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Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications

Anand Gokhale, Alexander Davydov, Francesco Bullo

Abstract

In this letter we study the proximal gradient dynamics. This recently-proposed continuous-time dynamics solves optimization problems whose cost functions are separable into a nonsmooth convex and a smooth component. First, we show that the cost function decreases monotonically along the trajectories of the proximal gradient dynamics. We then introduce a new condition that guarantees exponential convergence of the cost function to its optimal value, and show that this condition implies the proximal Polyak-Łojasiewicz condition. We also show that the proximal Polyak-Łojasiewicz condition guarantees exponential convergence of the cost function. Moreover, we extend these results to time-varying optimization problems, providing bounds for equilibrium tracking. Finally, we discuss applications of these findings, including the LASSO problem, certain matrix based problems and a numerical experiment on a feed-forward neural network.

Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications

Abstract

In this letter we study the proximal gradient dynamics. This recently-proposed continuous-time dynamics solves optimization problems whose cost functions are separable into a nonsmooth convex and a smooth component. First, we show that the cost function decreases monotonically along the trajectories of the proximal gradient dynamics. We then introduce a new condition that guarantees exponential convergence of the cost function to its optimal value, and show that this condition implies the proximal Polyak-Łojasiewicz condition. We also show that the proximal Polyak-Łojasiewicz condition guarantees exponential convergence of the cost function. Moreover, we extend these results to time-varying optimization problems, providing bounds for equilibrium tracking. Finally, we discuss applications of these findings, including the LASSO problem, certain matrix based problems and a numerical experiment on a feed-forward neural network.
Paper Structure (11 sections, 10 theorems, 27 equations, 1 figure)

This paper contains 11 sections, 10 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Proximal operator and the proximal gradient dynamics
  4. Polyak-Łojaciewicz and related conditions
  5. Global convergence of the proximal gradient dynamics
  6. Exponential convergence of the cost
  7. Extensions to time-varying optimization
  8. Applications and Examples
  9. Discussion
  10. Technical Results
  11. Proof of Theorem \ref{['thm:PL_cts_time']}

Key Result

Theorem 2

For the optimization problem eq:opt_problem, let the following assumptions hold true. Then, for the proximal gradient dynamics eq:prox_dynamics:

Figures (1)

  • Figure 2: Nonconvex loss landscape of a regularized feed-forward neural network with the trajectory of proximal gradient dynamics. The trajectory goes along a path where the cost function is monotonically decreasing, since the loss function is differentiable and the nonsmooth regularizer is CCP.

Theorems & Definitions (21)

  • Definition 1: Dini Derivative
  • Definition 2: L-smoothness
  • Definition 3: PL condition
  • Definition 4: Proximal PL Condition HK-JN-MS:16
  • Remark 1
  • Definition 5: Proximal KL condition HK-JN-MS:16
  • Theorem 2: Nonincreasing cost function under proximal gradient dynamics
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 11 more