## Line Chart: Accuracy vs. Thinking Compute ### Overview The image is a line chart comparing the accuracy of three different methods ("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 how accuracy changes with increasing computational effort for each method. ### Components/Axes * **X-axis:** Thinking Compute (thinking tokens in thousands). Scale ranges from approximately 10 to 120, with tick marks at intervals of 20. * **Y-axis:** Accuracy. Scale ranges from 0.74 to 0.81, with tick marks at intervals of 0.01. * **Legend:** Located in the bottom-right corner of the chart. * **Brown line with circle markers:** "majority@k" * **Blue line with square markers:** "short-1@k (Ours)" * **Cyan line with diamond markers:** "short-3@k (Ours)" ### Detailed Analysis * **majority@k (Brown line):** The line starts at approximately (15, 0.74) and slopes upward. * (15, 0.74) * (40, 0.77) * (60, 0.79) * (80, 0.80) * (100, 0.805) * (125, 0.81) * **short-1@k (Ours) (Blue line):** The line starts at approximately (15, 0.74) and increases to a peak, then decreases slightly. * (15, 0.74) * (30, 0.77) * (50, 0.774) * (70, 0.774) * (90, 0.772) * **short-3@k (Ours) (Cyan line):** The line starts at approximately (15, 0.74) and slopes upward, plateauing around 80. * (15, 0.74) * (25, 0.762) * (40, 0.79) * (60, 0.795) * (80, 0.798) * (100, 0.798) ### Key Observations * Initially, "short-3@k (Ours)" achieves higher accuracy with less compute compared to "majority@k" and "short-1@k (Ours)". * "short-1@k (Ours)" plateaus and even slightly decreases in accuracy after a certain compute level. * "majority@k" consistently increases in accuracy with increasing compute, eventually surpassing "short-3@k (Ours)". ### Interpretation The chart suggests that "short-3@k (Ours)" is more efficient in terms of accuracy gain for lower compute budgets. However, "majority@k" eventually outperforms the other methods with sufficient computational resources. "short-1@k (Ours)" appears to have diminishing returns and may not be as effective for higher compute levels. The data indicates a trade-off between initial efficiency and long-term performance depending on the available compute. The "Ours" label suggests that "short-1@k" and "short-3@k" are novel methods being compared against the baseline "majority@k".

## 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 (labeled as "Ours"), and short-3@k (also labeled as "Ours"). The chart shows 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 markers at 0.74, 0.76, 0.78, 0.80, and 0.81. * **Legend:** Located in the top-right corner. Contains the following labels and corresponding colors: * majority@k (Red-Brown) * short-1@k (Ours) (Pink) * short-3@k (Ours) (Light Blue) * **Gridlines:** A light gray grid is present to aid in reading values. ### Detailed Analysis * **majority@k (Red-Brown Line):** This line starts at approximately 0.745 at a Thinking Compute of 0, and slopes upward, reaching approximately 0.812 at a Thinking Compute of 120. * (0, 0.745) * (20, 0.765) * (40, 0.780) * (60, 0.790) * (80, 0.798) * (100, 0.805) * (120, 0.812) * **short-1@k (Ours) (Pink Line):** This line exhibits a steep initial increase, starting at approximately 0.74 at a Thinking Compute of 0, and quickly rises to approximately 0.795 at a Thinking Compute of 60. It then plateaus, reaching approximately 0.802 at a Thinking Compute of 120. * (0, 0.74) * (20, 0.775) * (40, 0.790) * (60, 0.795) * (80, 0.800) * (100, 0.801) * (120, 0.802) * **short-3@k (Ours) (Light Blue Line):** This line also shows a rapid initial increase, starting at approximately 0.74 at a Thinking Compute of 0, and reaching approximately 0.795 at a Thinking Compute of 60. It then levels off, with a slight decrease, reaching approximately 0.778 at a Thinking Compute of 120. * (0, 0.74) * (20, 0.780) * (40, 0.790) * (60, 0.795) * (80, 0.785) * (100, 0.782) * (120, 0.778) ### Key Observations * All three methods start with similar accuracy levels at low Thinking Compute. * The "short-1@k (Ours)" and "short-3@k (Ours)" methods demonstrate significantly faster initial accuracy gains compared to "majority@k". * The "short-1@k (Ours)" method achieves the highest accuracy overall, but its gains plateau after approximately 60 Thinking Compute. * The "short-3@k (Ours)" method shows a slight decrease in accuracy at higher Thinking Compute values (beyond 60). ### Interpretation The data suggests that the "short-1@k (Ours)" method is the most effective for achieving high accuracy with a relatively low amount of Thinking Compute. The initial rapid gains indicate that this method efficiently utilizes the available compute resources. The plateauing of accuracy suggests that there are diminishing returns beyond a certain point (around 60 Thinking Compute). The slight decline in accuracy for "short-3@k (Ours)" at higher compute levels could indicate overfitting or the introduction of noise with increased complexity. The "majority@k" method, while consistently improving, requires significantly more compute to reach comparable accuracy levels. This chart demonstrates a trade-off between computational cost and accuracy, and highlights the potential benefits of the "short-1@k (Ours)" approach for optimizing performance. The "Ours" label suggests these are novel methods being compared to a baseline.

## Line Chart: Accuracy vs. Thinking Compute for Different Reasoning Methods ### Overview The image is a line chart comparing the performance of three different computational reasoning methods. It plots model accuracy against the amount of "Thinking Compute" allocated, measured in thousands of thinking tokens. The chart demonstrates how accuracy scales with increased computational resources for each method. ### Components/Axes * **X-Axis (Horizontal):** Labeled "Thinking Compute (thinking tokens in thousands)". The scale runs from 0 to 120, with major tick marks at intervals of 20 (0, 20, 40, 60, 80, 100, 120). * **Y-Axis (Vertical):** Labeled "Accuracy". The scale runs from 0.74 to 0.81, with major tick marks at intervals of 0.01 (0.74, 0.75, 0.76, 0.77, 0.78, 0.79, 0.80, 0.81). * **Legend:** Located in the bottom-right quadrant of the chart area. It contains three entries: 1. `majority@k` - Represented by a solid red line with circular markers. 2. `short-1@k (Ours)` - Represented by a solid blue line with square markers. 3. `short-3@k (Ours)` - Represented by a solid cyan (light blue) line with diamond markers. * **Grid:** A light gray grid is present, aligning with the major tick marks on both axes. ### Detailed Analysis **1. `majority@k` (Red Line, Circle Markers):** * **Trend:** Shows a steady, near-linear upward trend across the entire range of compute. It has the steepest sustained slope. * **Data Points (Approximate):** * Starts at ~0.74 accuracy at 10k tokens. * Crosses 0.77 accuracy at ~40k tokens. * Crosses 0.79 accuracy at ~65k tokens. * Crosses 0.80 accuracy at ~95k tokens. * Ends at the highest point on the chart, ~0.808 accuracy at 120k tokens. **2. `short-1@k (Ours)` (Blue Line, Square Markers):** * **Trend:** Increases rapidly at low compute, then plateaus. The curve flattens significantly after ~50k tokens, showing diminishing returns. * **Data Points (Approximate):** * Starts at ~0.74 accuracy at 10k tokens. * Rises sharply to ~0.762 at 20k tokens. * Reaches ~0.771 at 40k tokens. * Peaks and plateaus around 0.774 between 50k and 80k tokens. * Shows a slight decline to ~0.773 at 90k tokens (the last data point for this series). **3. `short-3@k (Ours)` (Cyan Line, Diamond Markers):** * **Trend:** Exhibits the most rapid initial gain in accuracy, then also plateaus, but at a higher level than `short-1@k`. Its growth rate slows considerably after ~50k tokens. * **Data Points (Approximate):** * Starts at ~0.74 accuracy at 10k tokens. * Rises very steeply to ~0.763 at 20k tokens. * Reaches ~0.78 at 35k tokens. * Crosses 0.79 accuracy at ~45k tokens. * Plateaus near 0.799-0.80 between 80k and 100k tokens (the last data point for this series). ### Key Observations 1. **Initial Efficiency:** Both "Ours" methods (`short-1@k` and `short-3@k`) show a steeper initial slope than `majority@k`, indicating they achieve higher accuracy with very low compute budgets (below ~30k tokens). 2. **Crossover Point:** The `majority@k` line intersects and surpasses the `short-1@k` line at approximately 40k tokens. It intersects the `short-3@k` line at approximately 80k tokens. 3. **Plateau vs. Continuous Growth:** The two "Ours" methods plateau, suggesting a limit to the accuracy gains achievable by their specific approach with more compute. In contrast, `majority@k` continues to improve steadily, indicating its scaling behavior is different and potentially more robust at high compute levels. 4. **Performance Hierarchy:** At low compute (<40k tokens), the order is `short-3@k` > `short-1@k` ≈ `majority@k`. At high compute (>80k tokens), the order becomes `majority@k` > `short-3@k` > `short-1@k`. ### Interpretation This chart illustrates a classic trade-off in machine learning between **sample efficiency** and **scalability**. * The methods labeled "(Ours)" (`short-1@k` and `short-3@k`) are highly **sample/compute-efficient**. They extract maximum accuracy from a limited thinking budget, making them ideal for applications where computational cost or latency is a primary constraint. `short-3@k` is clearly the more effective of the two efficient methods. * The `majority@k` method represents a **scalable** approach. While less efficient initially, its performance continues to improve predictably with more resources. This suggests it may be a more reliable or powerful technique when computational constraints are relaxed, and maximum accuracy is the goal, regardless of cost. * The plateauing of the "Ours" methods could indicate a fundamental limitation in their architecture or strategy—they may be "thinking" in a way that quickly hits a ceiling of effectiveness. The continuous rise of `majority@k` suggests its "thinking" process (likely involving majority voting over multiple reasoning paths) benefits more consistently from additional computation. * **Practical Implication:** The choice between these methods depends on the deployment context. For a real-time chatbot, an efficient method like `short-3@k` is preferable. For a offline, high-stakes analysis where accuracy is paramount, `majority@k` would be the better choice given sufficient compute resources.

## Line Graph: Accuracy vs. Thinking Tokens (in Thousands) ### Overview The image is a line graph comparing the accuracy of three methods—**majority@k**, **short-1@k (Ours)**, and **short-3@k (Ours)**—as a function of the number of thinking tokens (in thousands). The x-axis represents the number of thinking tokens, and the y-axis represents accuracy (ranging from 0.74 to 0.81). Three distinct lines are plotted, each corresponding to a method, with the legend positioned in the bottom-right corner. --- ### Components/Axes - **X-axis**: "Thinking tokens in thousands" (range: 20 to 120, increments of 20). - **Y-axis**: "Accuracy" (range: 0.74 to 0.81, increments of 0.01). - **Legend**: Located in the bottom-right corner, with three entries: - **Red**: majority@k - **Blue**: short-1@k (Ours) - **Green**: short-3@k (Ours) --- ### Detailed Analysis #### 1. **majority@k (Red Line)** - **Trend**: Starts at the lowest point (0.74 at 20k tokens) and increases steadily. - **Key Data Points**: - 20k tokens: ~0.74 - 40k tokens: ~0.76 - 60k tokens: ~0.77 - 80k tokens: ~0.78 - 100k tokens: ~0.79 - 120k tokens: ~0.81 #### 2. **short-1@k (Ours) (Blue Line)** - **Trend**: Starts higher than majority@k but plateaus after 60k tokens. - **Key Data Points**: - 20k tokens: ~0.76 - 40k tokens: ~0.77 - 60k tokens: ~0.77 - 80k tokens: ~0.77 - 100k tokens: ~0.77 - 120k tokens: ~0.765 #### 3. **short-3@k (Ours) (Green Line)** - **Trend**: Starts at the lowest point (0.74 at 20k tokens), rises sharply, dips slightly, then surpasses majority@k after 100k tokens. - **Key Data Points**: - 20k tokens: ~0.74 - 40k tokens: ~0.78 - 60k tokens: ~0.79 - 80k tokens: ~0.785 - 100k tokens: ~0.795 - 120k tokens: ~0.81 --- ### Key Observations 1. **majority@k** shows a consistent upward trend, achieving the highest accuracy (0.81) at 120k tokens. 2. **short-1@k** plateaus at ~0.77 after 60k tokens, indicating diminishing returns. 3. **short-3@k** initially underperforms but surpasses majority@k after 100k tokens, suggesting potential for optimization. 4. The green line (short-3@k) dips slightly at 80k tokens but recovers by 100k tokens. --- ### Interpretation The data suggests that **majority@k** is the most reliable method for accuracy across all token ranges, while **short-3@k** demonstrates a non-linear improvement, possibly due to adaptive scaling or optimization. The **short-1@k** method’s plateau implies it may not benefit from additional tokens beyond 60k. The dip in short-3@k at 80k tokens could indicate a temporary inefficiency, but its recovery suggests robustness in larger-scale applications. This highlights the importance of method selection based on token availability and performance goals.