ADAPTIVE RANDOM FOURIER FEATURES WITH METROPOLIS SAMPLING

Abstract

The supervised learning problem to determine a neural network approximation Rd 3 x 7→ PKk=1 β̂keiωk·x with one hidden layer is studied as a random Fourier features algorithm. The Fourier features, i.e., the frequencies ωk ∈ Rd, are sampled using an adaptive Metropolis sampler. The Metropolis test accepts proposal frequencies ωk0 , having corresponding amplitudes β̂k0 , with the probability min {1, (|β̂k0 |/|β̂k|)γ}, for a certain positive parameter γ, determined by minimizing the approximation error for given computational work. This adaptive, non-parametric stochastic method leads asymptotically, as K → ∞, to equidistributed amplitudes |β̂k|, analogous to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods. Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is tested both on synthetic data and a real-world high-dimensional benchmark.