Metropolis-Hastings Dominates Importance Resampling for Policy Composition

Alexey Kurennoy; Ramil Yarullin; Fergal Reid

Home/Archive/arXiv

arXiv · 2610.03480

Metropolis-Hastings Dominates Importance Resampling for Policy Composition

Alexey Kurennoy·Ramil Yarullin·Fergal Reid

View original source ↗Read source document ↗Export metadata ↓

Abstract

Post-training a large language model (LLM) often requires exploring trade-offs between multiple rewards, but retraining for each trade-off is expensive. Decoding-time policy composition allows these trade-offs to be adjusted by combining reward-specific policies at inference time. This composition targets a weighted product of the policies' probabilities over complete responses, but standard implementations combine their next-token probabilities, generally introducing sampling bias. We analyze a known iterative correction based on independence Metropolis-Hastings (MH). Our main result shows that, for every rollout budget, MH produces an output distribution at least as close to the target as sampling-importance-resampling (SIR) with the same budget, as measured by every convex f-divergence. We also derive a lower bound on MH's improvement over the uncorrected decoder in a consensus objective measuring agreement with the supplied policies. We further characterize the correction's sampling error in two asymptotic regimes: when the reward-specific policies approach agreement, and when the log ratio between target and uncorrected-decoder probabilities fluctuates increasingly widely, as can happen for long responses. We complement our analysis with experiments in enumerable and LLM-scale settings.

Explore related subjects

cs.LG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Alexey Kurennoy, Ramil Yarullin, Fergal Reid. 2026-10-02. Metropolis-Hastings Dominates Importance Resampling for Policy Composition. https://arxiv.org/abs/2610.03480

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Robust Adversarial Quantification via Conflict-Aware Evidential Deep Learning

Reliability of deep learning models is critical for deployment in high-stakes applications, where out-of-distribution or adversarial inputs may lead to detrimental outcomes. Evidential Deep Learning, an efficient paradigm for uncertainty quantification, models predictions as Dirichlet distributions of a single forward pass. However, EDL is particularly vulnerable to adversarially perturbed inputs, making overconfident errors. Conflict-aware Evidential Deep Learning~\mbox{(C-EDL)} is a lightweight post-hoc uncertainty quantification approach that mitigates these issues, enhancing adversarial and OOD robustness without retraining. C-EDL generates diverse, task-preserving transformations per input and quantifies representational disagreement to calibrate uncertainty estimates when needed. C-EDL's conflict-aware prediction adjustment improves detection of OOD and adversarial inputs, maintaining high in-distribution accuracy and low computational overhead. Our experimental evaluation shows that C-EDL significantly outperforms state-of-the-art EDL variants and competitive baselines, achieving substantial reductions in coverage for OOD data (up to $\approx55\%$) and adversarial data (up to $\approx90\%$), across a range of datasets, attack types, and uncertainty metrics.

cs.LG↗

arXiv2026

Differential Privacy as a Perk: Federated Learning over Multiple-Access Fading Channels with a Multi-Antenna Base Station

Federated Learning (FL) is a distributed learning paradigm that preserves privacy by eliminating the need to exchange raw data during training. In its prototypical edge instantiation with underlying wireless transmissions enabled by analog over-the-air computing (AirComp), referred to as \emph{over-the-air FL (AirFL)}, the inherent channel noise plays a unique role of \emph{frenemy} in the sense that it degrades training due to noisy global aggregation while providing a natural source of randomness for privacy-preserving mechanisms, formally quantified by \emph{differential privacy (DP)}. It remains, nevertheless, challenging to effectively harness such channel impairments, as prior arts, under assumptions of either simple channel models or restricted types of loss functions, mostly considering (local) DP enhancement with a single-round or non-convergent bound on privacy loss. In this paper, we study AirFL over multiple-access fading channels with a multi-antenna base station (BS) subject to user-level DP requirements. Despite a recent study, which claimed in similar settings that artificial noise (AN) must be injected to ensure DP in general, we demonstrate, on the contrary, that DP can be gained as a \emph{perk} even \emph{without} employing any AN. Specifically, we derive a novel bound on DP that converges under general bounded-domain assumptions on model parameters, along with a convergence bound with general smooth and non-convex loss functions. Next, we optimize over receive beamforming and power allocations to characterize the optimal convergence-privacy trade-offs, which also reveal explicit conditions in which DP is achievable without compromising training. Finally, our theoretical findings are validated by extensive numerical results.

cs.LG↗

arXiv2026

Demystifying LLM-as-a-Judge: Analytically Tractable Model for Inference-Time Scaling

Recent developments in large language models have shown advantages in reallocating a notable share of computational resource from training time to inference time. However, the principles behind inference time scaling are not well understood. In this paper, we introduce an analytically tractable model of inference-time scaling: Bayesian linear regression with a reward-weighted sampler, where the reward is determined from a linear model, modeling LLM-as-a-judge scenario. We study this problem in the high-dimensional regime, where the deterministic equivalents dictate a closed-form expression for the posterior predictive mean and variance. We analyze the generalization error when training data are sampled from a teacher model. We draw $k$ inference-time samples and select via softmax at a temperature applied to a quadratic reward. When the reward is not too different from the teacher, the generalization error decreases monotonically with increasing inference time samples $k$. However, the specific reward that optimizes inference-time selection generally differs from the teacher. In contrast, substantial reward misspecification induces a finite optimal $k$ beyond which more sampling can increase the generalization error. For fixed $k$, there exists an optimal sampling temperature. We experimentally verify these facts in large language model inference with an additional large language model as a judge. In the "best-of-$k$" limit with the teacher as reward, we theoretically show that the generalization error decays as $Θ(1/k^2)$ and determine the leading coefficient via extreme value theory. These formulas delineate domains where scaling inference-time computation is provably preferable to collecting more data. Finally, we demonstrate that when task difficulty increases, the previously mentioned advantage of inference-time compute degrades.

cs.LG↗