Optimal Online Generalized Linear Regression with Stochastic Noise and Its Application to Heteroscedastic Bandits

• URL: http://arxiv.org/abs/2202.13603v2
• Date: Mon, 27 Mar 2023 17:52:52 GMT
• Title: Optimal Online Generalized Linear Regression with Stochastic Noise and Its Application to Heteroscedastic Bandits
• Authors: Heyang Zhao and Dongruo Zhou and Jiafan He and Quanquan Gu
• Abstract summary: We study the problem of online generalized linear regression in the setting of a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. We propose an algorithm based on FTRL to achieve the first variance-aware regret bound.
• Score: 88.6139446295537
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the problem of online generalized linear regression in the stochastic setting, where the label is generated from a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. More specifically, for $\sigma$-sub-Gaussian label noise, our analysis provides a regret upper bound of $O(\sigma^2 d \log T) + o(\log T)$, where $d$ is the dimension of the input vector, $T$ is the total number of rounds. We also prove a $\Omega(\sigma^2d\log(T/d))$ lower bound for stochastic online linear regression, which indicates that our upper bound is nearly optimal. In addition, we extend our analysis to a more refined Bernstein noise condition. As an application, we study generalized linear bandits with heteroscedastic noise and propose an algorithm based on FTRL to achieve the first variance-aware regret bound.

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