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On incremental and semi-global exponential stability of gradient flows satisfying generalized Łojasiewicz inequalities

Andreas Oliveira, Arthur C. B. de Oliveira, Mario Sznaier, Eduardo Sontag

Abstract

The Łojasiewicz inequality characterizes objective-value convergence along gradient flows and, in special cases, yields exponential decay of the cost. However, such results do not directly give rates of convergence in the state. In this paper, we use contraction theory to derive state-space guarantees for gradient systems satisfying generalized Łojasiewicz inequalities. We first show that, when the objective has a unique strongly convex minimizer, the generalized Łojasiewicz inequality implies semi-global exponential stability; on arbitrary compact subsets, this yields exponential stability. We then give two curvature-based sufficient conditions, together with constraints on the Łojasiewicz rate, under which the nonconvex gradient flow is globally incrementally exponentially stable.

On incremental and semi-global exponential stability of gradient flows satisfying generalized Łojasiewicz inequalities

Abstract

The Łojasiewicz inequality characterizes objective-value convergence along gradient flows and, in special cases, yields exponential decay of the cost. However, such results do not directly give rates of convergence in the state. In this paper, we use contraction theory to derive state-space guarantees for gradient systems satisfying generalized Łojasiewicz inequalities. We first show that, when the objective has a unique strongly convex minimizer, the generalized Łojasiewicz inequality implies semi-global exponential stability; on arbitrary compact subsets, this yields exponential stability. We then give two curvature-based sufficient conditions, together with constraints on the Łojasiewicz rate, under which the nonconvex gradient flow is globally incrementally exponentially stable.

Paper Structure

This paper contains 18 sections, 11 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stability Notions
  4. Gradient flows and the generalized Polyak-Łojasiewicz
  5. Contractivity Under Logarithmic Norms
  6. Contractivity under Riemannian Metrics
  7. $\mathcal{K}_\infty$-PŁI Functions and Contraction Metrics
  8. Semi-global exponential stability and Exponential Stability on Compacts
  9. Global Incremental Exponential Stability
  10. Discussion
  11. Example
  12. Conclusion
  13. Appendix
  14. Proof of Theorem \ref{['prop:global']}
  15. Proof of Corollary \ref{['cor:compact']}
  16. ...and 3 more sections

Key Result

Lemma 1

desoer2009feedback Let $A \in \mathbb{R}^{n \times n}$ and let $\|\cdot\|$ be any norm with associated logarithmic norm $\mu(\cdot)$. If $\alpha(A) = \max\{\Re(\lambda) : \lambda \text{ is an eigenvalue of } A\}$ denotes the spectral abscissa, then Further if $A \in \mathbb{R}^{n \times n}$ is a symmetric matrix then $\mu_2(A) = \lambda_{\max}(A)$.

Figures (2)

  • Figure 1: Schematic illustration of the assumptions and conclusion of Theorem \ref{['prop:concavity-on-annulus']}. A proper objective $f$ with satisfying $\mathcal{K}_\infty$-PŁI. The equilibrium $x^\ast$ lies in a strongly convex neighborhood, and everything outside an annulus is strongly convex. A bounded region where the function is strongly concave is allowed $\lambda_{\min}(\nabla^2 f(x))<0$ and is shown in red. Gradient flow trajectories $\dot{x}=-\nabla f(x)$ starting in different quadrants tend exponentially toward each other since the system is IES by Theorem \ref{['prop:concavity-on-annulus']}.
  • Figure 2: Schematic illustration of the assumptions and conclusion of Theorem \ref{['prop:above-PL-regime']}. The equilibrium $x^\ast$ lies in a strongly convex neighborhood (shaded blue region). A proper objective $f$ satisfying a $\mathcal{K}_\infty$-PŁI that is strictly stronger than the PŁI is shown in the lower right graph, and a global Hessian spectral abscissa lower bound (which need not be positive) is shown in the top right graph. Gradient flow trajectories $\dot{x}=-\nabla f(x)$ starting in different quadrants tend exponentially toward each other since the system is IES by Theorem \ref{['prop:above-PL-regime']}.

Theorems & Definitions (36)

  • Definition 1
  • Definition 2: Generalized Polyak-Łojasiewicz inequality
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1
  • Lemma 1
  • Proposition 1
  • proof
  • Remark 2
  • ...and 26 more