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# Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension

*Proceedings of Thirty Seventh Conference on Learning Theory*, PMLR 247:876-922, 2024.

#### Abstract

In the well-studied agnostic model of 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 in this model, we introduce a smoothed analysis framework where we require 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. Perhaps 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).