Estimating Stochastic Linear Combination of Non-Linear Regressions

Abstract

In this paper we study the problem of estimating stochastic linear combination of non-linear regressions, which has a close connection with many machine learning and statistical models such as non-linear regressions, the Single Index, Multi-index, Varying Coefficient Index Models and Two-layer Neural Networks. Specifically, we first show that with some mild assumptions, if the variate vector x is multivariate Gaussian, then there is an algorithm whose output vectors have ℓ₂-norm estimation errors of O(√p/n) with high probability, where p is the dimension of x and n is the number of samples. Then we extend our result to the case where x is sub-Gaussian using the zero-bias transformation, which could be seen as a generalization of the classic Stein's lemma. We also show that with some additional assumptions there is an algorithm whose output vectors have ℓ∞-norm estimation errors of O(1/√p + √p/n) with high probability. Finally, for both Gaussian and sub-Gaussian cases we propose a faster sub-sampling based algorithm and show that when the sub-sample sizes are large enough then the estimation errors will not be sacrificed by too much. Experiments for both cases support our theoretical results. To the best of our knowledge, this is the first work that studies and provides theoretical guarantees for the stochastic linear combination of non-linear regressions model.