# Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications

arXiv: https://arxiv.org/abs/2409.10664 · HTML: https://mlsanity.com/paper/2409.10664/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2024.

## Built on by (most cited first, 4 of 4)

- [Remarks on the Polyak-Lojasiewicz inequality and the convergence of gradient systems](https://mlsanity.com/paper/2503.23641/) (2025)
- [Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization](https://mlsanity.com/paper/2604.05042/) (2026)
- [Small-Gain Nash: Certified Contraction to Nash Equilibria in Differentiable Games](https://mlsanity.com/paper/2512.06791/) (2025)
- [Predictive Coding with Bayesian Priors via Proximal Gradients](https://mlsanity.com/paper/2606.08374/) (2026)

## Abstract (arXiv)

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-Lojasiewicz condition. We also show that the proximal Polyak-Lojasiewicz 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.