Mirror Descent Algorithms Achieve Optimal Convergence for Constrained Variational Inequalities

Variational inequalities are foundational in machine learning, underpinning generative adversarial networks, reinforcement learning, and adversarial training. A new paper by Alkousa, Stonyakin, Alashqar, and Ablaev (arXiv:2605.16262) tackles the challenging variant with functional constraints (inequality-type). They propose mirror descent-type algorithms that dynamically switch between productive and non-productive steps based on the values of functional constraints at each iteration. The work provides multiple step-size rules and stopping criteria, proving that the algorithms achieve optimal convergence rates for problems with bounded monotone operators and Lipschitz convex functional constraints.

The authors also introduce a practical modification: when handling many functional constraints, the algorithm only considers each constraint during a productive step or checks the first constraint that violates feasibility. This significantly reduces running time while maintaining convergence guarantees. Additionally, they extend the analysis to δ-monotone operators, allowing the algorithms to apply to constrained minimization problems where exact subgradient information is unavailable. Numerical experiments validate the performance, showing reliable convergence even under inexactness. The results offer a robust, theoretically grounded toolkit for training modern ML models with complex constraints.