# Avoiding strict saddle points of nonconvex regularized problems

arXiv: https://arxiv.org/abs/2401.09274 · HTML: https://mlsanity.com/paper/2401.09274/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 75 references that are in the map.
- First posted on arXiv 17 Jan 2024.

## Builds on (most cited first, 21 of 21)

- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [A Geometric Analysis of Phase Retrieval](https://mlsanity.com/paper/1602.06664/) (2016)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Accelerated Gradient Descent Escapes Saddle Points Faster than Gradient Descent](https://mlsanity.com/paper/1711.10456/) (2017)
- [Escaping Saddles with Stochastic Gradients](https://mlsanity.com/paper/1803.05999/) (2018)
- [First-order methods almost always avoid saddle points: the case of vanishing step-sizes](https://mlsanity.com/paper/1906.07772/) (2019)
- [Escaping from saddle points on Riemannian manifolds](https://mlsanity.com/paper/1906.07355/) (2019)
- [Efficiently escaping saddle points on manifolds](https://mlsanity.com/paper/1906.04321/) (2019)
- [Escaping Saddle Points in Constrained Optimization](https://mlsanity.com/paper/1809.02162/) (2018)
- [A proximal difference-of-convex algorithm with extrapolation](https://mlsanity.com/paper/1612.06265/) (2016)
- [A Newton-Based Method for Nonconvex Optimization with Fast Evasion of Saddle Points](https://mlsanity.com/paper/1707.08028/) (2017)
- [Active manifolds, stratifications, and convergence to local minima in nonsmooth optimization](https://mlsanity.com/paper/2108.11832/) (2021)
- [Nonconvex and Nonsmooth Sparse Optimization via Adaptively Iterative Reweighted Methods](https://mlsanity.com/paper/1810.10167/) (2018)
- [Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functions](https://mlsanity.com/paper/2108.02072/) (2021)
- [Escaping Saddle Points Faster with Stochastic Momentum](https://mlsanity.com/paper/2106.02985/) (2021)
- [Escaping strict saddle points of the Moreau envelope in nonsmooth optimization](https://mlsanity.com/paper/2106.09815/) (2021)
- [An Extrapolated Iteratively Reweighted l1 Method with Complexity Analysis](https://mlsanity.com/paper/2101.03763/) (2021)
- [Convergence Rate Analysis of Proximal Iteratively Reweighted ℓ1 Methods for ℓp Regularization Problems](https://mlsanity.com/paper/2007.05747/) (2020)
- [Iterative ℓ1 minimization for non-convex compressed sensing](https://mlsanity.com/paper/1604.07924/) (2016)
- [Linear Regularizers Enforce the Strict Saddle Property](https://mlsanity.com/paper/2205.09160/) (2022)

## Built on by (most cited first, 1 of 1)

- [Douglas-Rachford Splitting for Group-Sparse Feedback Linear-Quadratic Control](https://mlsanity.com/paper/2507.19895/) (2025)

## Abstract (arXiv)

In this paper, we consider a class of non-convex and non-smooth sparse optimization problems, which encompass most existing nonconvex sparsity-inducing terms. We show the second-order optimality conditions only depend on the nonzeros of the stationary points. We propose two damped iterative reweighted algorithms including the iteratively reweighted ℓ1 algorithm (DIRL1) and the iteratively reweighted ℓ2 (DIRL2) algorithm, to solve these problems. For DIRL1, we show the reweighted ℓ1 subproblem has support identification property so that DIRL1 locally reverts to a gradient descent algorithm around a stationary point. For DIRL2, we show the solution map of the reweighted ℓ2 subproblem is differentiable and Lipschitz continuous everywhere. Therefore, the map of DIRL1 and DIRL2 and their inverse are Lipschitz continuous, and the strict saddle points are their unstable fixed points. By applying the stable manifold theorem, these algorithms are shown to converge only to local minimizers with randomly initialization when the strictly saddle point property is assumed.