On the Geometric Ergodicity of Metropolis-Hastings Algorithms for Lattice Gaussian Sampling

TL;DR

This paper adapts the Metropolis-Hastings algorithm for lattice Gaussian sampling, proving uniform and geometric ergodicity of the proposed algorithms, which ensures fast and reliable convergence in cryptography and coding applications.

Contribution

It introduces two MH-based algorithms for lattice Gaussian sampling and proves their ergodicity properties, improving convergence guarantees over existing methods.

Findings

The independent MHK algorithm is uniformly ergodic with exponential convergence.

The SMK algorithm is geometrically ergodic, with convergence enhanced by initial state selection.

Explicit convergence rates are derived based on theta series.

Abstract

Sampling from the lattice Gaussian distribution is emerging as an important problem in coding and cryptography. In this paper, the classic Metropolis-Hastings (MH) algorithm from Markov chain Monte Carlo (MCMC) methods is adapted for lattice Gaussian sampling. Two MH-based algorithms are proposed, which overcome the restriction suffered by the default Klein's algorithm. The first one, referred to as the independent Metropolis-Hastings-Klein (MHK) algorithm, tries to establish a Markov chain through an independent proposal distribution. We show that the Markov chain arising from the independent MHK algorithm is uniformly ergodic, namely, it converges to the stationary distribution exponentially fast regardless of the initial state. Moreover, the rate of convergence is explicitly calculated in terms of the theta series, leading to a predictable mixing time. In order to further exploit the convergence potential, a symmetric Metropolis-Klein (SMK) algorithm is proposed. It is proven that the Markov chain induced by the SMK algorithm is geometrically ergodic, where a reasonable selection of the initial state is capable to enhance the convergence performance.