An adaptive scheme for real function optimization acting as a selection operator

We propose an adaptive scheme for real function optimization whose dynamics is driven by selection. The method is parametric and relies explicitly on the Gaussian density seen as an infinite search population. We define two gradient flows acting on the density parameters, in the spirit of neural network learning rules, which maximize either the function expectation relatively to the density or its logarithm. The first one leads to reinforcement learning and the second one leads to selection learning. Both can be understood as the effect of three operators acting on the density: translation, scaling, and rotation. Then we propose to approximate those systems with discrete time dynamical systems by means of three different methods: Monte Carlo integration, selection among a finite population, and reinforcement learning. This work synthesizes previously independent approaches and intends to show that evolutionary strategies and reinforcement learning are strongly related.