# Equilibrium bias and convergence in augmented primal--dual dynamics with sampled constraints

arXiv: https://arxiv.org/abs/2609.13925 · HTML: https://mlsanity.com/paper/2609.13925/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 20 references that are in the map.
- First posted on arXiv 12 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Stochastic Compositional Gradient Descent: Algorithms for Minimizing Compositions of Expected-Value Functions](https://mlsanity.com/paper/1411.3803/) (2014)
- [Solving Stochastic Compositional Optimization is Nearly as Easy as Solving Stochastic Optimization](https://mlsanity.com/paper/2008.10847/) (2020)
- [Accelerating Stochastic Composition Optimization](https://mlsanity.com/paper/1607.07329/) (2016)
- [On the Exponential Stability of Primal-Dual Gradient Dynamics](https://mlsanity.com/paper/1803.01825/) (2018)
- [Solving Stochastic Optimization with Expectation Constraints Efficiently by a Stochastic Augmented Lagrangian-Type Algorithm](https://mlsanity.com/paper/2106.11577/) (2021)
- [Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization](https://mlsanity.com/paper/1903.09580/) (2019)
- [Stochastic Compositional Optimization with Compositional Constraints](https://mlsanity.com/paper/2209.04086/) (2022)
- [Stochastic Compositional Gradient Descent under Compositional Constraints](https://mlsanity.com/paper/2012.09400/) (2020)
- [Stochastic Compositional Minimax Optimization with Provable Convergence Guarantees](https://mlsanity.com/paper/2408.12505/) (2024)

## Abstract (arXiv)

This work studies the stability and convergence of augmented primal-dual dynamics when constraint values are estimated from samples. Unbiased constraint observations can produce a biased augmented multiplier signal, shifting the equilibria of the mean dynamics. For componentwise inequalities, we give a necessary and sufficient condition for preserving the Karush-Kuhn-Tucker (KKT) equilibria and construct a convex example with a locally exponentially stable equilibrium that violates complementarity. To address this bias, constraint values are estimated recursively before forming the augmented multiplier signal. For smooth convex conic problems, a joint energy analysis establishes boundedness of the primal, dual, and estimation states, vanishing estimation error, and almost sure convergence of the primal-dual iterates to a single KKT point under global regularity and bounded conditional second moments. The result allows nonunique solutions and multipliers while keeping the number of samples per iteration fixed. Numerical studies illustrate the predicted equilibrium bias and examine convergence with nonunique KKT points and nonlinear constraints.