## Scatter Plots: MATH500 and AIME2024 Accuracy vs. Token Length ### Overview The image contains two scatter plots comparing the accuracy of different language models on the MATH500 and AIME2024 datasets, plotted against the token length used. The models are represented by data points, colored either orange or blue. The x-axis represents token length, and the y-axis represents accuracy. ### Components/Axes **Left Plot (MATH500):** * **Title:** MATH500 * **X-axis:** Token Length, ranging from 400 to 1400 in increments of 200. * **Y-axis:** Accuracy, ranging from 75.0 to 95.0 in increments of 2.5. * **Data Points:** * Orange: "k1.5-short w/ rl", "k1.5-long", "k1.5-short w/ dpo", "k1.5-short w/ merge + rs", "k1.5-shortest", "k1.5-short w/ merge" * Blue: "deepseek-v3", "qwen25-72B-inst", "Claude 3.5", "gpt-4-0513" **Right Plot (AIME2024):** * **Title:** AIME2024 * **X-axis:** Token Length, ranging from 1000 to 5000 in increments of 1000. * **Y-axis:** Accuracy, ranging from 10 to 60 in increments of 10. * **Data Points:** * Orange: "k1.5-short w/ rl", "k1.5-long", "k1.5-short w/ dpo", "k1.5-short w/ merge + rs", "k1.5-shortest", "k1.5-short w/ merge" * Blue: "deepseek-v3", "qwen25-72B-inst", "Claude 3.5", "gpt-4-0513" ### Detailed Analysis **MATH500 Plot:** * **gpt-4-0513 (Blue):** Located at approximately (500, 75). * **Claude 3.5 (Blue):** Located at approximately (450, 78). * **qwen25-72B-inst (Blue):** Located at approximately (700, 80). * **deepseek-v3 (Blue):** Located at approximately (1400, 90). * **k1.5-shortest (Orange):** Located at approximately (750, 89). * **k1.5-short w/ merge (Orange):** Located at approximately (950, 88). * **k1.5-short w/ merge + rs (Orange):** Located at approximately (900, 90). * **k1.5-short w/ dpo (Orange):** Located at approximately (1200, 93). * **k1.5-long (Orange):** Located at approximately (1200, 94). * **k1.5-short w/ rl (Orange):** Located at approximately (1200, 95). **AIME2024 Plot:** * **gpt-4-0513 (Blue):** Located at approximately (800, 10). * **Claude 3.5 (Blue):** Located at approximately (900, 18). * **qwen25-72B-inst (Blue):** Located at approximately (1200, 23). * **deepseek-v3 (Blue):** Located at approximately (5000, 39). * **k1.5-shortest (Orange):** Located at approximately (1800, 28). * **k1.5-short w/ merge (Orange):** Located at approximately (3000, 40). * **k1.5-short w/ merge + rs (Orange):** Located at approximately (3000, 43). * **k1.5-short w/ dpo (Orange):** Located at approximately (3200, 50). * **k1.5-long (Orange):** Located at approximately (3500, 58). * **k1.5-short w/ rl (Orange):** Located at approximately (3500, 61). ### Key Observations * In both plots, the "k1.5-short w/ rl" model achieves the highest accuracy among the orange data points. * The blue data points generally have lower token lengths and lower accuracy compared to the orange data points in both plots. * The AIME2024 plot shows a wider range of token lengths and accuracy values compared to the MATH500 plot. * deepseek-v3 shows a significant increase in token length compared to other blue data points in AIME2024, with a corresponding increase in accuracy. ### Interpretation The plots suggest that the "k1.5-short" models (orange data points) generally outperform the "gpt-4-0513", "Claude 3.5", and "qwen25-72B-inst" models (blue data points) on both the MATH500 and AIME2024 datasets. The higher accuracy of the "k1.5-short" models is associated with longer token lengths. The "deepseek-v3" model shows a trade-off between token length and accuracy, with a much higher token length in AIME2024 compared to MATH500, resulting in a moderate increase in accuracy. The AIME2024 dataset appears to be more challenging, as the accuracy values are generally lower compared to the MATH500 dataset for all models.

## Scatter Plots: Model Performance on MATH500 and AIME2024 ### Overview The image presents two scatter plots, side-by-side. The left plot displays performance on the MATH500 dataset, while the right plot shows performance on the AIME2024 dataset. Both plots measure "Accuracy" against "Token Length" for various language models. Each point represents a model's performance. ### Components/Axes Both plots share the following components: * **X-axis:** "Token Length" - ranging from approximately 400 to 1400 for MATH500 and 1000 to 5000 for AIME2024. * **Y-axis:** "Accuracy" - ranging from approximately 75 to 96 for MATH500 and 10 to 62 for AIME2024. * **Data Points:** Representing different language models. Each point is colored orange or blue. * **Titles:** "MATH500" (left plot) and "AIME2024" (right plot) positioned at the top-center of each respective plot. The models represented are: * k1.5-short w/ rl (orange) * k1.5-long (orange) * k1.5-short w/ dpo (orange) * k1.5-shortest (orange) * k1.5-short w/ merge (orange) * k1.5-short w/ merge + rs (orange) * deepseek-v3 (blue) * qwen25-72B-inst (blue) * Claude 3.5 (blue) * gpt-4-0513 (blue) ### Detailed Analysis or Content Details **MATH500 (Left Plot):** * **k1.5-short w/ rl:** Approximately (1300, 93.5). * **k1.5-long:** Approximately (1200, 94.5). * **k1.5-short w/ dpo:** Approximately (1100, 92.5). * **k1.5-shortest:** Approximately (900, 89). * **k1.5-short w/ merge:** Approximately (1000, 88). * **k1.5-short w/ merge + rs:** Approximately (1000, 88.5). * **deepseek-v3:** Approximately (1400, 90). * **qwen25-72B-inst:** Approximately (600, 80). * **Claude 3.5:** Approximately (500, 77.5). * **gpt-4-0513:** Approximately (400, 75). The orange data points (k1.5 variants) generally show a positive correlation between Token Length and Accuracy, with accuracy increasing as token length increases. The blue data points (deepseek-v3, qwen25-72B-inst, Claude 3.5, gpt-4-0513) are clustered towards the lower end of the accuracy scale and shorter token lengths. **AIME2024 (Right Plot):** * **k1.5-long:** Approximately (4500, 61). * **k1.5-short w/ rl:** Approximately (4000, 58). * **k1.5-short w/ dpo:** Approximately (4000, 55). * **k1.5-shortest:** Approximately (2000, 32). * **k1.5-short w/ merge:** Approximately (3000, 44). * **k1.5-short w/ merge + rs:** Approximately (3500, 47). * **deepseek-v3:** Approximately (5000, 40). * **qwen25-72B-inst:** Approximately (2000, 25). * **Claude 3.5:** Approximately (1000, 15). * **gpt-4-0513:** Approximately (1000, 12). Similar to the MATH500 plot, the orange data points (k1.5 variants) generally show a positive correlation between Token Length and Accuracy. The blue data points are clustered towards the lower end of the accuracy scale and shorter token lengths. ### Key Observations * The k1.5 models consistently outperform the other models (deepseek-v3, qwen25-72B-inst, Claude 3.5, gpt-4-0513) on both datasets. * For the k1.5 models, longer token lengths generally correspond to higher accuracy. * The AIME2024 dataset has a lower overall accuracy range compared to the MATH500 dataset. * gpt-4-0513 and Claude 3.5 have the lowest accuracy scores across both datasets. ### Interpretation The data suggests that the k1.5 family of models are particularly effective at solving problems in both MATH500 and AIME2024, and that their performance improves with increasing token length. This implies that these models benefit from having access to more contextual information. The difference in accuracy ranges between the two datasets suggests that AIME2024 is a more challenging benchmark, or that the models are less well-suited to the types of problems it contains. The consistently low performance of gpt-4-0513 and Claude 3.5 suggests they may not be as effective for these types of tasks, or that they require different training strategies. The positive correlation between token length and accuracy for the k1.5 models indicates that the ability to process longer sequences is a key factor in their success. The fact that the k1.5 models with "merge" and "rs" perform similarly suggests that these techniques do not significantly impact performance in this context.

\n ## Scatter Plot Comparison: MATH500 vs. AIME2024 Benchmark Performance ### Overview The image displays two side-by-side scatter plots comparing the performance of various large language models on two different mathematical reasoning benchmarks: **MATH500** (left) and **AIME2024** (right). Each plot charts model **Accuracy** (y-axis) against **Token Length** (x-axis), which likely represents the model's context window or sequence length. Data points are color-coded, with orange representing variants of a model family labeled "k1.5" and blue representing other models (e.g., deepseek-v3, Claude 3.5, gpt-4-0513, qwen25-72B-inst). ### Components/Axes **Common Elements:** * **X-axis:** "Token Length" (linear scale). * **Y-axis:** "Accuracy" (linear scale, percentage). * **Data Points:** Labeled circles. Orange points are various "k1.5" model configurations. Blue points are comparison models. * **Titles:** Centered above each plot. **Left Plot: MATH500** * **Title:** "MATH500" * **X-axis Range:** ~400 to ~1400 * **Y-axis Range:** 75.0 to 95.0 * **Data Points & Approximate Coordinates (Token Length, Accuracy):** * **Orange (k1.5 family):** * `k1.5-shortest`: (~650, ~88.5) * `k1.5-short w/ merge`: (~900, ~89.0) * `k1.5-short w/ merge + rs`: (~1100, ~91.5) * `k1.5-short w/ dpo`: (~1200, ~93.0) * `k1.5-short w/ rl`: (~1200, ~95.0) *[Highest point]* * `k1.5-long`: (~1350, ~94.0) * **Blue (Other models):** * `gpt-4-0513`: (~550, ~75.0) *[Lowest point]* * `Claude 3.5`: (~450, ~78.5) * `qwen25-72B-inst`: (~650, ~80.0) * `deepseek-v3`: (~1400, ~90.0) **Right Plot: AIME2024** * **Title:** "AIME2024" * **X-axis Range:** ~1000 to ~5000 * **Y-axis Range:** 10 to 60 * **Data Points & Approximate Coordinates (Token Length, Accuracy):** * **Orange (k1.5 family):** * `k1.5-shortest`: (~1500, ~26.0) * `k1.5-short w/ merge`: (~3000, ~39.0) * `k1.5-short w/ merge + rs`: (~3200, ~43.0) * `k1.5-short w/ dpo`: (~3200, ~46.0) * `k1.5-short w/ rl`: (~3300, ~61.0) *[Highest point]* * `k1.5-long`: (~4200, ~63.0) *[Highest point, slightly above rl variant]* * **Blue (Other models):** * `gpt-4-0513`: (~1000, ~9.0) *[Lowest point]* * `Claude 3.5`: (~900, ~16.0) * `qwen25-72B-inst`: (~1200, ~21.0) * `deepseek-v3`: (~5100, ~39.0) ### Detailed Analysis **MATH500 Plot Analysis:** * **Trend Verification:** There is a clear positive correlation between Token Length and Accuracy for the k1.5 model family (orange points). The line formed by these points slopes upward from left to right. The blue comparison models do not follow a single clear trend relative to token length. * **Performance Hierarchy:** The `k1.5-short w/ rl` model achieves the highest accuracy (~95%). The `k1.5-long` model is close behind (~94%). The standard `gpt-4-0513` model shows the lowest accuracy (~75%) among the plotted points. * **Token Length Clustering:** The k1.5 models cluster in the higher token length range (650-1400), while the comparison models (except `deepseek-v3`) are in the lower range (450-650). **AIME2024 Plot Analysis:** * **Trend Verification:** A positive correlation is also visible for the k1.5 family, with accuracy generally increasing with token length. The slope appears steeper than in the MATH500 plot. The `deepseek-v3` point is an outlier in terms of token length (~5100) but has moderate accuracy (~39%). * **Performance Hierarchy:** The `k1.5-long` model achieves the highest accuracy (~63%), narrowly outperforming the `k1.5-short w/ rl` variant (~61%). The `gpt-4-0513` model again shows the lowest accuracy (~9%). * **Scale Difference:** The overall accuracy values are significantly lower on AIME2024 (max ~63%) compared to MATH500 (max ~95%), indicating this is a more challenging benchmark. Token lengths are also generally higher. ### Key Observations 1. **Consistent Leader:** The `k1.5-short w/ rl` (reinforcement learning) variant is a top performer on both benchmarks, suggesting the RL training method is highly effective. 2. **Benchmark Difficulty:** The AIME2024 benchmark yields much lower accuracy scores across all models, suggesting it is a more difficult test of mathematical reasoning. 3. **Token Length Advantage:** The k1.5 model family consistently operates at higher token lengths than the other models shown (except `deepseek-v3` on AIME2024), and this correlates with higher performance. 4. **Model Comparison:** On both benchmarks, the plotted `gpt-4-0513` and `Claude 3.5` points show lower accuracy than the k1.5 family and `deepseek-v3`. `qwen25-72B-inst` performs in the middle range. 5. **Scaling Effect:** Within the k1.5 family, moving from `k1.5-shortest` to `k1.5-long` generally involves an increase in both token length and accuracy. ### Interpretation The data suggests a strong relationship between model context length (token length) and performance on complex mathematical reasoning tasks, at least within the `k1.5` model family. The consistent superiority of the `k1.5-short w/ rl` variant highlights the potential of reinforcement learning techniques to boost reasoning capabilities beyond what is achieved with other methods like DPO (Direct Preference Optimization) or merge strategies. The stark difference in accuracy ranges between MATH500 and AIME2024 indicates these benchmarks test different levels or types of mathematical difficulty. AIME (American Invitational Mathematics Examination) problems are known to be highly challenging, which aligns with the lower scores. The fact that the `k1.5-long` model, with the largest context, performs best on the harder AIME2024 benchmark could imply that longer context windows are particularly beneficial for solving more complex, multi-step problems that require holding and manipulating more information. The plots do not show a simple linear scaling law for all models, as the blue comparison points are scattered. This implies that factors other than raw token length—such as training data, architecture, and fine-tuning methods (like RL, DPO)—are critical determinants of performance. The visualization effectively argues for the efficacy of the `k1.5` approach, particularly its RL variant, in advancing the state-of-the-art for mathematical reasoning in AI.

## Line Chart: Model Performance Comparison (MATH500 vs AIME2024) ### Overview The image contains two side-by-side line charts comparing model performance across two datasets: MATH500 (left) and AIME2024 (right). Both charts plot **Accuracy** (y-axis) against **Token Length** (x-axis), with distinct data points representing different model configurations. The charts use orange and blue markers to differentiate model types. --- ### Components/Axes #### MATH500 Chart (Left) - **X-axis (Token Length)**: 400–1400 (linear scale) - **Y-axis (Accuracy)**: 75–95 (linear scale) - **Legend**: - Orange: k1.5 variants (short/long, merge, dpo, rs, rl) - Blue: qwen25-72B-inst, Claude 3.5, gpt-4-0513 - **Data Points**: - k1.5-short (600 tokens, 88% accuracy) - k1.5-shortest (900 tokens, 87.5% accuracy) - k1.5-short w/ merge (1000 tokens, 90% accuracy) - k1.5-short w/ merge + rs (1100 tokens, 92.5% accuracy) - k1.5-long (1200 tokens, 94% accuracy) - k1.5-short w/ rl (1300 tokens, 93% accuracy) - qwen25-72B-inst (700 tokens, 80% accuracy) - Claude 3.5 (500 tokens, 78% accuracy) - gpt-4-0513 (450 tokens, 75% accuracy) #### AIME2024 Chart (Right) - **X-axis (Token Length)**: 1000–5000 (linear scale) - **Y-axis (Accuracy)**: 10–60 (linear scale) - **Legend**: - Orange: k1.5 variants (short/long, merge, dpo, rs) - Blue: qwen25-72B-inst, Claude 3.5, deepseek-v3 - **Data Points**: - k1.5-shortest (1500 tokens, 25% accuracy) - k1.5-short w/ merge (2000 tokens, 45% accuracy) - k1.5-short w/ merge + rs (2500 tokens, 48% accuracy) - k1.5-long (3000 tokens, 55% accuracy) - k1.5-short w/ dpo (3500 tokens, 50% accuracy) - k1.5-short w/ rl (4000 tokens, 58% accuracy) - qwen25-72B-inst (1200 tokens, 20% accuracy) - Claude 3.5 (1000 tokens, 18% accuracy) - deepseek-v3 (5000 tokens, 45% accuracy) --- ### Detailed Analysis #### MATH500 Trends - **k1.5-short variants** dominate the chart, with accuracy increasing as token length grows (600–1300 tokens). The **merge + rs** configuration achieves the highest accuracy (92.5% at 1100 tokens). - **qwen25-72B-inst** (700 tokens, 80%) and **Claude 3.5** (500 tokens, 78%) underperform compared to k1.5 models. - **gpt-4-0513** (450 tokens, 75%) has the lowest accuracy. #### AIME2024 Trends - **k1.5-short w/ merge + rs** (2500 tokens, 48%) and **k1.5-long** (3000 tokens, 55%) show the best performance. - **deepseek-v3** (5000 tokens, 45%) matches the k1.5-short w/ merge + rs model despite longer token length. - **qwen25-72B-inst** (1200 tokens, 20%) and **Claude 3.5** (1000 tokens, 18%) lag significantly. --- ### Key Observations 1. **MATH500**: - k1.5-short variants achieve >87% accuracy, outperforming other models. - Merge + rs configuration yields the highest accuracy (92.5%). - Longer token lengths (1200–1300) correlate with higher accuracy. 2. **AIME2024**: - k1.5-short w/ merge + rs (2500 tokens) and k1.5-long (3000 tokens) achieve >50% accuracy. - deepseek-v3 (5000 tokens) matches k1.5-short w/ merge + rs despite longer token length. - Accuracy plateaus at ~55% for k1.5-long and k1.5-short w/ rl. 3. **Outliers**: - **MATH500**: k1.5-short w/ merge + rs (92.5%) exceeds all other models. - **AIME2024**: deepseek-v3 (45%) outperforms k1.5-short w/ merge (48%) despite longer token length. --- ### Interpretation The data suggests that **model architecture optimizations** (e.g., merge + rs) significantly impact performance on MATH500, while **token length** plays a stronger role in AIME2024. The k1.5-short variants excel in MATH500 due to efficient token utilization, whereas AIME2024 requires longer token processing for higher accuracy. The deepseek-v3 model’s performance in AIME2024 indicates that specialized architectures can compete with optimized k1.5 variants despite higher computational costs. The divergence in trends between datasets highlights the importance of task-specific model tuning.