Geometry, Computation, and Optimality in Stochastic Optimization

• URL: http://arxiv.org/abs/1909.10455v3
• Date: Fri, 11 Oct 2024 20:10:05 GMT
• Title: Geometry, Computation, and Optimality in Stochastic Optimization
• Authors: Chen Cheng, Daniel Levy, John C. Duchi,
• Abstract summary: We study computational and statistical consequences of problem geometry in and online optimization. By focusing on constraint set and gradient geometry, we characterize the problem families for which- and adaptive-gradient methods are (minimax) optimal.
• Score: 24.154336772159745
• License:
• Abstract: We study computational and statistical consequences of problem geometry in stochastic and online optimization. By focusing on constraint set and gradient geometry, we characterize the problem families for which stochastic- and adaptive-gradient methods are (minimax) optimal and, conversely, when nonlinear updates -- such as those mirror descent employs -- are necessary for optimal convergence. When the constraint set is quadratically convex, diagonally pre-conditioned stochastic gradient methods are minimax optimal. We provide quantitative converses showing that the ``distance'' of the underlying constraints from quadratic convexity determines the sub-optimality of subgradient methods. These results apply, for example, to any $\ell_p$-ball for $p < 2$, and the computation/accuracy tradeoffs they demonstrate exhibit a striking analogy to those in Gaussian sequence models.

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