## Line Chart: Accuracy vs. Thinking Compute ### Overview The image is a line chart comparing the accuracy of three different models ("majority@k", "short-1@k (Ours)", and "short-3@k (Ours)") as a function of "Thinking Compute" (measured in thousands of thinking tokens). The chart displays accuracy on the y-axis, ranging from 0.74 to 0.81, and thinking compute on the x-axis, ranging from 20,000 to 120,000 tokens. ### Components/Axes * **X-axis:** "Thinking Compute (thinking tokens in thousands)". The axis ranges from 20 to 120 in increments of 20. * **Y-axis:** "Accuracy". The axis ranges from 0.74 to 0.81 in increments of 0.01. * **Legend:** Located in the bottom-right corner of the chart. * **Brown line with circle markers:** "majority@k" * **Light blue line with square markers:** "short-1@k (Ours)" * **Teal line with diamond markers:** "short-3@k (Ours)" ### Detailed Analysis * **majority@k (Brown line with circle markers):** * Trend: The line generally slopes upward, indicating increasing accuracy with increasing thinking compute. * Data Points: * (20, 0.74) * (40, 0.77) * (60, 0.788) * (80, 0.798) * (100, 0.805) * (120, 0.809) * **short-1@k (Ours) (Light blue line with square markers):** * Trend: The line increases initially, plateaus, and then slightly decreases. * Data Points: * (20, 0.74) * (40, 0.772) * (60, 0.774) * (80, 0.774) * (100, 0.772) * **short-3@k (Ours) (Teal line with diamond markers):** * Trend: The line increases and then plateaus. * Data Points: * (20, 0.74) * (40, 0.78) * (60, 0.794) * (80, 0.796) * (100, 0.799) ### Key Observations * All three models start with the same accuracy at a thinking compute of 20,000 tokens (0.74). * The "majority@k" model consistently increases in accuracy as thinking compute increases. * The "short-1@k (Ours)" model plateaus and slightly decreases after a certain point. * The "short-3@k (Ours)" model plateaus after an initial increase. * The "majority@k" model has the highest accuracy at the highest thinking compute (120,000 tokens). ### Interpretation The chart compares the performance of three different models as a function of thinking compute. The "majority@k" model appears to benefit most from increased thinking compute, as its accuracy consistently increases. The "short-1@k (Ours)" and "short-3@k (Ours)" models show diminishing returns, with accuracy plateauing or even decreasing after a certain point. This suggests that the "majority@k" model may be more efficient or better suited for higher levels of thinking compute compared to the other two models. The "short-3@k (Ours)" model performs better than "short-1@k (Ours)" overall.

## 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 three different methods: "majority@k", "short-1@k (Ours)", and "short-3@k (Ours)". The chart displays how accuracy changes as the amount of thinking compute increases. ### Components/Axes * **X-axis:** "Thinking Compute (thinking tokens in thousands)". Scale ranges from approximately 0 to 120, with markers at 20, 40, 60, 80, 100, and 120. * **Y-axis:** "Accuracy". Scale ranges from approximately 0.74 to 0.81, with gridlines at 0.75, 0.76, 0.77, 0.78, 0.79, 0.80, and 0.81. * **Legend:** Located in the bottom-right corner. Contains the following labels and corresponding colors: * "majority@k" - Dark Red * "short-1@k (Ours)" - Orange-Red * "short-3@k (Ours)" - Light Blue ### Detailed Analysis * **majority@k (Dark Red):** The line slopes upward, indicating increasing accuracy with increasing thinking compute. * At 20 (Thinking Compute), Accuracy is approximately 0.745. * At 40, Accuracy is approximately 0.775. * At 60, Accuracy is approximately 0.788. * At 80, Accuracy is approximately 0.795. * At 100, Accuracy is approximately 0.803. * At 120, Accuracy is approximately 0.808. * **short-1@k (Ours) (Orange-Red):** The line initially rises sharply, then plateaus. * At 20, Accuracy is approximately 0.748. * At 40, Accuracy is approximately 0.776. * At 60, Accuracy is approximately 0.791. * At 80, Accuracy is approximately 0.797. * At 100, Accuracy is approximately 0.801. * At 120, Accuracy is approximately 0.802. * **short-3@k (Ours) (Light Blue):** The line rises rapidly initially, then levels off and slightly decreases. * At 20, Accuracy is approximately 0.752. * At 40, Accuracy is approximately 0.785. * At 60, Accuracy is approximately 0.795. * At 80, Accuracy is approximately 0.796. * At 100, Accuracy is approximately 0.796. * At 120, Accuracy is approximately 0.793. ### Key Observations * "short-3@k (Ours)" achieves the highest accuracy at lower thinking compute values (up to 80). * "majority@k" shows a consistent, albeit slower, increase in accuracy across all thinking compute values. * "short-1@k (Ours)" demonstrates a rapid initial improvement, but its accuracy plateaus quickly. * The accuracy of "short-3@k (Ours)" slightly decreases at the highest thinking compute value (120). ### Interpretation The chart demonstrates the trade-off between computational cost ("Thinking Compute") and accuracy for different methods. "short-3@k (Ours)" appears to be the most efficient method, achieving high accuracy with relatively low computational cost. However, its performance plateaus and even slightly declines at higher compute levels, suggesting diminishing returns. "majority@k" provides a more stable, though slower, improvement in accuracy as compute increases. The plateauing of "short-1@k (Ours)" suggests that it reaches its performance limit quickly. The data suggests that for optimal performance, the choice of method depends on the available computational resources and the desired level of accuracy. The slight decrease in "short-3@k (Ours)" at 120 could indicate overfitting or the need for further optimization at higher compute levels.

## Line Chart: Accuracy vs. Thinking Compute for Different Methods ### Overview The image is a line chart comparing the performance of three different methods ("majority@k", "short-1@k (Ours)", and "short-3@k (Ours)") in terms of accuracy as a function of thinking compute, measured in thousands of thinking tokens. The chart demonstrates how accuracy scales with increased computational resources for each method. ### Components/Axes * **Chart Type:** Line chart with markers. * **X-Axis:** * **Title:** "Thinking Compute (thinking tokens in thousands)" * **Scale:** Linear, ranging from approximately 10 to 125. * **Major Tick Marks:** 20, 40, 60, 80, 100, 120. * **Y-Axis:** * **Title:** "Accuracy" * **Scale:** Linear, ranging from 0.74 to 0.81. * **Major Tick Marks:** 0.74, 0.75, 0.76, 0.77, 0.78, 0.79, 0.80, 0.81. * **Legend:** * **Position:** Bottom-right corner of the chart area. * **Entries:** 1. `majority@k` - Represented by a red line with circular markers. 2. `short-1@k (Ours)` - Represented by a blue line with square markers. 3. `short-3@k (Ours)` - Represented by a cyan (light blue) line with diamond markers. ### Detailed Analysis The chart plots three distinct data series. Below is an analysis of each, including approximate data points extracted from the grid. **1. Data Series: `majority@k` (Red line, circle markers)** * **Trend:** Shows a steady, near-linear upward slope across the entire range of thinking compute. It starts as the lowest-performing method at low compute but eventually surpasses the others. * **Approximate Data Points:** * ~10k tokens: Accuracy ≈ 0.740 * ~40k tokens: Accuracy ≈ 0.770 * ~60k tokens: Accuracy ≈ 0.790 * ~80k tokens: Accuracy ≈ 0.795 * ~100k tokens: Accuracy ≈ 0.802 * ~120k tokens: Accuracy ≈ 0.808 **2. Data Series: `short-1@k (Ours)` (Blue line, square markers)** * **Trend:** Exhibits a rapid initial increase in accuracy, which then plateaus and slightly declines after approximately 60k thinking tokens. It shows diminishing returns. * **Approximate Data Points:** * ~10k tokens: Accuracy ≈ 0.740 * ~20k tokens: Accuracy ≈ 0.762 * ~30k tokens: Accuracy ≈ 0.769 * ~40k tokens: Accuracy ≈ 0.772 * ~60k tokens: Accuracy ≈ 0.774 (peak) * ~80k tokens: Accuracy ≈ 0.774 * ~90k tokens: Accuracy ≈ 0.773 **3. Data Series: `short-3@k (Ours)` (Cyan line, diamond markers)** * **Trend:** Shows a very steep initial increase, followed by a continued but more gradual rise. It maintains the highest accuracy for most of the middle range (approx. 30k to 80k tokens) before being overtaken by `majority@k`. * **Approximate Data Points:** * ~10k tokens: Accuracy ≈ 0.740 * ~20k tokens: Accuracy ≈ 0.763 * ~30k tokens: Accuracy ≈ 0.780 * ~40k tokens: Accuracy ≈ 0.789 * ~60k tokens: Accuracy ≈ 0.795 * ~80k tokens: Accuracy ≈ 0.797 * ~100k tokens: Accuracy ≈ 0.799 ### Key Observations 1. **Convergence at Low Compute:** All three methods start at approximately the same accuracy (≈0.740) when thinking compute is very low (~10k tokens). 2. **Performance Crossover:** There is a notable crossover point between 80k and 100k tokens where the steadily rising `majority@k` line surpasses the `short-3@k` line. 3. **Plateau Behavior:** The `short-1@k` method clearly plateaus, suggesting a limit to its performance gain from additional compute. In contrast, `majority@k` shows no sign of plateauing within the charted range. 4. **Efficiency of "Ours" Methods:** Both methods labeled "(Ours)" achieve higher accuracy than the baseline (`majority@k`) at lower to medium compute budgets (e.g., at 40k tokens, `short-3@k` is ~0.789 vs. `majority@k`'s ~0.770). ### Interpretation This chart likely comes from a research paper on efficient inference or reasoning in language models, where "thinking tokens" represent intermediate computation steps. The data suggests: * **Trade-off Between Efficiency and Peak Performance:** The proposed methods (`short-1@k`, `short-3@k`) are more **compute-efficient**, reaching high accuracy levels with fewer thinking tokens. `short-3@k` is particularly effective in the mid-range. However, the baseline `majority@k` method, while less efficient, appears to have a higher **ultimate performance ceiling** if given sufficient compute resources. * **Methodological Insight:** The "short" methods might involve techniques that truncate or summarize reasoning chains, leading to quick gains but eventual saturation. The `majority@k` method (possibly a form of majority voting over many reasoning paths) scales more predictably with compute, implying it can leverage additional resources to refine answers further without an obvious limit in this range. * **Practical Implication:** The choice of method depends on the operational constraint. For applications with a strict budget on inference compute (tokens), `short-3@k` is optimal. If maximum accuracy is the goal and compute is less constrained, `majority@k` becomes preferable at higher token counts. The chart provides a clear empirical basis for making this trade-off decision.

```markdown ## Line Chart: Accuracy vs. Thinking Compute (Tokens in Thousands) ### Overview The chart compares the accuracy of three methods—**majority@k**, **short-1@k (Ours)**, and **short-3@k (Ours)**—as a function of "Thinking Compute" (measured in thousands of tokens). The y-axis represents accuracy (0.74–0.81), and the x-axis ranges from 20 to 120 thousand tokens. The data shows distinct trends for each method, with **majority@k** demonstrating the most consistent improvement. --- ### Components/Axes - **X-axis**: "Thinking Compute (thinking tokens in thousands)" (20–120, in increments of 20). - **Y-axis**: "Accuracy" (0.74–0.81, in increments of 0.01). - **Legend**: - **Red**: majority@k - **Blue**: short-1@k (Ours) - **Green**: short-3@k (Ours) - **Placement**: Legend is positioned at the **bottom-right** of the chart. --- ### Detailed Analysis #### 1. **majority@k (Red Line)** - **Trend**: Steadily increases from 0.74 (x=20) to 0.81 (x=120). - **Key Points**: - x=20: 0.74 - x=40: 0.77 - x=60: 0.78 - x=80: 0.79 - x=100: 0.80 - x=120: 0.81 #### 2. **short-1@k (Blue Line)** - **Trend**: Sharp initial rise, then plateaus. - **Key Points**: - x=20: 0.74 - x=40: 0.77 - x=60: 0.77 - x=80: 0.77 - x=100: 0.77 - x=120: 0.77 #### 3. **short-3@k (Green Line)** - **Trend**: Gradual increase, outperforming short-1@k but lagging behind majority@k. - **Key Points**: - x=20: 0.74 - x=40: 0.78 - x=60: 0.79 - x=80: 0.79 - x=100: 0.80 - x=120: 0.80 --- ### Key Observations 1. **majority@k** shows the most significant improvement, with a **0.07 accuracy gain** over 100 tokens. 2. **short-1@k** plateaus at 0.77 after x