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arXiv · 2610.17735

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

Sungjun Lim·Adrian Robert Minut·Nico Carlsen Daheim·Mohammad Emtiyaz Khan·Kyungwoo Song·Thomas Möllenhoff
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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.

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Sungjun Lim, Adrian Robert Minut, Nico Carlsen Daheim, Mohammad Emtiyaz Khan, Kyungwoo Song, Thomas Möllenhoff. 2026-10-13. Train for Many, Update with One: Improving Multi-Sample Reasoning via Fenchel-Bregman Posterior Learning. https://arxiv.org/abs/2610.17735

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