriff' being more associated with men. • Race/religion/country sentiment: Prompting our model to discuss some groups leads to sentences with lower or higher sentiment, likely reflecting text in our dataset. | |-------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| Intersectional Results We did not investigate intersectional biases. | | Ethical Considerations | |-----------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| | Data | The data is the same as described in Rae et al. (2021). | | Human Life | The model is not intended to inform decisions about matters central to human life or flourishing. | | Mitigations | We considered filtering the dataset to remove toxic content but decided against it due to the observation that this can introduce new biases as studied by Welbl et al. (2021). More work is needed on mitigation approaches to toxic content and other types of risks associated with language models, such as those discussed in Weidinger et al. (2021). | | Risks and Harms | The data is collected from the internet, and thus undoubtedly there is toxic/biased content in our training dataset. Fur- thermore, it is likely that personal information is also in the dataset that has been used to train our models. We defer to the more detailed discussion in Weidinger et al. (2021). | | Use Cases | Especially fraught use cases include the generation of fac- tually incorrect information with the intent of distributing it or using the model to generate racist, sexist or otherwise toxic text with harmful intent. Many more use cases that could cause harm exist. Such applications to malicious use are discussed in detail in Weidinger et al. (2021). | Table A8 j Chinchilla model card. We follow the framework presented in Mitchell et al. (2019). ## J. List of trained models In Table A9 we list the model size and configuration of all models used in this study. Many models have been trained multiple times, for a different number of training steps. Table A7 j Chinchilla BIG-bench results. For each subset of BIG-bench (BIG-bench collaboration, 2021), we show Chinchilla and Gopher 's accuracy. | Task | Chinchilla | Gopher | Task | Chinchilla | Gopher | |---------------------------------|--------------|----------|------------------------------|--------------|----------| | hyperbaton | 54.2 | 51.7 | movie_dialog_same_or_diff | 54.5 | 50.7 | | causal_judgment | 57.4 | 50.8 | winowhy | 62.5 | 56.7 | | formal_fallacies_syllogisms_neg | 52.1 | 50.7 | movie_recommendation | 75.6 | 50.5 | | crash_blossom | 47.6 | 63.6 | moral_permissibility | 57.3 | 55.1 | | discourse_marker_prediction | 13.1 | 11.7 | strategyqa | 68.3 | 61 | | general_knowledge_json | 94.3 | 93.9 | nonsense_words_grammar | 78 | 61.4 | | sports_understanding | 71 | 54.9 | metaphor_boolean | 93.1 | 59.3 | | implicit_relations | 49.4 | 36.4 | navigate | 52.6 | 51.1 | | penguins_in_a_table | 48.7 | 40.6 | presuppositions_as_nli | 49.9 | 34 | | intent_recognition | 92.8 | 88.7 | temporal_sequences | 32 | 19 | | reasoning_about_colored_objects | 59.7 | 49.2 | question_selection | 52.6 | 41.4 | | logic_grid_puzzle | 44 | 35.1 | logical_fallacy_detection | 72.1 | 58.9 | | timedial | 68.8 | 50.9 | physical_intuition | 79 | 59.7 | | epistemic_reasoning | 60.6 | 56.4 | physics_mc | 65.5 | 50.9 | | ruin_names | 47.1 | 38.6 | identify_odd_metaphor | 68.8 | 38.6 | | hindu_knowledge | 91.4 | 80 | understanding_fables | 60.3 | 39.6 | | misconceptions | 65.3 | 61.7 | logical_sequence | 64.1 | 36.4 | | implicatures | 75 | 62 | mathematical_induction | 47.3 | 57.6 | | disambiguation_q | 54.7 | 45.5 | fantasy_reasoning | 69 | 64.1 | | known_unknowns | 65.2 | 63.6 | SNARKS | 58.6 | 48.3 | | dark_humor_detection | 66.2 | 83.1 | crass_ai | 75 | 56.8 | | analogical_similarity | 38.1 | 17.2 | entailed_polarity | 94 | 89.5 | | sentence_ambiguity | 71.7 | 69.1 | irony_identification | 73 | 69.7 | | riddle_sense | 85.7 | 68.2 | evaluating_info_essentiality | 17.6 | 16.7 | | date_understanding | 52.3 | 44.1 | phrase_relatedness | 94 | 81.8 | | analytic_entailment | 67.1 | 53 | novel_concepts | 65.6 | 59.1 | | odd_one_out | 70.9 | 32.5 | empirical_judgments | 67.7 | 52.5 | | logical_args | 56.2 | 59.1 | figure_of_speech_detection | 63.3 | 52.7 | | alignment_questionnaire | 91.3 | 79.2 | english_proverbs | 82.4 | 57.6 | | similarities_abstraction | 87 | 81.8 | Human_organs_senses_mcc | 85.7 | 84.8 | | anachronisms | 69.1 | 56.4 | gre_reading_comprehension | 53.1 | 27.3 | Table A9 j All models. We list the hyperparameters and size of all models trained as part of this work. Many shown models have been trained with multiple learning rate schedules/number of training tokens. | Parameters (million) | d_model | ffw_size | kv_size | n_heads | n_layers | |------------------------|-----------|------------|-----------|-----------|------------| | 44 | 512 | 2048 | 64 | 8 | 8 | | 57 | 576 | 2304 | 64 | 9 | 9 | | 74 | 640 | 2560 | 64 | 10 | 10 | | 90 | 640 | 2560 | 64 | 10 | 13 | | 106 | 640 | 2560 | 64 | 10 | 16 | | 117 | 768 | 3072 | 64 | 12 | 12 | | 140 | 768 | 3072 | 64 | 12 | 15 | | 163 | 768 | 3072 | 64 | 12 | 18 | | 175 | 896 | 3584 | 64 | 14 | 14 | | 196 | 896 | 3584 | 64 | 14 | 16 | | 217 | 896 | 3584 | 64 | 14 | 18 | | 251 | 1024 | 4096 | 64 | 16 | 16 | | 278 | 1024 | 4096 | 64 | 16 | 18 | | 306 | 1024 | 4096 | 64 | 16 | 20 | | 425 | 1280 | 5120 | 128 | 10 | 18 | | 489 | 1280 | 5120 | 128 | 10 | 21 | | 509 | 1408 | 5632 | 128 | 11 | 18 | | 552 | 1280 | 5120 | 128 | 10 | 24 | | 587 | 1408 | 5632 | 128 | 11 | 21 | | 632 | 1536 | 6144 | 128 | 12 | 19 | | 664 | 1408 | 5632 | 128 | 11 | 24 | | 724 | 1536 | 6144 | 128 | 12 | 22 | | 816 | 1536 | 6144 | 128 | 12 | 25 | | 893 | 1792 | 7168 | 128 | 14 | 20 | | 1,018 | 1792 | 7168 | 128 | 14 | 23 | | 1,143 | 1792 | 7168 | 128 | 14 | 26 | | 1,266 | 2048 | 8192 | 128 | 16 | 22 | | 1,424 | 2176 | 8704 | 128 | 17 | 22 | | 1,429 | 2048 | 8192 | 128 | 16 | 25 | | 1,593 | 2048 | 8192 | 128 | 16 | 28 | | 1,609 | 2176 | 8704 | 128 | 17 | 25 | | 1,731 | 2304 | 9216 | 128 | 18 | 24 | | 1,794 | 2176 | 8704 | 128 | 17 | 28 | | 2,007 | 2304 | 9216 | 128 | 18 | 28 | | 2,283 | 2304 | 9216 | 128 | 18 | 32 | | 2,298 | 2560 | 10240 | 128 | 20 | 26 | | 2,639 | 2560 | 10240 | 128 | 20 | 30 | | 2,980 | 2560 | 10240 | 128 | 20 | 34 | | 3,530 | 2688 | 10752 | 128 | 22 | 36 | | 3,802 | 2816 | 11264 | 128 | 22 | 36 | | 4,084 | 2944 | 11776 | 128 | 22 | 36 | | 4,516 | 3072 | 12288 | 128 | 24 | 36 | | 6,796 | 3584 | 14336 | 128 | 28 | 40 | | 9,293 | 4096 | 16384 | 128 | 32 | 42 | | 11,452 | 4352 | 17408 | 128 | 32 | 47 | | 12,295 | 4608 | 18432 | 128 | 36 | 44 | | 12,569 | 4608 | 18432 | 128 | 32 | 47 | | 13,735 | 4864 | 19456 | 128 | 32 | 47 | | 14,940 | 4992 | 19968 | 128 | 32 | 49 | | 16,183 | 5120 | 20480 | 128 | 40 | 47 |