Sample-Optimal PAC Learning of Halfspaces with Malicious Noise

TL;DR

This paper improves the sample complexity bounds for PAC learning of halfspaces under malicious noise, achieving near-optimal bounds with a novel analysis and extending results to stronger noise models.

Contribution

It presents a new analysis of an existing algorithm that achieves near-optimal sample complexity for learning halfspaces with malicious noise, and extends the approach to more challenging noise models.

Findings

Achieves near-optimal sample complexity of d in isotropic log-concave distributions.

Introduces a matrix Chernoff-type inequality for covariance matrix analysis.

Extends the algorithm to handle stronger nasty noise models.

Abstract

We study efficient PAC learning of homogeneous halfspaces in $\mathbb{R}^d$ in the presence of malicious noise of Valiant (1985). This is a challenging noise model and only until recently has near-optimal noise tolerance bound been established under the mild condition that the unlabeled data distribution is isotropic log-concave. However, it remains unsettled how to obtain the optimal sample complexity simultaneously. In this work, we present a new analysis for the algorithm of Awasthi et al. (2017) and show that it essentially achieves the near-optimal sample complexity bound of $\tilde{O}(d)$, improving the best known result of $\tilde{O}(d^2)$. Our main ingredient is a novel incorporation of a matrix Chernoff-type inequality to bound the spectrum of an empirical covariance matrix for well-behaved distributions, in conjunction with a careful exploration of the localization schemes of Awasthi et al. (2017). We further extend the algorithm and analysis to the more general and stronger nasty noise model of Bshouty et al. (2002), showing that it is still possible to achieve near-optimal noise tolerance and sample complexity in polynomial time.