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There has been a large amount of interest, both in the past and particularlyrecently, into the power of different families of universal approximators, e.g.ReLU networks, polynomials, rational functions. However, current research hasfocused almost exclusively on understanding this problem in a worst-casesetting, e.g. bounding the error of the best infinity-norm approximation in abox. In this setting a high-degree polynomial is required to even approximate asingle ReLU.  However, in real applications with high dimensional data we expect it is onlyimportant to approximate the desired function well on certain relevant parts ofits domain. With this motivation, we analyze the ability of neural networks andpolynomial kernels of bounded degree to achieve good statistical performance ona simple, natural inference problem with sparse latent structure. We givealmost-tight bounds on the performance of both neural networks and low degreepolynomials for this problem. Our bounds for polynomials involve new techniqueswhich may be of independent interest and show major qualitative differenceswith what is known in the worst-case setting.

Representational Power of ReLU Networks and Polynomial Kernels: Beyond Worst-Case Analysis