Train for Many, Update with One: Improving Multi-Sample Reasoning via Fenchel-Bregman Posterior Learning

Abstract:Multi-sample reasoning measures candidate coverage, the probability that at least one of $K$ candidates is correct. However, minimizing the average loss of individual attempts does not directly optimize candidate coverage. We therefore formulate finite-$K$ posterior learning, which trains a distribution over solver parameters to reduce the probability that all $K$ independently generated candidates fail. To optimize this objective efficiently, we propose a Fenchel-Bregman (FB) approach that requires only one posterior draw per update, rather than the $K$ used by direct Monte Carlo optimization. FB first uses an exact Fenchel reformulation to separate problem-specific weights from the posterior update, enabling one-draw posterior updates. It then applies a closed-form Bregman update to track these weights through an exponential running average of observed failures, reusing the same draw. We instantiate FB with IVON, a variational optimizer for Gaussian posteriors over solver parameters, yielding Fenchel-Bregman IVON (FBI). Across diverse recursive and autoregressive reasoning tasks, FBI improves candidate coverage. In a matched comparison, it achieves comparable Pass@10 using 12.3% of the direct Monte Carlo baseline's measured training time per update.

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2610.17735 [cs.LG]

(or arXiv:2610.17735v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.17735 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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