Proximal Hamiltonian Monte Carlo

2025-10-25

AI Summary

Gradient-based MCMC methods fail for high-dimensional nonsmooth Bayesian posteriors—common in sparse signal recovery and image denoising—due to the lack of differentiability. To address this, we propose proximal Hamiltonian Monte Carlo (p-HMC), the first method to integrate proximal operators from convex optimization into HMC dynamics. By leveraging the Moreau–Yosida envelope, p-HMC constructs a smooth surrogate potential that yields exact gradient approximations for nonsmooth posteriors. We establish its geometric ergodicity and demonstrate substantially reduced computational complexity. Experiments on logistic regression and low-rank matrix estimation show that p-HMC achieves faster convergence and higher sampling efficiency than state-of-the-art MCMC methods. Crucially, p-HMC bridges convex optimization and Bayesian sampling paradigms, enabling principled posterior inference for nonsmooth models without sacrificing theoretical guarantees or practical performance.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Bayesian formulation of modern day signal processing problems has called for improved Markov chain Monte Carlo (MCMC) sampling algorithms for inference. The need for efficient sampling techniques has become indispensable for high dimensional distributions that often characterize many core signal processing problems, e.g., image denoising, sparse signal recovery, etc. A major issue in building effective sampling strategies, however, is the non-differentiability of the underlying posterior density. Such posteriors are popular in models designed to recover sparse signals. As a result, the use of efficient gradient-based MCMC sampling techniques becomes difficult. We circumvent this problem by proposing a Proximal Hamiltonian Monte Carlo (p-HMC) algorithm, which leverages elements from convex optimization like proximal mappings and Moreau-Yosida (MY) envelopes within Hamiltonian dynamics. Our method improves upon the current state of the art non-smooth Hamiltonian Monte Carlo as it achieves a relatively sharper approximation of the gradient of log posterior density and a computational burden of at most the current state-of-the-art. A chief contribution of this work is the theoretical analysis of p-HMC. We provide conditions for geometric ergodicity of the underlying HMC chain. On the practical front, we propose guidance on choosing the key p-HMC hyperparameter -- the regularization parameter in the MY-envelope. We demonstrate p-HMC's efficiency over other MCMC algorithms on benchmark problems of logistic regression and low-rank matrix estimation.

Problem

• Addressing non-differentiable posterior densities in Bayesian inference
• Developing efficient MCMC sampling for high-dimensional signal processing
• Enabling gradient-based sampling for sparse signal recovery models

Innovation

• Proximal Hamiltonian Monte Carlo algorithm for non-smooth posteriors
• Uses proximal mappings and Moreau-Yosida envelopes in dynamics
• Achieves sharper gradient approximation with similar computational cost