Optimization via First-Order Switching Methods: Skew-Symmetric Dynamics and Optimistic Discretization
Antesh Upadhyay, Sang Bin Moon, Abolfazl Hashemi
TL;DR
This paper investigates constrained optimization with functional constraints using Switching Gradient Methods (SGM). It shows that even under $L$-smoothness, SGM cannot surpass the $O(\epsilon^{-2})$ rate, due to skew-symmetric dynamics revealed through a continuous-time analysis. To overcome this, the authors introduce soft-switching and optimistic discretization frameworks, notably Soft Switching Gradient Method (SSGM) and its variants SSPPM and SSPPM-E, which retain the same theoretical rate but demonstrate improved practical convergence and stability. They provide both theoretical guarantees under mild assumptions and empirical evidence of skew-symmetry and improved performance, highlighting potential for scalable, dual-free methods in large-scale constrained problems.
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.