Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension

TL;DR

This paper introduces a smoothed-analysis framework for learning low-dimensional concepts with bounded Gaussian surface area, enabling efficient algorithms for classes like intersections of halfspaces and convex sets under mild perturbations.

Contribution

It presents a novel smoothed-analysis approach that improves learnability results for low-intrinsic-dimension concepts, including new algorithms for agnostically learning intersections of halfspaces.

Findings

Efficient learning algorithms for low-dimensional concepts with Gaussian surface area.

First polynomial-time algorithm for agnostically learning intersections of halfspaces.

Enhanced understanding of robustness in learning under small Gaussian perturbations.

Abstract

In traditional models of supervised learning, the goal of a learner -- given examples from an arbitrary joint distribution on $\mathbb{R}^d \times \{\pm 1\}$ -- is to output a hypothesis that is competitive (to within $\epsilon$) of the best fitting concept from some class. In order to escape strong hardness results for learning even simple concept classes, we introduce a smoothed-analysis framework that requires a learner to compete only with the best classifier that is robust to small random Gaussian perturbation. This subtle change allows us to give a wide array of learning results for any concept that (1) depends on a low-dimensional subspace (aka multi-index model) and (2) has a bounded Gaussian surface area. This class includes functions of halfspaces and (low-dimensional) convex sets, cases that are only known to be learnable in non-smoothed settings with respect to highly structured distributions such as Gaussians. Surprisingly, our analysis also yields new results for traditional non-smoothed frameworks such as learning with margin. In particular, we obtain the first algorithm for agnostically learning intersections of $k$-halfspaces in time $k^{poly(\frac{\log k}{\epsilon \gamma}) }$ where $\gamma$ is the margin parameter. Before our work, the best-known runtime was exponential in $k$ (Arriaga and Vempala, 1999).