Saddle-point dynamics: conditions for asymptotic stability of saddle points

TL;DR

This paper analyzes the stability of saddle points in differentiable functions using saddle-point dynamics, providing conditions for asymptotic convergence based on convexity-concavity and linearization properties.

Contribution

It introduces new conditions under which saddle points are asymptotically stable, extending convergence analysis beyond convex-concave functions to more general cases.

Findings

Saddle points are asymptotically stable under convexity-concavity conditions.

Linearization and proximal normal properties ensure convergence in non-convex cases.

Global convergence results are established for broader classes of functions.

Abstract

This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradient-ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics, the function along the proximal normals to the saddle set, and the linearity of the function in one variable. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.