# Variance reduction for additive functional of Markov chains via martingale representations

## TL;DR

This paper introduces a new variance reduction technique for additive functionals of Markov chains using a discrete-time martingale representation, improving efficiency without requiring ergodicity or stationary distribution knowledge.

## Contribution

The paper presents a novel non-asymptotic variance reduction method for Markov chains based on martingale representations, applicable to MCMC methods.

## Key findings

- Cost-to-variance ratio is improved over naive algorithms.
- Method does not require stationary distribution or ergodicity.
- Numerical tests show enhanced performance in Langevin MCMC.

## Abstract

In this paper we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete time martingale representation. Our approach is fully non-asymptotic and does not require the knowledge of the stationary distribution (and even any type of ergodicity) or specific structure of the underlying density. By rigorously analyzing the convergence properties of the proposed algorithm, we show that its cost-to-variance product is indeed smaller than one of the naive algorithm. The numerical performance of the new method is illustrated for the Langevin-type Markov Chain Monte Carlo (MCMC) methods.