  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.
