Residual-Controlled Multiplier Learning for Stochastic Constrained Decision-Making

2026-06-05

AI Summary

This work addresses the instability of multiplier updates and compromised constraint feasibility in standard primal-dual methods under stochastic minibatch feedback, where noise accumulation degrades performance. To overcome this, the authors propose Residual-Controlled Multiplier Learning (RCML), which innovatively models multiplier dynamics using a pressure–residual structure. By integrating projected pressure feedback with a residual-integral backbone, RCML achieves finite-gain tracking, while modular stochastic stabilization components mitigate heterogeneous noise. The study establishes, for the first time, a stochastic residual bound under minibatch feedback and reveals that, near regular KKT points of nonconvex problems, the residual-based feedback law admits a local interpretation in terms of KKT residuals. Experiments demonstrate that RCML significantly enhances constraint feasibility and multiplier stability across optimization, resource allocation, and fair ranking tasks, without sacrificing objective performance.

Abstract

Stochastic constrained decision-making requires optimizing performance objectives while enforcing statistical requirements such as safety or fairness. However, standard primal--dual methods struggle to update multipliers robustly under stochastic mini-batch feedback, as the noise of mini-batch gradients and constraint estimates can be directly accumulated into the multiplier memory. To address this issue, we propose Residual-Controlled Multiplier Learning (RCML), which reformulates multiplier updating as projected-pressure feedback. The central idea is to decompose the projected multiplier into an effective pressure signal for primal descent and a pressure-memory residual for finite-gain multiplier tracking. To handle heterogeneous and noisy observations, we further augment this residual-integral backbone with modular stochastic stabilization components. For the convex-affine backbone, we establish finite-gain convergence, derive a stochastic residual bound under mini-batch feedback, and show that the residual feedback law admits a local KKT-residual interpretation near regular KKT points of nonconvex problems. Experiments across optimization, allocation, and fair-ranking tasks show that RCML improves feasibility control and multiplier stability while maintaining competitive objective performance. Code is available here.

Problem

• stochastic constrained decision-making
• multiplier learning
• primal-dual methods
• mini-batch feedback
• multiplier stability

Innovation

• Residual-Controlled Multiplier Learning
• stochastic constrained optimization
• projected-pressure feedback
• multiplier stability
• KKT-residual interpretation