Generalized EXTRA stochastic gradient Langevin dynamics

TL;DR

This paper introduces a generalized decentralized Langevin dynamics algorithm that removes bias in Bayesian learning over networks, improving performance bounds and efficiency compared to existing methods.

Contribution

It proposes a novel generalized EXTRA-based decentralized SGLD algorithm that eliminates bias in full-batch settings and enhances performance bounds in mini-batch scenarios.

Findings

Bias is eliminated in full-batch decentralized SGLD.

Performance bounds are significantly improved in mini-batch settings.

Numerical results demonstrate the efficiency of the proposed method.

Abstract

Langevin algorithms are popular Markov Chain Monte Carlo methods for Bayesian learning, particularly when the aim is to sample from the posterior distribution of a parametric model, given the input data and the prior distribution over the model parameters. Their stochastic versions such as stochastic gradient Langevin dynamics (SGLD) allow iterative learning based on randomly sampled mini-batches of large datasets and are scalable to large datasets. However, when data is decentralized across a network of agents subject to communication and privacy constraints, standard SGLD algorithms cannot be applied. Instead, we employ decentralized SGLD (DE-SGLD) algorithms, where Bayesian learning is performed collaboratively by a network of agents without sharing individual data. Nonetheless, existing DE-SGLD algorithms induce a bias at every agent that can negatively impact performance; this bias persists even when using full batches and is attributable to network effects. Motivated by the EXTRA algorithm and its generalizations for decentralized optimization, we propose the generalized EXTRA stochastic gradient Langevin dynamics, which eliminates this bias in the full-batch setting. Moreover, we show that, in the mini-batch setting, our algorithm provides performance bounds that significantly improve upon those of standard DE-SGLD algorithms in the literature. Our numerical results also demonstrate the efficiency of the proposed approach.