Efficient Sampling for Learning Sparse Additive Models in High Dimensions

Abstract

We consider the problem of learning sparse additive models, i.e., functions of the form: f() = _l S _l(x_l), ^d from point queries of f. Here S is an unknown subset of coordinate variables with S = k d. Assuming _l's to be smooth, we propose a set of points at which to sample f and an efficient randomized algorithm that recovers a uniform approximation to each unknown _l. We provide a rigorous theoretical analysis of our scheme along with sample complexity bounds. Our algorithm utilizes recent results from compressive sensing theory along with a novel convex quadratic program for recovering robust uniform approximations to univariate functions, from point queries corrupted with arbitrary bounded noise. Lastly we theoretically analyze the impact of noise -- either arbitrary but bounded, or stochastic -- on the performance of our algorithm.

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Reproductions