# Preconditioned Primal-Dual Gradient Methods for Nonconvex Composite and Finite-Sum Optimization

arXiv: https://arxiv.org/abs/2309.13416 · HTML: https://mlsanity.com/paper/2309.13416/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2023.

## Built on by (most cited first, 2 of 2)

- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [DualHash: A Stochastic Primal-Dual Algorithm with Theoretical Guarantee for Deep Hashing](https://mlsanity.com/paper/2510.18218/) (2025)

## Abstract (arXiv)

In this paper, we first introduce a preconditioned primal-dual gradient algorithm based on conjugate duality theory. This algorithm is designed to solve composite optimization problem whose objective function consists of two summands: a continuously differentiable nonconvex function and the composition of a nonsmooth nonconvex function with a linear operator. In contrast to existing nonconvex primal-dual algorithms, our proposed algorithm, through the utilization of conjugate duality, does not require the calculation of proximal mapping of nonconvex functions. Under mild conditions, we prove that any cluster point of the generated sequence is a critical point of the composite optimization problem. In the context of Kurdyka-Lojasiewicz property, we establish global convergence and convergence rates for the iterates. Secondly, for nonconvex finite-sum optimization, we propose a stochastic algorithm that combines the preconditioned primal-dual gradient algorithm with a class of variance reduced stochastic gradient estimators. Almost sure global convergence and expected convergence rates are derived relying on the Kurdyka-Lojasiewicz inequality. Finally, some preliminary numerical results are presented to demonstrate the effectiveness of the proposed algorithms.