# A Stochastic Semismooth Newton Method for Nonsmooth Nonconvex Optimization

arXiv: https://arxiv.org/abs/1803.03466 · HTML: https://mlsanity.com/paper/1803.03466/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2018.

## Built on by (most cited first, 3 of 3)

- [A Semismooth Newton Stochastic Proximal Point Algorithm with Variance Reduction](https://mlsanity.com/paper/2204.00406/) (2022)
- [A Normal Map-Based Proximal Stochastic Gradient Method: Convergence and Identification Properties](https://mlsanity.com/paper/2305.05828/) (2023)
- [A New Random Reshuffling Method for Nonsmooth Nonconvex Finite-sum Optimization](https://mlsanity.com/paper/2312.01047/) (2023)

## Abstract (arXiv)

In this work, we present a globalized stochastic semismooth Newton method for solving stochastic optimization problems involving smooth nonconvex and nonsmooth convex terms in the objective function. We assume that only noisy gradient and Hessian information of the smooth part of the objective function is available via calling stochastic first and second order oracles. The proposed method can be seen as a hybrid approach combining stochastic semismooth Newton steps and stochastic proximal gradient steps. Two inexact growth conditions are incorporated to monitor the convergence and the acceptance of the semismooth Newton steps and it is shown that the algorithm converges globally to stationary points in expectation. Moreover, under standard assumptions and utilizing random matrix concentration inequalities, we prove that the proposed approach locally turns into a pure stochastic semismooth Newton method and converges r-superlinearly with high probability. We present numerical results and comparisons on ℓ1-regularized logistic regression and nonconvex binary classification that demonstrate the efficiency of our algorithm.