A Unified Analysis of Saddle Flow Dynamics: Stability and Algorithm Design

Pengcheng You; Yingzhu Liu; Enrique Mallada

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arXiv · 2409.05290

A Unified Analysis of Saddle Flow Dynamics: Stability and Algorithm Design

Pengcheng You·Yingzhu Liu·Enrique Mallada

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Abstract

This work examines the conditions for asymptotic and exponential convergence of saddle flow dynamics of convex-concave functions. First, we propose an observability-based certificate for asymptotic convergence, directly bridging the gap between the invariant set in a LaSalle argument and the equilibrium set of saddle flows. This certificate generalizes conventional conditions for convergence, e.g., strict convexity-concavity, and leads to a novel state-augmentation method that requires minimal assumptions for asymptotic convergence. We also show that global exponential stability follows from strong convexity-strong concavity, providing a lower-bound estimate of the convergence rate. This insight also explains the convergence of proximal saddle flows for strongly convex-concave objective functions. Our results generalize to dynamics with projections on the vector field and have applications in solving constrained convex optimization via primal-dual methods. Based on these insights, we study four algorithms built upon different Lagrangian function transformations. We validate our work by applying these methods to solve a network flow optimization and a Lasso regression problem.

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BibTeX RIS

Pengcheng You, Yingzhu Liu, Enrique Mallada. 2024-09-09. A Unified Analysis of Saddle Flow Dynamics: Stability and Algorithm Design. https://arxiv.org/abs/2409.05290

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