We prove that Fisher-Rao natural gradient descent (FR-NGD) optimally approximates the continuous time replicator equation (an essential model of evolutionary dynamics), and term this correspondence "conjugate natural selection". This correspondence promises alternative approaches for evolutionary computation over continuous or high-dimensional hypothesis spaces. As a special case, FR-NGD also provides the optimal approximation of continuous Bayesian inference when hypotheses compete on the basis of predicting actual observations. In this case, the method avoids the need to compute prior probabilities. We demonstrate our findings on a non-convex optimization problem and a system identification task for a stochastic process with time-varying parameters.

翻译：我们证明了Fisher-Rao自然梯度下降（FR-NGD）能最优逼近连续时间复制动力方程（进化动力学的核心模型），并将这种对应关系称为"共轭自然选择"。该对应关系为连续或高维假设空间上的进化计算提供了新范式。作为特例，当假设基于实际观测预测能力进行竞争时，FR-NGD还能实现连续贝叶斯推断的最优近似——该方法无需计算先验概率。我们在非凸优化问题及含时变参数的随机过程系统辨识任务中验证了上述发现。