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Pretrained language models have shown superior performance on many naturallanguage processing tasks, yet they still struggle at multi-step formalreasoning tasks like grade school math problems. One key challenge offinetuning them to solve such math reasoning problems is that many existingdatasets only contain one reference solution for each problem, despite the factthat there are often alternative solutions resembling different reasoning pathsto the final answer. This way, the finetuned models are biased towards thelimited reference solutions, which limits their generalization to unseenexamples. To mitigate this issue, we propose to let the model perform samplingduring training and learn from both self-sampled fully-correct solutions, whichyield the correct answer upon execution, and partially-correct solutions, whoseintermediate state matches an intermediate state of a known correct solution.We show that our use of self-sampled correct and partially-correct solutionscan benefit learning and help guide the sampling process, leading to moreefficient exploration of the solution space. Additionally, we explore varioustraining objectives to support learning from multiple solutions per example andfind they greatly affect the performance. Experiments on two math reasoningdatasets show the effectiveness of our method compared to learning from asingle reference solution with MLE, where we improve PASS@100 from 35.5% to44.5% for GSM8K, and 27.6% to 36.2% PASS@80 for MathQA. Such improvements arealso consistent across different model sizes. Our code is available athttps://github.com/microsoft/TraceCodegen.

Learning Math Reasoning from Self-Sampled Correct and Partially-Correct Solutions