Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension

• URL: http://arxiv.org/abs/2407.00966v1
• Date: Mon, 1 Jul 2024 04:58:36 GMT
• Title: Smoothed Analysis for Learning Concepts with Low Intrinsic Dimension
• Authors: Gautam Chandrasekaran, Adam Klivans, Vasilis Kontonis, Raghu Meka, Konstantinos Stavropoulos,
• Abstract summary: In traditional models of supervised learning, the goal of a learner is to output a hypothesis that is competitive (to within $epsilon$) of the best fitting concept from some class. We introduce a smoothed-analysis framework that requires a learner to compete only with the best agnostic. We obtain the first algorithm forally learning intersections of $k$-halfspaces in time.
• Score: 17.485243410774814
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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).

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