Operator Calculus for Population-Based Optimization: A Mean-Field Convergence Theory

2026-06-12

AI Summary

This work addresses the lack of a unified convergence analysis framework for population-based optimization algorithms, which hinders systematic comparison and generalization. The authors propose an operator calculus framework that models diverse algorithms as compositions of three fundamental operators—mutation, selection, and recombination—acting on probability measures. By leveraging mean-field limits, they derive a continuous-time transport-reaction-jump partial differential equation governing the algorithmic dynamics. Building upon operator semigroup theory and functional analysis on spaces of probability measures, they develop a modular Lyapunov method that enables dissipativity verification operator by operator. Under explicit stability and regularity conditions, they establish exponential decay of both a state-space Lyapunov functional and the search error, thereby providing a unified guarantee of exponential convergence for a broad class of distributed optimization algorithms.

Abstract

Population-based and distributional optimization methods, from evolution strategies and consensus-based optimization to covariance-matrix adaptation and stochastic gradient methods viewed as distributional dynamics, are widely used for nonconvex or black-box problems, yet their convergence analyses remain fragmented across algorithm-specific techniques. We introduce an operator calculus in which a broad class of such methods, after choosing an appropriate state space and, where necessary, augmenting the state by memory or strategy variables, is described as a composition of three elementary operators (mutation, selection, and recombination) acting on probability measures. Under explicit stability and regularity conditions, the composite operator admits a pre-generator whose continuous-time limit is a transport-reaction-jump (TRJ) PDE that preserves the operator splitting. On this foundation we establish a modular Lyapunov principle. If a state-space Lyapunov function both dissipates under the full generator and controls the relevant search-space gauges, then the state-space Lyapunov functional and the induced search errors decay exponentially. The additive generator structure allows dissipation estimates to be assembled operator by operator, providing a toolkit for certifying convergence of composite mean-field algorithms.

Problem

• population-based optimization
• convergence analysis
• mean-field theory
• nonconvex optimization
• black-box optimization

Innovation

• operator calculus
• mean-field convergence
• transport-reaction-jump PDE
• modular Lyapunov principle
• population-based optimization