Optimal Online Generalized Linear Regression with Stochastic Noise and Its Application to Heteroscedastic Bandits
This work addresses the problem of improving regret bounds in online learning with noisy data for researchers in machine learning and optimization, offering incremental theoretical advancements.
The paper tackles online generalized linear regression with stochastic noise, providing a nearly optimal regret bound of O(σ²d log T) + o(log T) for σ-sub-Gaussian noise and extending it to Bernstein noise conditions, with an application to heteroscedastic bandits achieving the first variance-aware regret bound.
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 $σ$-sub-Gaussian label noise, our analysis provides a regret upper bound of $O(σ^2 d \log T) + o(\log T)$, where $d$ is the dimension of the input vector, $T$ is the total number of rounds. We also prove a $Ω(σ^2d\log(T/d))$ 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.