Toward L_∞-recovery of Nonlinear Functions: A Polynomial Sample Complexity Bound for Gaussian Random Fields
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Many machine learning applications require learning a function with a small worst-case error over the entire input domain, that is, the L_∞-error, whereas most existing theoretical works only guarantee recovery in average errors such as the L_2-error. L_∞-recovery from polynomial samples is even impossible for seemingly simple function classes such as constant-norm infinite-width two-layer neural nets. This paper makes some initial steps beyond the impossibility results by leveraging the randomness in the ground-truth functions. We prove a polynomial sample complexity bound for random ground-truth functions drawn from Gaussian random fields. Our key technical novelty is to prove that the degree-k spherical harmonics components of a function from Gaussian random field cannot be spiky in that their L_∞/L_2 ratios are upperbounded by O(d √(ln k)) with high probability. In contrast, the worst-case L_∞/L_2 ratio for degree-k spherical harmonics is on the order of Ω(min{d^k/2,k^d/2}).
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