## Chart: Accuracy vs. Thinking Compute ### Overview The image is a line chart comparing the accuracy of different models (pass@k (Oracle), majority@k, short-1@k (Ours), and short-3@k (Ours)) against the thinking compute (thinking tokens in thousands). The chart shows how accuracy increases with thinking compute for each model. ### Components/Axes * **X-axis:** Thinking Compute (thinking tokens in thousands). Scale ranges from 20 to 140, with tick marks at 20, 40, 60, 80, 100, 120, and 140. * **Y-axis:** Accuracy. Scale ranges from 0.40 to 0.65, with tick marks at 0.40, 0.45, 0.50, 0.55, 0.60, and 0.65. * **Legend:** Located in the bottom-right corner of the chart. * `pass@k (Oracle)`: Black dotted line with triangle markers. * `majority@k`: Brown solid line with circle markers. * `short-1@k (Ours)`: Blue solid line with square markers. * `short-3@k (Ours)`: Teal solid line with diamond markers. ### Detailed Analysis * **pass@k (Oracle):** The black dotted line with triangle markers shows a steep upward trend, indicating a rapid increase in accuracy with increasing thinking compute. * At 20k tokens, accuracy is approximately 0.40. * At 40k tokens, accuracy is approximately 0.50. * At 60k tokens, accuracy is approximately 0.58. * At 80k tokens, accuracy is approximately 0.63. * At 85k tokens, accuracy is approximately 0.65. * **majority@k:** The brown solid line with circle markers shows a gradual upward trend, indicating a slower increase in accuracy with increasing thinking compute. * At 20k tokens, accuracy is approximately 0.40. * At 40k tokens, accuracy is approximately 0.43. * At 60k tokens, accuracy is approximately 0.47. * At 80k tokens, accuracy is approximately 0.50. * At 100k tokens, accuracy is approximately 0.51. * At 120k tokens, accuracy is approximately 0.515. * At 140k tokens, accuracy is approximately 0.52. * **short-1@k (Ours):** The blue solid line with square markers shows an upward trend, with accuracy increasing with thinking compute. * At 20k tokens, accuracy is approximately 0.40. * At 40k tokens, accuracy is approximately 0.49. * At 60k tokens, accuracy is approximately 0.52. * At 80k tokens, accuracy is approximately 0.54. * **short-3@k (Ours):** The teal solid line with diamond markers shows an upward trend, with accuracy increasing with thinking compute. * At 20k tokens, accuracy is approximately 0.40. * At 40k tokens, accuracy is approximately 0.48. * At 60k tokens, accuracy is approximately 0.51. * At 80k tokens, accuracy is approximately 0.54. ### Key Observations * The `pass@k (Oracle)` model achieves the highest accuracy for a given thinking compute value. * The `majority@k` model has the lowest accuracy compared to the other models. * The `short-1@k (Ours)` and `short-3@k (Ours)` models perform similarly, with `short-1@k` slightly outperforming `short-3@k`. * All models show an increase in accuracy with increasing thinking compute, but the rate of increase varies. ### Interpretation The chart demonstrates the relationship between thinking compute and accuracy for different models. The `pass@k (Oracle)` model serves as an upper bound or ideal performance, while the `majority@k` model represents a baseline. The `short-1@k (Ours)` and `short-3@k (Ours)` models show improved performance compared to the baseline, suggesting that the "Ours" models are effective in leveraging thinking compute to improve accuracy. The diminishing returns observed in the `majority@k` model suggest that simply increasing compute may not always lead to significant gains in accuracy, and more sophisticated models like `pass@k` and the "Ours" models are needed to effectively utilize higher compute budgets.

## Line Chart: Accuracy vs. Thinking Compute ### Overview This image presents a line chart illustrating the relationship between "Thinking Compute" (measured in thousands of tokens) and "Accuracy" for several different methods. The chart compares the performance of an "Oracle" method ("pass@k") against three other methods: "majority@k", "short-1@k", and "short-3@k". The methods "short-1@k" and "short-3@k" are identified as "Ours", indicating they are the results of the study. ### Components/Axes * **X-axis:** "Thinking Compute (thinking tokens in thousands)". Scale ranges from approximately 10 to 140, with markers at 20, 40, 60, 80, 100, 120, and 140. * **Y-axis:** "Accuracy". Scale ranges from approximately 0.40 to 0.65, with markers at 0.40, 0.45, 0.50, 0.55, 0.60, and 0.65. * **Legend:** Located in the top-right corner of the chart. Contains the following labels and corresponding line styles/colors: * "pass@k (Oracle)" - Black dashed line with diamond markers. * "majority@k" - Brown solid line with circle markers. * "short-1@k (Ours)" - Blue solid line with square markers. * "short-3@k (Ours)" - Cyan solid line with triangle markers. ### Detailed Analysis * **pass@k (Oracle):** This line exhibits a steep upward slope, starting at approximately 0.42 at a compute of 20 and reaching approximately 0.63 at a compute of 80. The line plateaus after 80, with minimal increase in accuracy. * (20, 0.42) * (40, 0.56) * (60, 0.61) * (80, 0.63) * (100, 0.63) * (120, 0.63) * **majority@k:** This line shows a gradual upward slope, starting at approximately 0.40 at a compute of 20 and reaching approximately 0.52 at a compute of 120. * (20, 0.40) * (40, 0.45) * (60, 0.48) * (80, 0.50) * (100, 0.51) * (120, 0.52) * **short-1@k (Ours):** This line demonstrates a moderate upward slope, starting at approximately 0.40 at a compute of 20 and reaching approximately 0.53 at a compute of 120. * (20, 0.40) * (40, 0.47) * (60, 0.50) * (80, 0.52) * (100, 0.53) * (120, 0.53) * **short-3@k (Ours):** This line shows a similar trend to "short-1@k", but with slightly higher accuracy values. It starts at approximately 0.40 at a compute of 20 and reaches approximately 0.55 at a compute of 120. * (20, 0.40) * (40, 0.49) * (60, 0.52) * (80, 0.54) * (100, 0.55) * (120, 0.55) ### Key Observations * The "pass@k (Oracle)" method significantly outperforms all other methods across the entire range of "Thinking Compute". * "short-3@k (Ours)" consistently achieves higher accuracy than "short-1@k (Ours)". * The performance gains for all methods diminish as "Thinking Compute" increases beyond 80. * "majority@k" has the lowest accuracy across all compute values. ### Interpretation The chart demonstrates the impact of "Thinking Compute" on the accuracy of different methods. The "Oracle" method, representing an ideal scenario, shows the potential maximum accuracy achievable with increasing compute. The "Ours" methods ("short-1@k" and "short-3@k") represent practical approaches that achieve reasonable accuracy, with "short-3@k" being more effective. The diminishing returns observed at higher compute values suggest that there is a limit to the benefits of increasing compute beyond a certain point. The relatively low performance of "majority@k" indicates that it is not a suitable method for this task. The data suggests that increasing the model's "thinking" capacity (as measured by tokens) improves performance, but the improvement is not linear and plateaus. The comparison between the "Ours" methods and the "Oracle" method highlights the gap between current practical approaches and the theoretical maximum performance.

\n ## Line Chart: Accuracy vs. Thinking Compute ### Overview The image is a line chart comparing the performance (Accuracy) of four different methods as a function of computational effort (Thinking Compute). The chart demonstrates how accuracy scales with increased compute for an "Oracle" method and three alternative approaches, two of which are labeled as "(Ours)". ### Components/Axes * **Y-Axis:** Labeled "Accuracy". The scale ranges from 0.40 to 0.65, with major grid lines at intervals of 0.05. * **X-Axis:** Labeled "Thinking Compute (thinking tokens in thousands)". The scale ranges from 20 to 140, with major grid lines at intervals of 20 (20, 40, 60, 80, 100, 120, 140). * **Legend:** Located in the bottom-right quadrant of the chart area. It contains four entries: 1. `pass@k (Oracle)`: Represented by a black dotted line with upward-pointing triangle markers. 2. `majority@k`: Represented by a solid dark red line with circle markers. 3. `short-1@k (Ours)`: Represented by a solid blue line with square markers. 4. `short-3@k (Ours)`: Represented by a solid cyan line with diamond markers. ### Detailed Analysis All four data series originate from the same approximate starting point at the lowest compute value shown. **1. pass@k (Oracle)** * **Trend:** Exhibits the steepest, near-linear upward slope. It demonstrates the highest accuracy for any given compute level above the starting point. * **Data Points (Approximate):** * (20, 0.40) * (30, 0.485) * (40, 0.535) * (50, 0.565) * (60, 0.59) * (70, 0.615) * (80, 0.65) **2. majority@k** * **Trend:** Shows the most gradual, concave upward slope. It has the lowest accuracy of all methods for compute values above ~25. * **Data Points (Approximate):** * (20, 0.40) * (40, 0.43) * (60, 0.46) * (80, 0.49) * (100, 0.505) * (120, 0.515) * (140, 0.52) **3. short-1@k (Ours)** * **Trend:** Shows a moderate, concave upward slope, positioned between the Oracle and majority methods. * **Data Points (Approximate):** * (20, 0.40) * (30, 0.475) * (40, 0.49) * (50, 0.51) * (60, 0.525) * (70, 0.54) **4. short-3@k (Ours)** * **Trend:** Follows a very similar trajectory to `short-1@k (Ours)`, with a nearly identical slope, but is consistently positioned slightly to the right (requiring more compute for similar accuracy) or slightly below (lower accuracy for similar compute). * **Data Points (Approximate):** * (20, 0.40) * (30, 0.45) * (40, 0.48) * (50, 0.50) * (60, 0.515) * (70, 0.525) * (80, 0.535) * (90, 0.54) ### Key Observations 1. **Universal Starting Point:** All methods begin at approximately 0.40 accuracy with 20k thinking tokens. 2. **Performance Hierarchy:** A clear and consistent hierarchy is established: `pass@k (Oracle)` >> `short-1@k (Ours)` ≈ `short-3@k (Ours)` > `majority@k`. 3. **Diminishing Returns:** All curves show signs of diminishing returns (concavity), but the degree varies drastically. The Oracle method's returns diminish the least within the plotted range. 4. **Proximity of "Ours" Methods:** The two proposed methods (`short-1` and `short-3`) perform very similarly, with `short-1` having a slight edge in efficiency (achieving the same accuracy with less compute). 5. **Compute Range:** The Oracle method is only plotted up to 80k tokens, while `majority@k` extends to 140k, suggesting the Oracle may not require or was not tested at higher compute levels. ### Interpretation This chart likely comes from research on scaling inference compute for language models or reasoning systems. The "Thinking Compute" axis represents the resource (in tokens) allocated to a problem-solving process. * **The "Oracle" as an Upper Bound:** The `pass@k (Oracle)` line represents a theoretical or idealized best-case scenario (perhaps using ground-truth information or an unbounded verifier). It serves as a performance ceiling, showing the maximum achievable accuracy for a given compute budget under perfect conditions. * **Efficiency of Proposed Methods:** The core message is that the authors' methods (`short-1@k` and `short-3@k`) offer a significant efficiency improvement over the `majority@k` baseline. They achieve substantially higher accuracy for the same compute, or the same accuracy with much less compute. For example, to reach 0.50 accuracy, `majority@k` requires ~100k tokens, while `short-1@k` requires only ~45k tokens. * **The Cost of "Short" Strategies:** The names `short-1` and `short-3` imply these methods use shorter or more constrained reasoning chains. The chart quantifies the trade-off: these constrained strategies are less accurate than the ideal Oracle but are far more compute-efficient than a simple majority vote approach, striking a practical balance for real-world applications where compute is limited. * **Scalability Insight:** The steep slope of the Oracle line suggests that with perfect verification, accuracy scales very favorably with compute. The flatter slopes of the other methods indicate they hit practical limits or inefficiencies in how they utilize additional compute. The research likely aims to close the gap between practical methods and the Oracle bound.

## Line Graph: Accuracy vs. Thinking Tokens (in Thousands) ### Overview The image is a line graph comparing the accuracy of four different models as a function of "thinking tokens in thousands." The y-axis represents accuracy (ranging from 0.40 to 0.65), and the x-axis represents the number of thinking tokens (in thousands, from 20 to 140). Four data series are plotted, each with distinct markers and colors, as defined in the legend. --- ### Components/Axes - **X-axis (Horizontal)**: "Thinking tokens in thousands" (20 to 140, in increments of 20). - **Y-axis (Vertical)**: "Accuracy" (0.40 to 0.65, in increments of 0.05). - **Legend**: Located on the right side of the graph. Entries include: - **pass@k (Oracle)**: Black dashed line with triangle markers. - **majority@k**: Red solid line with circle markers. - **short-1@k (Ours)**: Blue solid line with square markers. - **short-3@k (Ours)**: Cyan solid line with diamond markers. --- ### Detailed Analysis #### 1. **pass@k (Oracle)** - **Trend**: Steep upward slope, starting at 0.40 (20k tokens) and rising to 0.65 (140k tokens). - **Key Data Points**: - 20k tokens: 0.40 - 40k tokens: 0.55 - 80k tokens: 0.60 - 120k tokens: 0.63 - 140k tokens: 0.65 #### 2. **majority@k** - **Trend**: Gradual upward slope, starting at 0.40 (20k tokens) and rising to 0.52 (140k tokens). - **Key Data Points**: - 20k tokens: 0.40 - 40k tokens: 0.43 - 80k tokens: 0.47 - 120k tokens: 0.51 - 140k tokens: 0.52 #### 3. **short-1@k (Ours)** - **Trend**: Moderate upward slope, starting at 0.40 (20k tokens) and rising to 0.54 (140k tokens). - **Key Data Points**: - 20k tokens: 0.40 - 40k tokens: 0.47 - 80k tokens: 0.52 - 120k tokens: 0.53 - 140k tokens: 0.54 #### 4. **short-3@k (Ours)** - **Trend**: Slightly steeper than short-1@k, starting at 0.40 (20k tokens) and rising to 0.53 (140k tokens). - **Key Data Points**: - 20k tokens: 0.40 - 40k tokens: 0.45 - 80k tokens: 0.51 - 120k tokens: 0.53 - 140k tokens: 0.53 --- ### Key Observations 1. **pass@k (Oracle)** consistently outperforms all other models, achieving the highest accuracy across all token counts. 2. **majority@k** has the lowest accuracy, showing minimal improvement with increased tokens. 3. **short-1@k** and **short-3@k** (both labeled "Ours") demonstrate similar performance, with short-3@k slightly outperforming short-1@k at higher token counts. 4. All models show diminishing returns as token counts increase beyond 80k. --- ### Interpretation The graph highlights the relationship between computational resources (thinking tokens) and model performance. The **pass@k (Oracle)** model, likely representing a ground-truth or idealized system, achieves the highest accuracy, suggesting that increased computational capacity directly improves performance. In contrast, the **majority@k** model (a baseline or simple heuristic) shows limited gains, indicating its inefficiency. The **short-1@k** and **short-3@k** models (labeled "Ours") represent optimized or constrained approaches. While both outperform majority@k, their performance plateaus at higher token counts, suggesting that further resource allocation yields diminishing returns. The slight edge of short-3@k over short-1@k implies that the 3k-token configuration may be more efficient or effective than the 1k-token variant. The data underscores the trade-off between computational cost and accuracy, with the Oracle model serving as a benchmark for ideal performance. The short models, while resource-efficient, still lag behind the Oracle, highlighting the need for further optimization or alternative strategies to bridge this gap.