Information-constrained optimization: can adaptive processing of gradients help?

TL;DR

This paper investigates whether adaptive processing of gradients can improve optimization under local information constraints like privacy and quantization, showing that adaptivity often does not outperform nonadaptive methods, except in specific cases.

Contribution

The paper provides tight lower bounds for convergence rates in constrained optimization, demonstrating the limited advantage of adaptive gradient processing under various information constraints.

Findings

Adaptive processing often does not outperform nonadaptive methods in constrained settings.

Lower bounds are established for convex and strongly convex functions under privacy, quantization, and coordinate access constraints.

A specific problem is shown where adaptive processing outperforms nonadaptive approaches.

Abstract

We revisit first-order optimization under local information constraints such as local privacy, gradient quantization, and computational constraints limiting access to a few coordinates of the gradient. In this setting, the optimization algorithm is not allowed to directly access the complete output of the gradient oracle, but only gets limited information about it subject to the local information constraints. We study the role of adaptivity in processing the gradient output to obtain this limited information from it.We consider optimization for both convex and strongly convex functions and obtain tight or nearly tight lower bounds for the convergence rate, when adaptive gradient processing is allowed. Prior work was restricted to convex functions and allowed only nonadaptive processing of gradients. For both of these function classes and for the three information constraints mentioned above, our lower bound implies that adaptive processing of gradients cannot outperform nonadaptive processing in most regimes of interest. We complement these results by exhibiting a natural optimization problem under information constraints for which adaptive processing of gradient strictly outperforms nonadaptive processing.