# Adaptive Primal-Dual Stochastic Gradient Method for Expectation-constrained Convex Stochastic Programs

arXiv: https://arxiv.org/abs/2012.14943 · HTML: https://mlsanity.com/paper/2012.14943/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2020.

## Built on by (most cited first, 10 of 10)

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- [Stochastic Methods for AUC Optimization subject to AUC-based Fairness Constraints](https://mlsanity.com/paper/2212.12603/) (2022)
- [A single-loop SPIDER-type stochastic subgradient method for expectation-constrained nonconvex nonsmooth optimization](https://mlsanity.com/paper/2501.19214/) (2025)
- [Solving Stochastic Optimization with Expectation Constraints Efficiently by a Stochastic Augmented Lagrangian-Type Algorithm](https://mlsanity.com/paper/2106.11577/) (2021)
- [Stochastic Smoothed Primal-Dual Algorithms for Nonconvex Optimization with Linear Inequality Constraints](https://mlsanity.com/paper/2504.07607/) (2025)
- [Local AdaGrad-Type Algorithm for Stochastic Convex-Concave Optimization](https://mlsanity.com/paper/2106.10022/) (2021)
- [Same Flow, Different Paths: Variance Reduction in Flow Matching](https://mlsanity.com/paper/2609.17287/) (2026)

## Abstract (arXiv)

Stochastic gradient methods (SGMs) have been widely used for solving stochastic optimization problems. A majority of existing works assume no constraints or easy-to-project constraints. In this paper, we consider convex stochastic optimization problems with expectation constraints. For these problems, it is often extremely expensive to perform projection onto the feasible set. Several SGMs in the literature can be applied to solve the expectation-constrained stochastic problems. We propose a novel primal-dual type SGM based on the Lagrangian function. Different from existing methods, our method incorporates an adaptiveness technique to speed up convergence. At each iteration, our method inquires an unbiased stochastic subgradient of the Lagrangian function, and then it renews the primal variables by an adaptive-SGM update and the dual variables by a vanilla-SGM update. We show that the proposed method has a convergence rate of O(1/√k) in terms of the objective error and the constraint violation. Although the convergence rate is the same as those of existing SGMs, we observe its significantly faster convergence than an existing non-adaptive primal-dual SGM and a primal SGM on solving the Neyman-Pearson classification and quadratically constrained quadratic programs. Furthermore, we modify the proposed method to solve convex-concave stochastic minimax problems, for which we perform adaptive-SGM updates to both primal and dual variables. A convergence rate of O(1/√k) is also established to the modified method for solving minimax problems in terms of primal-dual gap.