Variance reduction for Random Coordinate Descent-Langevin Monte Carlo

TL;DR

This paper introduces a variance reduction technique called RCAD for Random Coordinate Descent-Langevin Monte Carlo, significantly improving efficiency in sampling from log-concave distributions by reducing variance without increasing iteration count.

Contribution

The paper proposes the RCAD method that reduces variance in RCD-LMC, enabling faster convergence and computational savings in high-dimensional sampling tasks.

Findings

RCAD reduces variance in RCD-LMC methods.

RCAD-O-LMC and RCAD-U-LMC achieve convergence with fewer total computations.

The methods outperform traditional RCD-LMC in efficiency.

Abstract

Sampling from a log-concave distribution function is one core problem that has wide applications in Bayesian statistics and machine learning. While most gradient free methods have slow convergence rate, the Langevin Monte Carlo (LMC) that provides fast convergence requires the computation of gradients. In practice one uses finite-differencing approximations as surrogates, and the method is expensive in high-dimensions. A natural strategy to reduce computational cost in each iteration is to utilize random gradient approximations, such as random coordinate descent (RCD) or simultaneous perturbation stochastic approximation (SPSA). We show by a counter-example that blindly applying RCD does not achieve the goal in the most general setting. The high variance induced by the randomness means a larger number of iterations are needed, and this balances out the saving in each iteration. We then introduce a new variance reduction approach, termed Randomized Coordinates Averaging Descent (RCAD), and incorporate it with both overdamped and underdamped LMC. The methods are termed RCAD-O-LMC and RCAD-U-LMC respectively. The methods still sit in the random gradient approximation framework, and thus the computational cost in each iteration is low. However, by employing RCAD, the variance is reduced, so the methods converge within the same number of iterations as the classical overdamped and underdamped LMC. This leads to a computational saving overall.