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We study the problem of online generalized linear regression in thestochastic setting, where the label is generated from a generalized linearmodel with possibly unbounded additive noise. We provide a sharp analysis ofthe classical follow-the-regularized-leader (FTRL) algorithm to cope with thelabel noise. More specifically, for $\sigma$-sub-Gaussian label noise, ouranalysis 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 ofrounds. We also prove a $\Omega(\sigma^2d\log(T/d))$ lower bound for stochasticonline linear regression, which indicates that our upper bound is nearlyoptimal. In addition, we extend our analysis to a more refined Bernstein noisecondition. As an application, we study generalized linear bandits withheteroscedastic noise and propose an algorithm based on FTRL to achieve thefirst variance-aware regret bound.

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