Efficient and Near-Optimal Smoothed Online Learning for Generalized Linear Functions

• URL: http://arxiv.org/abs/2205.13056v1
• Date: Wed, 25 May 2022 21:31:36 GMT
• Title: Efficient and Near-Optimal Smoothed Online Learning for Generalized Linear Functions
• Authors: Adam Block and Max Simchowitz
• Abstract summary: We give a computationally efficient algorithm that is the first to enjoy the statistically optimal log(T/sigma) regret for realizable K-wise linear classification. We develop a novel characterization of the geometry of the disagreement region induced by generalized linear classifiers.
• Score: 28.30744223973527
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Due to the drastic gap in complexity between sequential and batch statistical learning, recent work has studied a smoothed sequential learning setting, where Nature is constrained to select contexts with density bounded by 1/{\sigma} with respect to a known measure {\mu}. Unfortunately, for some function classes, there is an exponential gap between the statistically optimal regret and that which can be achieved efficiently. In this paper, we give a computationally efficient algorithm that is the first to enjoy the statistically optimal log(T/{\sigma}) regret for realizable K-wise linear classification. We extend our results to settings where the true classifier is linear in an over-parameterized polynomial featurization of the contexts, as well as to a realizable piecewise-regression setting assuming access to an appropriate ERM oracle. Somewhat surprisingly, standard disagreement-based analyses are insufficient to achieve regret logarithmic in 1/{\sigma}. Instead, we develop a novel characterization of the geometry of the disagreement region induced by generalized linear classifiers. Along the way, we develop numerous technical tools of independent interest, including a general anti-concentration bound for the determinant of certain matrix averages.

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