Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise
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We study the problem of learning general (i.e., not necessarily homogeneous) halfspaces with Random Classification Noise under the Gaussian distribution. We establish nearly-matching algorithmic and Statistical Query (SQ) lower bound results revealing a surprising information-computation gap for this basic problem. Specifically, the sample complexity of this learning problem is Θ(d/ϵ), where d is the dimension and ϵ is the excess error. Our positive result is a computationally efficient learning algorithm with sample complexity Õ(d/ϵ + d/(max{p, ϵ})^2), where p quantifies the bias of the target halfspace. On the lower bound side, we show that any efficient SQ algorithm (or low-degree test) for the problem requires sample complexity at least Ω(d^1/2/(max{p, ϵ})^2). Our lower bound suggests that this quadratic dependence on 1/ϵ is inherent for efficient algorithms.
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