Independent Metropolis-Hastings

Reward-guided inference-time alignment can improve joint audio-visual generation, but optimizing multiple alignment objectives requires many expensive model evaluations, making existing methods costly under limited compute. Particle-based search enables global exploration but incurs estimation error with a finite population, while Metropolis-Hastings (MH) can refine candidates locally but benefits from a strong initialization. We introduce Spend Where It Matters (\modelname), a two-stage, training-free, reward-agnostic framework that combines particle search with MH refinement under a fixed compute budget. In the first stage, we adaptively allocate particles across denoising checkpoints using online variance estimates, minimizing estimation variance while concentrating compute where it is most useful. This search provides a promising candidate for MH refinement, which uses local noise-and-denoise proposals on completed outputs to further improve reward. By starting from a strong candidate, MH can recover additional reward without requiring a larger particle population or costly global exploration. We show that, at matched compute, combining particle search with MH refinement achieves higher rewards than either approach alone. On JavisBench-Mini, at closely matched compute, \modelname achieves mean relative gains across seven metrics of $30.7\%$ over BoN and $14.1\%$ over EvoSearch with JavisDiT++, and $9.2\%$ over BoN and $17.0\%$ over EvoSearch with LTX-2. Beyond joint audio-visual generation, we also demonstrate the benefits of adaptive particle allocation on SD1.5 image generation, improving AestheticScore by $2.2\%$ and ImageReward by $6.0\%$ over uniform allocation.

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.

Bayesian quantum tomography requires efficient inference while preserving a posterior fixed by the prior and Born likelihood. Learned transport provides fast amortized samples, but reward tuning can reshape the generated distribution rather than improve exploration of this fixed target. We introduce GRPO-QPS, a target-preserving framework in which GRPO learns proposal behavior and an exact Metropolis correction preserves the posterior after training. Across the evaluated reconstruction benchmarks, GRPO-QPS improves over BuresTomFlow and Flow-GRPO on thermal, cat, Dicke, and cluster families, and it closely matches an exact two-qubit reference posterior. Tuned conventional MCMC is slightly stronger on several original continuous benchmarks where the available fixed proposals already match the posterior geometry well. To test whether this reflects a fundamental limitation of learned exploration, we evaluate a more challenging multimodal thermal posterior. At six qubits and 800 shots, the learned proposal achieves a minimum effective sample size of 102 per $1{,}000$ likelihood calls, compared with 28 for prior independence, 27 for a tuned fixed mixture, and 20 for Haario adaptive Metropolis. A record-conditioned policy also transfers to unseen 3,000-shot records, matching or exceeding the strongest conventional baseline in all nine held-out seed-record comparisons. These results show that GRPO-QPS combines target-preserving Bayesian inference with broad gains over learned transport baselines and a sampling advantage when efficient exploration requires proposal geometry beyond the evaluated conventional kernels.

This work shows that diffusion models learned with standard denoising loss can provide effective global MCMC proposals for complex high-dimensional target densities. The method is motivated by the observation that sequentially applying a forward and reverse diffusion process defines a Markov chain with a target stationary distribution for an ideal denoiser trained on samples of the target distribution. This observation can be made exact for any denoiser by applying a Metropolis-Hastings step whose acceptance ratio includes the density of the forward and reverse paths of a discrete time SDE approximation. We therefore propose to train denoising diffusion models on locally convergent MALA samples to learn global MCMC proposals. We call the composition of the global denoiser-based path sampler and a local MALA sampler Denoising Diffusion Monte Carlo (DDMC). Experiments show that DDMC can provide global proposals with high acceptance across a variety of complex target densities. Our results offer preliminary evidence that the established scaling behavior of standard diffusion training transfers directly to exact sampling from high-dimensional unnormalized densities.

We present a simple, yet general approach to study the scaling properties as the dimensionality of Metropolised MCMC sampling algorithms increases. The study relies on the symmetries of the Hamiltonian formalism and ultimately on the symmetry of the Metropolis-Hastings formula. Our findings contain, as particular cases, many known results for the Random Walk Metropolis, MALA and other algorithms. In addition, they provide, in an easy way, new optimal scaling results for a variety of proposal mechanisms, including implicit proposals and proposals generated with the help of differential equation integrators. The analysis applies to targets that are products of a given, not necessarily univariate distribution, and also to cases where the different terms in the product are scaled differently. We show how to construct gradient-based MALA-like proposals where the variance of the proposal as the dimension $d$ increases may be taken as $O(1/d^μ)$, with $μ>0$ arbitrarily small, to be compared with the values $μ= 1$ for Random Walk Metropolis and $μ=1/3$ for MALA.

The Normalized Maximum Likelihood (NML) codelength, or stochastic complexity, represents a principled criterion for universal coding. While recent coarea-based formulations provided a calculation method for smooth models, this framework collapses for the non-smooth estimators ubiquitous in modern machine learning (e.g., Lasso, Sparse SVMs). In this work, we provide a rigorous framework for computing the NML for regular path-differentiable Lipschitz (PDL) estimators. By applying classical geometric measure theory and bridging the coarea formula with conservative Jacobians, we prove that the stochastic complexity for non-smooth models is well-posed and theoretically consistent with the outputs of modern Automatic Differentiation. To compute this quantity exactly, we introduce the Propose-and-Project Metropolis-Hastings (PDL-PPMH) sampler, a geometric MCMC algorithm capable of traversing the non-differentiable level sets of the maximum likelihood estimator. We theoretically justify its components, including a stochastic tangent space proposal and a provably convergent non-smooth projection solver. We demonstrate the method's robustness by sampling from a high-dimensional Lasso posterior ($P=2000$), while simultaneously quantifying the computational scaling that governs the trade-off between exactness and mixing time. Crucially, we empirically demonstrate that our exact NML criterion provides a highly data-efficient alternative to cross-validation, achieving statistically indistinguishable predictive optima without requiring data splitting. Altogether, our work paves the way for the theoretical analysis of the NML codelength for regular non-smooth models.

We consider the problem of sampling from a posterior distribution arising in Bayesian inverse problems in science, engineering, and imaging. Our method belongs to the family of independence Metropolis-Hastings (IMH) sampling algorithms, which are common in Bayesian inference. Relying on the existence of an approximate posterior distribution that is cheaper to sample from but may have significant bias, we introduce Proximal-IMH, a scheme that removes this bias by correcting samples from the approximate posterior through an auxiliary optimization problem. This yields a local adjustment that trades off adherence to the exact model against stability around the approximate reference point. For idealized settings, we prove that the proximal correction tightens the match between approximate and exact posteriors, thereby improving acceptance rates and mixing. The method applies to both linear and nonlinear input-output operators and is particularly suitable for inverse problems where exact posterior sampling is too expensive. We present numerical experiments including multimodal and data-driven priors with nonlinear input-output operators. The results show that Proximal-IMH reliably outperforms existing IMH variants.

Assume that we would like to estimate the expected value of a function $F$ with respect to an intractable density $π$, which is specified up to some unknown normalising constant. We prove that if $π$ is close enough under KL divergence to another density $q$, an independent Metropolis sampler estimator that obtains samples from $π$ with proposal density $q$, enriched with a variance reduction computational strategy based on control variates, achieves smaller asymptotic variance than i.i.d. sampling from $π$. The control variates construction requires no extra computational effort but assumes that the expected value of $F$ under $q$ is analytically available. We illustrate this result by calculating the marginal likelihood in a linear regression model with prior-likelihood conflict and a non-conjugate prior. Furthermore, we propose an adaptive independent Metropolis algorithm that adapts the proposal density such that its KL divergence with the target is being reduced. We demonstrate its applicability in a Bayesian logistic and Gaussian process regression problems and we rigorously justify our asymptotic arguments under easily verifiable and essentially minimal conditions.