Thinking Outside the Ball: Optimal Learning with Gradient Descent for Generalized Linear Stochastic Convex Optimization

Abstract

We consider linear prediction with a convex Lipschitz loss, or more generally, stochastic convex optimization problems of generalized linear form, i.e.~where each instantaneous loss is a scalar convex function of a linear function. We show that in this setting, early stopped Gradient Descent (GD), without any explicit regularization or projection, ensures excess error at most $\epsilon$ (compared to the best possible with unit Euclidean norm) with an optimal, up to logarithmic factors, sample complexity of $\tilde{O}(1/\epsilon^2)$ and only $\tilde{O}(1/\epsilon^2)$ iterations. This contrasts with general stochastic convex optimization, where $\Omega(1/\epsilon^4)$ iterations are needed Amir et al. [2021b]. The lower iteration complexity is ensured by leveraging uniform convergence rather than stability. But instead of uniform convergence in a norm ball, which we show can guarantee suboptimal learning using $\Theta(1/\epsilon^4)$ samples, we rely on uniform convergence in a distribution-dependent ball.