Stochastic Inexact Augmented Lagrangian Method for Nonconvex Expectation Constrained Optimization

TL;DR

This paper introduces a stochastic inexact augmented Lagrangian method for nonconvex expectation constrained problems, achieving better complexity bounds and demonstrating superior performance on real data tasks.

Contribution

It develops a novel stochastic inexact augmented Lagrangian algorithm using variance reduction, improving complexity bounds for nonconvex constrained optimization.

Findings

Achieves an $O( ext{epsilon}^{-5})$ oracle complexity, better than the previous $O( ext{epsilon}^{-6})$.

Outperforms existing methods on fairness and Neyman-Pearson classification problems.

Demonstrates effectiveness through numerical experiments on real datasets.

Abstract

Many real-world problems not only have complicated nonconvex functional constraints but also use a large number of data points. This motivates the design of efficient stochastic methods on finite-sum or expectation constrained problems. In this paper, we design and analyze stochastic inexact augmented Lagrangian methods (Stoc-iALM) to solve problems involving a nonconvex composite (i.e. smooth+nonsmooth) objective and nonconvex smooth functional constraints. We adopt the standard iALM framework and design a subroutine by using the momentum-based variance-reduced proximal stochastic gradient method (PStorm) and a postprocessing step. Under certain regularity conditions (assumed also in existing works), to reach an $\varepsilon$-KKT point in expectation, we establish an oracle complexity result of $O(\varepsilon^{-5})$, which is better than the best-known $O(\varepsilon^{-6})$ result. Numerical experiments on the fairness constrained problem and the Neyman-Pearson classification problem with real data demonstrate that our proposed method outperforms an existing method with the previously best-known complexity result.