# Projective Proximal Gradient Descent for A Class of Nonconvex Nonsmooth Optimization Problems: Fast Convergence Without Kurdyka-Lojasiewicz (KL) Property

arXiv: https://arxiv.org/abs/2304.10499 · HTML: https://mlsanity.com/paper/2304.10499/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 18 references that are in the map.
- First posted on arXiv 20 Apr 2023.

## Builds on (most cited first, 6 of 6)

- [Accelerated Gradient Methods for Nonconvex Nonlinear and Stochastic Programming](https://mlsanity.com/paper/1310.3787/) (2013)
- [Accelerated Gradient Descent Escapes Saddle Points Faster than Gradient Descent](https://mlsanity.com/paper/1711.10456/) (2017)
- [Statistical consistency and asymptotic normality for high-dimensional robust M-estimators](https://mlsanity.com/paper/1501.00312/) (2015)
- [Convergence Analysis of Proximal Gradient with Momentum for Nonconvex Optimization](https://mlsanity.com/paper/1705.04925/) (2017)
- [Restarted Nonconvex Accelerated Gradient Descent: No More Polylogarithmic Factor in the O(ε-7/4) Complexity](https://mlsanity.com/paper/2201.11411/) (2022)
- [Learning Sparse Visual Representations with Leaky Capped Norm Regularizers](https://mlsanity.com/paper/1711.02857/) (2017)

## Abstract (arXiv)

Nonconvex and nonsmooth optimization problems are important and challenging for statistics and machine learning. In this paper, we propose Projected Proximal Gradient Descent (PPGD) which solves a class of nonconvex and nonsmooth optimization problems, where the nonconvexity and nonsmoothness come from a nonsmooth regularization term which is nonconvex but piecewise convex. In contrast with existing convergence analysis of accelerated PGD methods for nonconvex and nonsmooth problems based on the Kurdyka-Lojasiewicz (KL) property, we provide a new theoretical analysis showing local fast convergence of PPGD. It is proved that PPGD achieves a fast convergence rate of cO(1/k2) when the iteration number k ≥ k0 for a finite k0 on a class of nonconvex and nonsmooth problems under mild assumptions, which is locally Nesterov's optimal convergence rate of first-order methods on smooth and convex objective function with Lipschitz continuous gradient. Experimental results demonstrate the effectiveness of PPGD.