Quantum-Inspired Hamiltonian Monte Carlo for Bayesian Sampling

TL;DR

This paper introduces a quantum-inspired variant of Hamiltonian Monte Carlo that employs a random mass matrix to improve sampling efficiency for complex distributions, with applications in large-scale machine learning.

Contribution

It proposes a novel quantum-inspired HMC algorithm with a random mass matrix and develops a stochastic gradient version for large datasets, providing theoretical and empirical validation.

Findings

QHMC improves sampling stability and accuracy.

QSGNHT effectively handles large-scale data.

The methods outperform traditional HMC in experiments.

Abstract

Hamiltonian Monte Carlo (HMC) is an efficient Bayesian sampling method that can make distant proposals in the parameter space by simulating a Hamiltonian dynamical system. Despite its popularity in machine learning and data science, HMC is inefficient to sample from spiky and multimodal distributions. Motivated by the energy-time uncertainty relation from quantum mechanics, we propose a Quantum-Inspired Hamiltonian Monte Carlo algorithm (QHMC). This algorithm allows a particle to have a random mass matrix with a probability distribution rather than a fixed mass. We prove the convergence property of QHMC and further show why such a random mass can improve the performance when we sample a broad class of distributions. In order to handle the big training data sets in large-scale machine learning, we develop a stochastic gradient version of QHMC using Nos{\'e}-Hoover thermostat called QSGNHT, and we also provide theoretical justifications about its steady-state distributions. Finally in the experiments, we demonstrate the effectiveness of QHMC and QSGNHT on synthetic examples, bridge regression, image denoising and neural network pruning. The proposed QHMC and QSGNHT can indeed achieve much more stable and accurate sampling results on the test cases.