Robins-Monro Augmented Lagrangian Method for Stochastic Convex Optimization

TL;DR

This paper introduces a Robbins-Monro augmented Lagrangian method (RMALM) for constrained stochastic convex optimization, achieving linear convergence and demonstrating superior performance over existing algorithms on synthetic and real data.

Contribution

The paper proposes a novel hybrid algorithm combining Robbins-Monro stochastic approximation with augmented Lagrangian methods, with proven convergence rates and complexity analysis.

Findings

The RMALM algorithm exhibits linear convergence under mild conditions.

The method has a global complexity of $ ext{O}(1/ extvarepsilon^{1+q})$ for $ extvarepsilon$-solutions.

Numerical experiments show RMALM outperforms existing algorithms on synthetic and real datasets.

Abstract

In this paper, we propose a Robbins-Monro augmented Lagrangian method (RMALM) to solve a class of constrained stochastic convex optimization, which can be regarded as a hybrid of the Robbins-Monro type stochastic approximation method and the augmented Lagrangian method of convex optimizations. Under mild conditions, we show that the proposed algorithm exhibits a linear convergence rate. Moreover, instead of verifying a computationally intractable stopping criteria, we show that the RMALM with the increasing subproblem iteration number has a global complexity $\mathcal{O}(1/\varepsilon^{1+q})$ for the $\varepsilon$-solution (i.e., $\mathbb{E}\left(\|x^k-x^*\|^2\right) < \varepsilon$), where $q$ is any positive number. Numerical results on synthetic and real data demonstrate that the proposed algorithm outperforms the existing algorithms.