Sampling from distributions conditioned on desired semantic properties is an emerging challenge in modern generative modeling. Metropolis-Hastings (MH) provides a principled route to conditional sampling, but requires access to exact pointwise target-density evaluations, which are not available in generative settings. Meanwhile, pairwise comparisons by humans or model"judge"are highly accessible and have proved valuable across diverse applications. We introduce Pref-MH, a general exact MH sampler for judge-induced conditional distributions using only stochastic binary pairwise comparisons. Our key observation is that the MH unnormalized density ratio matches the preference odds of the Bradley-Terry (BT) choice model. The central challenge is that while MH requires precise ratio computation, BT judges provide only sampled binary feedback. To this end, we develop a valid accept/reject rule whose resulting Markov chain provably converges to the target distribution. We further show that, for a fixed proposal kernel and budget, Pref-MH is optimal in the Peskun-Tierney sense among this class of exact reversible acceptance rules. Experiments on text generation and molecular design with LLM judges, as well as image generation with VLM judges, demonstrate that Pref-MH provides a practical and flexible approach to conditional sampling when comparative feedback is relatively easy to obtain.
Inference-time selection methods, such as Best-of-N, improve generation by sampling a pool of candidates and selecting the top-ranked completion according to a reward model. Distillation seeks to amortize this procedure into a single policy by replacing raw rewards with in-pool ranks and learning a policy that upweights higher-ranked completions. However, existing rank-based policies typically use smooth full-support reweighting, so low-ranked completions receive less mass but remain in the target support. Although a sharper reweighting reduces lower-tail mass, it also increases reliance on brittle ranking at the top made by a single reward model. We propose TUP: a Truncate-bad, Upweight-good Policy that removes low-ranked completions from the support and reweights only the retained upper tail with a tunable sharpness. TUP admits a closed-form, prompt-independent normalization and can be trained fully offline via binary cross-entropy, using shifted-truncated win-rates as soft labels and distilled-to-reference log-likelihood ratios as logits. Theoretically, under certain assumptions, we show that for any unknown oracle reward, the best monotone rank-reweighting can be matched by a lower-tail truncation rule, providing formal support for removing the lower tail rather than merely downweighting it. Empirically, we show that TUP is competitive with strong offline alignment baselines.
Yarin Bar, Yaniv Romano· 0 citations
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