Shape of Thought: When Distribution Matters More than Correctness in Reasoning Tasks

TL;DR

Shape of Thought challenges the conventional emphasis on final-answer correctness by showing that training on synthetic CoT traces, even when they yield incorrect final answers, can improve reasoning performance beyond human-curated data. The key insight is that closer alignment between the training data distribution and the model’s native distribution, coupled with the presence of reusable intermediate reasoning steps in imperfect CoTs, drives these gains. Paraphrasing human CoTs to reduce distribution mismatch further enhances performance, and controlled degradation studies reveal the model’s tolerance to certain levels of error. Across math, counting, and code-generation tasks and multiple model families, the work highlights data distribution as a critical dimension for scaling reasoning capabilities in LLMs.

Abstract

We present the surprising finding that a language model's reasoning capabilities can be improved by training on synthetic datasets of chain-of-thought (CoT) traces from more capable models, even when all of those traces lead to an incorrect final answer. Our experiments show this approach can yield better performance on reasoning tasks than training on human-annotated datasets. We hypothesize that two key factors explain this phenomenon: first, the distribution of synthetic data is inherently closer to the language model's own distribution, making it more amenable to learning. Second, these `incorrect' traces are often only partially flawed and contain valid reasoning steps from which the model can learn. To further test the first hypothesis, we use a language model to paraphrase human-annotated traces -- shifting their distribution closer to the model's own distribution -- and show that this improves performance. For the second hypothesis, we introduce increasingly flawed CoT traces and study to what extent models are tolerant to these flaws. We demonstrate our findings across various reasoning domains like math, algorithmic reasoning and code generation using MATH, GSM8K, Countdown and MBPP datasets on various language models ranging from 1.5B to 9B across Qwen, Llama, and Gemma models. Our study shows that curating datasets that are closer to the model's distribution is a critical aspect to consider. We also show that a correct final answer is not always a reliable indicator of a faithful reasoning process.

Figures (16)

• Figure 1: Shape of Thought -Fine-tuning on synthetic CoTs, even those with incorrect final answers, can outperform training on human-written data. We generate two synthetic datasets using a stronger model: G, containing CoT traces with correct final answers, and W, containing traces with incorrect final answers. Our results shows that fine-tuning a weaker model on both G and W datasets leads to higher downstream accuracy compared to the baseline of training on human-written CoTs (H) due to the distributional differences in the data.
• Figure 2: Performance on MATH. Gemma-2-2B performance on MATH500 test dataset clearly shows gains from both G and W synthetic dataset over H. Similar trends are also seen in scaling experiments over G-9B (see Table \ref{['tab:scale_and_para_joint']}). (a) G and W outperforms H across most of the iterations. (b) Starting higher train loss for H compared to the synthetic datasets suggests the crucial importance of data distribution in reasoning performance and not just final answer correctness.
• Figure 3: Performance on GSM8K. Results of Gemma-2-2B model on GSM8K task after SFT on Gemma-2-27B-It generated G and W datasets. Here W CoTs match and even slightly surpass G and both clearly outperform H CoTs, showing that they contain useful signals to learn from. For W and G, We show absolute accuracy gains with respect to H performance in Table \ref{['tab:gsm8k_max']}. (a) G and W outperform H across all iterations. (b) Higher starting train loss for H compared to the synthetic datasets which measures data distribution's proximity to model, suggests the importance of data distribution in post-training even beyond correctness.
• Figure 4: Performance on Countdown. In harder tasks like Countdown where base model has near zero performance, even Qwen-2.5-1.5B learns even from W CoTs. Learning from G is better as compared to W. We see similar trends in other models as shown in Table \ref{['tab:countdown_max']}.
• Figure 5: Reasoning performance on code generation on MBPP test data. G and W outperforms H across all three models. Paraphrased human written CoTs being distributionally closer improves performance over H. Training losses below clearly show how H being further away from the model's distribution as compared to the synthetic datasets lead to similar results in other reasoning domains like code generation.