# Mirror Descent-Type Algorithms for the Variational Inequality Problem with Functional Constraints

arXiv: https://arxiv.org/abs/2605.16262 · HTML: https://mlsanity.com/paper/2605.16262/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 18 of its 43 references that are in the map.
- First posted on arXiv 24 Feb 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 18 of 18)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [A Variational Inequality Perspective on Generative Adversarial Networks](https://mlsanity.com/paper/1802.10551/) (2018)
- [CRPO: A New Approach for Safe Reinforcement Learning with Convergence Guarantee](https://mlsanity.com/paper/2011.05869/) (2020)
- [The limits of min-max optimization algorithms: convergence to spurious non-critical sets](https://mlsanity.com/paper/2006.09065/) (2020)
- [A Forward-Backward Splitting Method for Monotone Inclusions Without Cocoercivity](https://mlsanity.com/paper/1808.04162/) (2018)
- [A Stochastic Optimization Framework for Fair Risk Minimization](https://mlsanity.com/paper/2102.12586/) (2021)
- [Primal-dual stochastic gradient method for convex programs with many functional constraints](https://mlsanity.com/paper/1802.02724/) (2018)
- [Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints](https://mlsanity.com/paper/1710.06612/) (2017)
- [Independent Learning in Constrained Markov Potential Games](https://mlsanity.com/paper/2402.17885/) (2024)
- [First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems](https://mlsanity.com/paper/2204.03132/) (2022)
- [Convergence of the Iterates in Mirror Descent Methods](https://mlsanity.com/paper/1805.01526/) (2018)
- [Provably Learning Nash Policies in Constrained Markov Potential Games](https://mlsanity.com/paper/2306.07749/) (2023)
- [On Modification of an Adaptive Stochastic Mirror Descent Algorithm for Convex Optimization Problems with Functional Constraints](https://mlsanity.com/paper/1904.09513/) (2019)
- [Primal Methods for Variational Inequality Problems with Functional Constraints](https://mlsanity.com/paper/2403.12859/) (2024)
- [A Primal-Dual Approach to Solving Variational Inequalities with General Constraints](https://mlsanity.com/paper/2210.15659/) (2022)
- [An extra gradient Anderson-accelerated algorithm for pseudomonotone variational inequalities](https://mlsanity.com/paper/2408.06606/) (2024)

## Abstract (arXiv)

Variational inequalities play a key role in machine learning research, such as generative adversarial networks, reinforcement learning, adversarial training, and generative models. This paper is devoted to the constrained variational inequality problems with functional constraints (inequality-type constraints). We propose some mirror descent-type algorithms that switch between productive and non-productive steps depending on the values of the functional constraints at iterations, with many different step size rules and stopping criteria. We analyze the proposed algorithms and prove their optimal convergence rate to achieve a solution with desired accuracy, for problems with bounded and monotone operators and Lipschitz convex functional constraints. In addition, we propose a modification of the proposed algorithms by considering each functional constraint in the calculation when we have a productive step, as well as the first constraint that violates the feasibility. This modification can save the running time of algorithms when we have many functional constraints. In addition, we provide an analysis of the proposed algorithms for δ-monotone operators, allowing us to apply the proposed algorithms, as a special case, to constrained minimization problems when we do not have access to the exact information about the subgradient of the objective function. Numerical experiments that illustrate the work and performance of the proposed algorithms are also given.