Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension

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 $ε$) 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. Our definition of smoothed agnostic learning is an interpolation between the case where the instance distribution $D$ and the optimal classifier can be arbitrarily coupled (which corresponds to agnostic learning and $σ= 0$) and completely decoupled (when $σ= \infty$). This decoupling allows us to avoid worst-case concepts that can encode complexity-theoretic primitives. 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 $k^{\mathrm{poly}(\frac{\log k}{εγ}) }$ where $γ$ is the margin parameter. Before our work, the best-known runtime was exponential in $k$ (Arriaga and Vempala, FOCS' 99).