Learning a Single Neuron Robustly to Distributional Shifts and Adversarial Label Noise

Part of Advances in Neural Information Processing Systems 37 (NeurIPS 2024) Main Conference Track

Authors

Shuyao Li, Sushrut Karmalkar, Ilias Diakonikolas, Jelena Diakonikolas

Abstract

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial distribution shifts, where the labels can be arbitrary, and the goal is to find a "best-fit" function.More precisely, given training samples from a reference distribution $p_0$, the goal is to approximate the vector $\mathbf{w}^*$which minimizes the squared loss with respect to the worst-case distribution that is close in $\chi^2$-divergence to $p_{0}$.We design a computationally efficient algorithm that recovers a vector $\hat{\mathbf{w}}$satisfying $\mathbb{E}\_{p^*} (\sigma(\hat{\mathbf{w}} \cdot \mathbf{x}) - y)^2 \leq C \hspace{0.2em} \mathbb{E}\_{p^*} (\sigma(\mathbf{w}^* \cdot \mathbf{x}) - y)^2 + \epsilon$, where $C>1$ is a dimension-independent constant and $(\mathbf{w}^*, p^*)$ is the witness attaining the min-max risk$\min_{\mathbf{w}:\|\mathbf{w}\| \leq W} \max\_{p} \mathbb{E}\_{(\mathbf{x}, y) \sim p} (\sigma(\mathbf{w} \cdot \mathbf{x}) - y)^2 - \nu \chi^2(p, p_0)$.Our algorithm follows the primal-dual framework and is designed by directly bounding the risk with respect to the original, nonconvex $L_2^2$ loss.From an optimization standpoint, our work opens new avenues for the design of primal-dual algorithms under structured nonconvexity.