Sample-Optimal PAC Learning of Halfspaces with Malicious Noise
This work addresses a challenging noise model in PAC learning for machine learning researchers, offering incremental improvements in sample complexity and noise tolerance.
The paper tackles the problem of efficiently learning homogeneous halfspaces with malicious noise, achieving a near-optimal sample complexity of ˜O(d), which improves upon the previous best of ˜O(d^2). It also extends these results to the nasty noise model while maintaining near-optimal noise tolerance and polynomial-time efficiency.
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.