Comparison-Based Natural Gradient Optimization in High Dimension

Abstract : We propose a novel natural gradient based stochastic search algorithm, VD-CMA, for the optimization of high dimensional numerical functions. The algorithm is comparison-based and hence invariant to monotonic transformations of the objective function. It adapts a multivariate normal distribution with a restricted covariance matrix with twice the dimension as degrees of freedom, representing an arbitrarily oriented long axis and additional axis-parallel scaling. We derive the different components of the algorithm and show linear internal time and space complexity. We find empirically that the algorithm adapts its covariance matrix to the inverse Hessian on convex-quadratic functions with an Hessian with one short axis and different scaling on the diagonal. We then evaluate VD-CMA on test functions and compare it to different methods. On functions covered by the internal model of VD-CMA and on the Rosenbrock function, VD-CMA outperforms CMA-ES (having quadratic internal time and space complexity) not only in internal complexity but also in number of function calls with increasing dimension.