A Retrospective Approximation Approach for Smooth Stochastic Optimization

• URL: http://arxiv.org/abs/2103.04392v3
• Date: Wed, 6 Mar 2024 23:17:04 GMT
• Title: A Retrospective Approximation Approach for Smooth Stochastic Optimization
• Authors: David Newton, Raghu Bollapragada, Raghu Pasupathy, Nung Kwan Yip
• Abstract summary: Gradient (SG) is the defactorandom iterative technique to solve optimization (SO) problems with a smooth (non-fimation) objective $imation.
• Score: 0.2867517731896504
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Stochastic Gradient (SG) is the defacto iterative technique to solve stochastic optimization (SO) problems with a smooth (non-convex) objective $f$ and a stochastic first-order oracle. SG's attractiveness is due in part to its simplicity of executing a single step along the negative subsampled gradient direction to update the incumbent iterate. In this paper, we question SG's choice of executing a single step as opposed to multiple steps between subsample updates. Our investigation leads naturally to generalizing SG into Retrospective Approximation (RA) where, during each iteration, a "deterministic solver" executes possibly multiple steps on a subsampled deterministic problem and stops when further solving is deemed unnecessary from the standpoint of statistical efficiency. RA thus rigorizes what is appealing for implementation -- during each iteration, "plug in" a solver, e.g., L-BFGS line search or Newton-CG, as is, and solve only to the extent necessary. We develop a complete theory using relative error of the observed gradients as the principal object, demonstrating that almost sure and $L_1$ consistency of RA are preserved under especially weak conditions when sample sizes are increased at appropriate rates. We also characterize the iteration and oracle complexity (for linear and sub-linear solvers) of RA, and identify a practical termination criterion leading to optimal complexity rates. To subsume non-convex $f$, we present a certain "random central limit theorem" that incorporates the effect of curvature across all first-order critical points, demonstrating that the asymptotic behavior is described by a certain mixture of normals. The message from our numerical experiments is that the ability of RA to incorporate existing second-order deterministic solvers in a strategic manner might be important from the standpoint of dispensing with hyper-parameter tuning.

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