Provable Robustness of Adversarial Training for Learning Halfspaces with Noise
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We analyze the properties of adversarial training for learning adversarially robust halfspaces in the presence of agnostic label noise. Denoting 𝖮𝖯𝖳_p,r as the best robust classification error achieved by a halfspace that is robust to perturbations of ℓ_p balls of radius r, we show that adversarial training on the standard binary cross-entropy loss yields adversarially robust halfspaces up to (robust) classification error Õ(√(𝖮𝖯𝖳_2,r)) for p=2, and Õ(d^1/4√(𝖮𝖯𝖳_∞, r) + d^1/2𝖮𝖯𝖳_∞,r) when p=∞. Our results hold for distributions satisfying anti-concentration properties enjoyed by log-concave isotropic distributions among others. We additionally show that if one instead uses a nonconvex sigmoidal loss, adversarial training yields halfspaces with an improved robust classification error of O(𝖮𝖯𝖳_2,r) for p=2, and O(d^1/4𝖮𝖯𝖳_∞, r) when p=∞. To the best of our knowledge, this is the first work to show that adversarial training provably yields robust classifiers in the presence of noise.
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