Asymptotic Optimality in Stochastic Optimization
This work addresses fundamental limitations in stochastic optimization algorithms, particularly for constrained problems, which is important for researchers and practitioners in machine learning and optimization.
The paper tackles the problem of achieving optimal convergence guarantees in stochastic convex optimization by developing a local minimax theory and fully online adaptive methods. It shows that standard stochastic gradient procedures fail for constraint identification and are suboptimal for nonlinear constraints, necessitating asymptotically optimal Riemannian stochastic gradient methods.
We study local complexity measures for stochastic convex optimization problems, providing a local minimax theory analogous to that of Hájek and Le Cam for classical statistical problems. We give complementary optimality results, developing fully online methods that adaptively achieve optimal convergence guarantees. Our results provide function-specific lower bounds and convergence results that make precise a correspondence between statistical difficulty and the geometric notion of tilt-stability from optimization. As part of this development, we show how variants of Nesterov's dual averaging---a stochastic gradient-based procedure---guarantee finite time identification of constraints in optimization problems, while stochastic gradient procedures fail. Additionally, we highlight a gap between problems with linear and nonlinear constraints: standard stochastic-gradient-based procedures are suboptimal even for the simplest nonlinear constraints, necessitating the development of asymptotically optimal Riemannian stochastic gradient methods.