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Optimal Online Generalized Linear Regression with Stochastic Noise and Its Application to Heteroscedastic Bandits
Heyang Zhao · Dongruo Zhou · Jiafan He · Quanquan Gu

Wed Jul 26 02:00 PM -- 03:30 PM (PDT) @ Exhibit Hall 1 #328
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 an $\Omega(\sigma^2d\log(T/d))$ 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 heterogeneous noise and propose an algorithm based on FTRL to achieve the first variance-aware regret bound.

Author Information

Heyang Zhao (Computer Science Department, University of California, Los Angeles)
Dongruo Zhou (UCLA)
Jiafan He (University of California, Los Angeles)
Quanquan Gu (University of California, Los Angeles)

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