Near-Optimal SQ Lower Bounds for Agnostically Learning Halfspaces and ReLUs under Gaussian Marginals
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We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. In the former problem, given labeled examples (𝐱, y) from an unknown distribution on ℝ^d ×{± 1}, whose marginal distribution on 𝐱 is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+ϵ, where OPT is the 0-1 loss of the best-fitting halfspace. In the latter problem, given labeled examples (𝐱, y) from an unknown distribution on ℝ^d ×ℝ, whose marginal distribution on 𝐱 is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with square loss OPT+ϵ, where OPT is the square loss of the best-fitting ReLU. We prove Statistical Query (SQ) lower bounds of d^poly(1/ϵ) for both of these problems. Our SQ lower bounds provide strong evidence that current upper bounds for these tasks are essentially best possible.
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