Momentum Accelerates Evolutionary Dynamics

TL;DR

This paper introduces momentum into evolutionary dynamics, demonstrating it accelerates convergence, can alter stability, and affects cycling behavior, with broad applicability beyond gradient flows.

Contribution

It is the first to incorporate momentum into evolutionary dynamics, showing how it accelerates convergence and modifies stability and cycling behaviors.

Findings

Momentum accelerates convergence of evolutionary dynamics.

Momentum can break cycles like rock-paper-scissors.

Convergence properties depend on momentum parameters.

Abstract

We combine momentum from machine learning with evolutionary dynamics, where momentum can be viewed as a simple mechanism of intergenerational memory. Using information divergences as Lyapunov functions, we show that momentum accelerates the convergence of evolutionary dynamics including the replicator equation and Euclidean gradient descent on populations. When evolutionarily stable states are present, these methods prove convergence for small learning rates or small momentum, and yield an analytic determination of the relative decrease in time to converge that agrees well with computations. The main results apply even when the evolutionary dynamic is not a gradient flow. We also show that momentum can alter the convergence properties of these dynamics, for example by breaking the cycling associated to the rock-paper-scissors landscape, leading to either convergence to the ordinarily non-absorbing equilibrium, or divergence, depending on the value and mechanism of momentum.