# Adaptive and Universal Algorithms for Variational Inequalities with Optimal Convergence

arXiv: https://arxiv.org/abs/2010.07799 · HTML: https://mlsanity.com/paper/2010.07799/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Oct 2020.

## Built on by (most cited first, 12 of 12)

- [No-Regret Learning in Games with Noisy Feedback: Faster Rates and Adaptivity via Learning Rate Separation](https://mlsanity.com/paper/2206.06015/) (2022)
- [Nest Your Adaptive Algorithm for Parameter-Agnostic Nonconvex Minimax Optimization](https://mlsanity.com/paper/2206.00743/) (2022)
- [On the Convergence of AdaGrad(Norm) on Rd: Beyond Convexity, Non-Asymptotic Rate and Acceleration](https://mlsanity.com/paper/2209.14827/) (2022)
- [Smooth Monotone Stochastic Variational Inequalities and Saddle Point Problems: A Survey](https://mlsanity.com/paper/2208.13592/) (2022)
- [Adaptive Accelerated (Extra-)Gradient Methods with Variance Reduction](https://mlsanity.com/paper/2201.12302/) (2022)
- [A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games](https://mlsanity.com/paper/2311.10859/) (2023)
- [Lean and Mean Adaptive Optimization via Subset-Norm and Subspace-Momentum with Convergence Guarantees](https://mlsanity.com/paper/2411.07120/) (2024)
- [Achieving Near-Optimal Convergence for Distributed Minimax Optimization with Adaptive Stepsizes](https://mlsanity.com/paper/2406.02939/) (2024)
- [Layer-wise Quantization for Quantized Optimistic Dual Averaging](https://mlsanity.com/paper/2505.14371/) (2025)
- [Parameter-Free Non-Ergodic Extragradient Algorithms for Solving Monotone Variational Inequalities](https://mlsanity.com/paper/2604.07662/) (2026)
- [A regret minimization approach to fixed-point iterations](https://mlsanity.com/paper/2509.21653/) (2025)
- [Adaptive Delayed-Update Cyclic Algorithm for Variational Inequalities](https://mlsanity.com/paper/2603.29128/) (2026)

## Abstract (arXiv)

We develop new adaptive algorithms for variational inequalities with monotone operators, which capture many problems of interest, notably convex optimization and convex-concave saddle point problems. Our algorithms automatically adapt to unknown problem parameters such as the smoothness and the norm of the operator, and the variance of the stochastic evaluation oracle. We show that our algorithms are universal and simultaneously achieve the optimal convergence rates in the non-smooth, smooth, and stochastic settings. The convergence guarantees of our algorithms improve over existing adaptive methods by a Omega(√ln T) factor, matching the optimal non-adaptive algorithms. Additionally, prior works require that the optimization domain is bounded. In this work, we remove this restriction and give algorithms for unbounded domains that are adaptive and universal. Our general proof techniques can be used for many variants of the algorithm using one or two operator evaluations per iteration. The classical methods based on the ExtraGradient/MirrorProx algorithm require two operator evaluations per iteration, which is the dominant factor in the running time in many settings.