On the Mixing Time of Markov Chain Monte Carlo for Integer Least-Square Problems

TL;DR

This paper investigates how the structure of the lattice in integer least-square problems affects the mixing time of MCMC algorithms, revealing dependencies on local minima and SNR, with theoretical and empirical insights.

Contribution

It provides a detailed analysis of the factors influencing MCMC mixing times in integer LS problems, including lattice structure and local minima, and offers guidelines for setting temperature parameters.

Findings

Mixing time depends on lattice structure and local minima presence.

For some lattices, mixing time is SNR-independent and polynomial in dimension.

For others, mixing time increases unboundedly with SNR.

Abstract

In this paper, we study the mixing time of Markov Chain Monte Carlo (MCMC) for integer least-square (LS) optimization problems. It is found that the mixing time of MCMC for integer LS problems depends on the structure of the underlying lattice. More specifically, the mixing time of MCMC is closely related to whether there is a local minimum in the lattice structure. For some lattices, the mixing time of the Markov chain is independent of the signal-to-noise ($SNR$) ratio and grows polynomially in the problem dimension; while for some lattices, the mixing time grows unboundedly as $SNR$ grows. Both theoretical and empirical results suggest that to ensure fast mixing, the temperature for MCMC should often grow positively as the $SNR$ increases. We also derive the probability that there exist local minima in an integer least-square problem, which can be as high as $1/3-\frac{1}{\sqrt{5}}+\frac{2\arctan(\sqrt{5/3})}{\sqrt{5}\pi}$.