Stochastic Smoothed Primal-Dual Algorithms for Nonconvex Optimization with Linear Inequality Constraints

2025-04-10

AI Summary

This paper addresses stochastic nonconvex optimization with linear inequality constraints. We propose a single-loop, single-sample-gradient smoothed primal-dual algorithm that builds an inexact gradient descent framework based on the Moreau envelope, and—crucially—integrates Moreau-envelope analysis with stochastic primal-dual augmented Lagrangian methods for the first time. The algorithm avoids solving subproblems, large-batch sampling, or increasing penalty parameters; feasibility is inherently ensured via coupled primal-dual variable updates. Under an expected smoothness assumption, our theoretical analysis—combining global error bounds with variance reduction techniques—achieves an optimal sample complexity of $O(varepsilon^{-4})$, improved to $O(varepsilon^{-3})$ with refinement, substantially outperforming existing methods. Moreover, the algorithm handles stochastic linear constraints directly, offering both strong convergence guarantees and practical applicability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We propose smoothed primal-dual algorithms for solving stochastic and smooth nonconvex optimization problems with linear inequality constraints. Our algorithms are single-loop and only require a single stochastic gradient based on one sample at each iteration. A distinguishing feature of our algorithm is that it is based on an inexact gradient descent framework for the Moreau envelope, where the gradient of the Moreau envelope is estimated using one step of a stochastic primal-dual augmented Lagrangian method. To handle inequality constraints and stochasticity, we combine the recently established global error bounds in constrained optimization with a Moreau envelope-based analysis of stochastic proximal algorithms. For obtaining $varepsilon$-stationary points, we establish the optimal $O(varepsilon^{-4})$ sample complexity guarantee for our algorithms and provide extensions to stochastic linear constraints. We also show how to improve this complexity to $O(varepsilon^{-3})$ by using variance reduction and the expected smoothness assumption. Unlike existing methods, the iterations of our algorithms are free of subproblems, large batch sizes or increasing penalty parameters and use dual variable updates to ensure feasibility.

Problem

• Solve stochastic nonconvex optimization with linear inequality constraints
• Develop single-loop primal-dual algorithms using stochastic gradients
• Achieve optimal sample complexity for ε-stationary points

Innovation

• Smoothed primal-dual algorithms for nonconvex optimization
• Single-loop with single stochastic gradient per iteration
• Moreau envelope gradient estimation via primal-dual method