On the Generalization Error of Stochastic Mirror Descent for Quadratically-Bounded Losses: an Improved Analysis.

In this work, we revisit the generalization error of stochastic mirror descent for quadratically bounded losses studied in Telgarsky (2022). Quadratically bounded losses is a broad class of loss functions, capturing both Lipschitz and smooth functions, for both regression and classification problems. We study the high probability generalization for this class of losses on linear predictors in both realizable and non-realizable cases when the data are sampled IID or from a Markov chain. The prior work relies on an intricate coupling argument between the iterates of the original problem and those projected onto a bounded domain. This approach enables blackbox application of concentration inequalities, but also leads to suboptimal guarantees due in part to the use of a union bound across all iterations. In this work, we depart significantly from the prior work of Telgarsky (2022), and introduce a novel approach for establishing high probability generalization guarantees. In contrast to the prior work, our work directly analyzes the moment generating function of a novel supermartingale sequence and leverages the structure of stochastic mirror descent. As a result, we obtain improved bounds in all aforementioned settings. Specifically, in the realizable case and non-realizable case with light-tailed sub-Gaussian data, we improve the bounds by a $\log T$ factor, matching the correct rates of $1/T$ and $1/\sqrt{T}$, respectively. In the more challenging case of heavy-tailed polynomial data, we improve the existing bound by a $\mathrm{poly}\ T$ factor.