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 for 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 (an essential model of evolutionary dynamics) by minimizing the mean-squared error for the relative fitness of competing hypotheses. 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 when absolute fitness corresponds to the negative KL-divergence of a hypothesis's predictions from actual observations, FR-NGD provides the optimal approximation of continuous Bayesian inference. We use this result to demonstrate a novel method for estimating the parameters of stochastic processes.