Convergence analysis of the discrete consensus-based optimization algorithm with random batch interactions and heterogeneous noises

TL;DR

This paper rigorously analyzes the convergence of a generalized discrete consensus-based optimization algorithm with random batch interactions and heterogeneous noises, proving exponential convergence under certain conditions.

Contribution

It introduces a generalized discrete CBO algorithm with random batch interactions and heterogeneous noises, providing the first rigorous convergence analysis in this setting.

Findings

Proves exponential convergence of the discrete CBO algorithm under suitable conditions.

Recasts the algorithm as a consensus model with a random switching network topology.

Improves upon previous analyses by handling heterogeneous noises and random batch interactions.

Abstract

We present stochastic consensus and convergence of the discrete consensus-based optimization (CBO) algorithm with random batch interactions and heterogeneous external noises. Despite the wide applications and successful performance in many practical simulations, the convergence of the discrete CBO algorithm was not rigorously investigated in such a generality. In this work, we introduce a generalized discrete CBO algorithm with a weighted representative point and random batch interactions, and show that the proposed discrete CBO algorithm exhibits stochastic consensus and convergence toward the common equilibrium state exponentially fast under suitable assumptions on system parameters. For this, we recast the given CBO algorithm with random batch interactions as a discrete consensus model with a random switching network topology, and then we use the mixing property of interactions over sufficiently long time interval to derive stochastic consensus and convergence estimates in mean square and almost sure senses. Our proposed analysis significantly improves earlier works on the convergence analysis of CBO models with full batch interactions and homogeneous external noises.