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| <title>Small But Mighty: Analyzing Compact Math Models</title> | |
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| <h1>Small But Mighty: Compact Mathematical Reasoning Models</h1> | |
| <div class="key-finding"> | |
| <h3>Key Insight</h3> | |
| <p>The data shows that some small language models (under 200M parameters) can demonstrate significant mathematical abilities, challenging the assumption that large parameter counts are required for basic math reasoning. Models tested using this <a href="https://github.com/FlameF0X/Simple-Math-Benchmark">link</a>.</p> | |
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| <h2>Model Overview</h2> | |
| <p>This analysis compares the performance of various language models on mathematical problems, with a focus on compact models:</p> | |
| <div class="model-card"> | |
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| <span class="model-name">FlameF0X/MathGPT2</span> | |
| <span class="model-params">81.9M parameters</span> | |
| </div> | |
| <p>The highest performer among tested models, demonstrating remarkable mathematical abilities despite its tiny parameter count. Shows particular strength in addition operations with 63.1% accuracy.</p> | |
| <div class="performance-highlight"> | |
| <strong>Overall math accuracy:</strong> 42.7% on 1000 test questions | |
| </div> | |
| </div> | |
| <div class="model-card"> | |
| <div class="model-info"> | |
| <span class="model-name">PingVortex/VLM-1</span> | |
| <span class="model-params">124M parameters</span> | |
| </div> | |
| <p>The second best performer, scoring 4.8% overall accuracy. Shows more balanced performance across operations with particular strength in division (14.2%).</p> | |
| <div class="performance-highlight"> | |
| <strong>Operation strength:</strong> 14.2% accuracy on division | |
| </div> | |
| </div> | |
| <div class="model-card"> | |
| <div class="model-info"> | |
| <span class="model-name">aquiffoo/aquif-r1-1b</span> | |
| <span class="model-params">1.14B parameters + CoT reasoning</span> | |
| </div> | |
| <p>Despite being significantly larger and incorporating chain-of-thought reasoning capabilities, this model shows no measurable performance on the tested mathematical problems.</p> | |
| </div> | |
| <h2>Performance Analysis</h2> | |
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| <img src="6818abac-ba0b-4fae-aaf4-d42a9d4ebc04.png" alt="Chart showing model accuracy by operation type"> | |
| <div class="chart-caption">Figure 1: Accuracy by Operation Type (%)</div> | |
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| <img src="284d12f0-c0f1-4e2f-8455-1ad7fefc3e1e.png" alt="Chart showing model performance on math problems"> | |
| <div class="chart-caption">Figure 2: Correct vs Incorrect Answers (1000 questions each)</div> | |
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| <h2>Detailed Accuracy Results</h2> | |
| <table> | |
| <thead> | |
| <tr> | |
| <th>Model</th> | |
| <th>Addition (+)</th> | |
| <th>Subtraction (-)</th> | |
| <th>Multiplication (*)</th> | |
| <th>Division (/)</th> | |
| <th>Exponentiation (**)</th> | |
| <th>Overall</th> | |
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| <tr class="highlight"> | |
| <td>MathGPT2 (81.9M)</td> | |
| <td>63.1%</td> | |
| <td>59.3%</td> | |
| <td>34.9%</td> | |
| <td>22.3%</td> | |
| <td>8.8%</td> | |
| <td>42.7%</td> | |
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| <tr> | |
| <td>VLM-1 (124M)</td> | |
| <td>2.6%</td> | |
| <td>3.1%</td> | |
| <td>0.0%</td> | |
| <td>14.2%</td> | |
| <td>0.0%</td> | |
| <td>4.8%</td> | |
| </tr> | |
| <tr> | |
| <td>aquif-r1-1b (1.14B)</td> | |
| <td>0.0%</td> | |
| <td>0.0%</td> | |
| <td>0.0%</td> | |
| <td>0.0%</td> | |
| <td>0.0%</td> | |
| <td>0.0%</td> | |
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| </table> | |
| <h2>Key Observations</h2> | |
| <ul> | |
| <li><strong>Size doesn't always matter:</strong> MathGPT2 with only 81.9M parameters demonstrates impressive mathematical abilities, outperforming larger models.</li> | |
| <li><strong>Operation specialization:</strong> MathGPT2 excels at addition (63.1%) and subtraction (59.3%), while VLM-1 shows particular strength in division operations (14.2%).</li> | |
| <li><strong>Architectural importance:</strong> The results suggest that architecture design and training approach may be more important than raw parameter count for specialized tasks.</li> | |
| <li><strong>Zero performance:</strong> Four of the tested models showed no measurable mathematical ability on this test set.</li> | |
| <li><strong>Chain-of-thought limitations:</strong> Despite having CoT capabilities, aquif-r1-1b did not demonstrate mathematical reasoning abilities in this benchmark.</li> | |
| </ul> | |
| <div class="key-finding"> | |
| <h3>Research Implications</h3> | |
| <p>These results challenge conventional wisdom about language model scaling laws for mathematical reasoning capabilities. They suggest that highly specialized, compact models might offer a more efficient approach for specific reasoning tasks compared to general-purpose large models.</p> | |
| </div> | |
| <h2>Conclusion</h2> | |
| <p>This analysis demonstrates that extremely small language models can exhibit significant mathematical reasoning abilities, with models as small as 81.9M parameters showing the ability to solve basic arithmetic problems. The standout performer, MathGPT2 with only 81.9M parameters, achieved an impressive 42.7% accuracy on a diverse set of 1000 mathematical questions.</p> | |
| <p>These findings suggest that efficient architectural design and specialized training approaches may be more important than raw parameter count when optimizing for specific reasoning capabilities. This could have significant implications for resource-constrained applications where deploying massive models is impractical.</p> | |
| <p>Future research directions could include investigating what specific architectural choices enable these compact models to perform mathematical operations, and how these insights might be applied to develop more efficient specialized models for other reasoning tasks.</p> | |
| <div class="footer"> | |
| <p>Data analysis based on benchmark results for MathGPT2 (81.9M), VLM-1 (124M), aquif-r1-1b (1.14B), and other models</p> | |
| <p>© 2025 • Created for educational purposes</p> | |
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