Asymptotic optimality in stochastic optimization

We study local complexity measures for stochastic convex optimization problems, providing a minimax theory analogous to that of Hájek and Le Cam classical statistical problems. give complementary optimality results, developing fully online methods adaptively achieve optimal convergence guarantees. Our results provide function-specific lower bounds make precise correspondence between difficulty the geometric notion tilt-stability from optimization. As part this development, we show how variants Nesterov’s dual averaging—a gradient-based procedure—guarantee finite time identification constraints in while gradient procedures fail. Additionally, highlight gap problems with linear nonlinear constraints: standard stochastic-gradient-based are suboptimal even simplest constraints, necessitating development asymptotically Riemannian methods.