Adaptive Stochastic Natural Gradient Method for Optimizing Functions with Low Effective Dimensionality

Abstract

Black-box optimization algorithms, such as evolutionary algorithms, have been recognized as useful tools for real-world applications. Several efficient probabilistic model-based evolutionary algorithms, such as the compact genetic algorithm (cGA) and the covariance matrix adaptation evolution strategy (CMA-ES), can be regarded as a stochastic natural gradient ascent on statistical manifolds. Our baseline algorithm is the adaptive stochastic natural gradient (ASNG) method which automatically adapts the learning rate based on the signal-to-noise ratio (SNR) of the approximated natural gradient. ASNG has shown effectiveness in a practical application, but the convergence speed of ASNG deteriorates on objective functions with low effective dimensionality (LED), where LED means that part of the design variables is ineffective or does not affect the objective value significantly. In this paper, we propose an element-wise adjustment method for the approximated natural gradient based on the element-wise SNR and introduce the proposed adjustment method into ASNG. The proposed method suppresses the natural gradient elements with the low SNRs, helping to accelerate the learning rate adaptation in ASNG. We incorporate the proposed method into the cGA and demonstrate the effectiveness of the proposed method on the benchmark functions of binary optimization.