Optimization via First-Order Switching Methods: Skew-Symmetric Dynamics and Optimistic Discretization

TL;DR

This paper investigates the convergence properties of the Switching Gradient Method (SGM) for constrained optimization, revealing limitations under smoothness and proposing optimistic and soft switching variants for improved performance.

Contribution

The paper provides a continuous-time analysis of SGM's limitations under smoothness and introduces optimistic and soft switching methods to enhance convergence.

Findings

SGM does not automatically benefit from smoothness for faster rates.

Continuous-time analysis reveals fundamental limitations of SGM's dynamics.

Optimistic and soft switching methods can potentially improve convergence rates.

Abstract

Large-scale constrained optimization problems are at the core of many tasks in control, signal processing, and machine learning. Notably, problems with functional constraints arise when, beyond a performance{\nobreakdash-}centric goal (e.g., minimizing the empirical loss), one desires to satisfy other requirements such as robustness, fairness, etc. A simple method for such problems, which remarkably achieves optimal rates for non-smooth, convex, strongly convex, and weakly convex functions under first-order oracle, is Switching Gradient Method (SGM): in each iteration depending on a predetermined constraint violation tolerance, use the gradient of objective or the constraint as the update vector. While the performance of SGM is well-understood for non-smooth functions and in fact matches its unconstrained counterpart, i.e., Gradient Descent (GD), less is formally established about its convergence properties under the smoothness of loss and constraint functions. In this work, we aim to fill this gap. First, we show that SGM may not benefit from faster rates under smoothness, in contrast to improved rates for GD under smoothness. By taking a continuous-time limit perspective, we show the issue is fundamental to SGM's dynamics and not an artifact of our analysis. Our continuous-time limit perspective further provides insights towards alleviating SGM's shortcomings. Notably, we show that leveraging the idea of optimism, a well-explored concept in variational inequalities and min-max optimization, could lead to faster methods. This perspective further enables designing a new class of ``soft'' switching methods, for which we further analyze their iteration complexity under mild assumptions.