Conjugate Natural Selection: Fisher-Rao Natural Gradient Descent Optimally Approximates Evolutionary Dynamics and Continuous Bayesian Inference

Rather than refining individual candidate solutions for a general non-convex optimization problem, by analogy to evolution, we consider minimizing the average loss of a parametric distribution over hypotheses. In this setting, we prove that Fisher-Rao natural gradient descent (FR-NGD) optimally approximates the continuous-time replicator equation, which is an essential model for evolutionary dynamics, by minimizing the mean-squared error of relative fitness. We term this finding "conjugate natural selection" and demonstrate its utility by numerically solving an example non-convex optimization problem over a continuous strategy space. Next, by developing known connections between discrete-time replicator dynamics and Bayes's rule, we show that FR-NGD of the KL-divergence of modeled predictions from observations in continuous time provides the optimal approximation of continuous Bayesian inference. We use this result to demonstrate a novel method for estimating the parameters of a stochastic processes.