# Second-Order Mirror Descent: Convergence in Games Beyond Averaging and Discounting

arXiv: https://arxiv.org/abs/2111.09982 · HTML: https://mlsanity.com/paper/2111.09982/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 81 references that are in the map.
- First posted on arXiv 18 Nov 2021.

## Builds on (most cited first, 16 of 16)

- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [No-regret learning and mixed Nash equilibria: They do not mix](https://mlsanity.com/paper/2010.09514/) (2020)
- [On Gradient-Based Learning in Continuous Games](https://mlsanity.com/paper/1804.05464/) (2018)
- [Exponentiated Gradient Meets Gradient Descent](https://mlsanity.com/paper/1902.01903/) (2019)
- [Survival of the strictest: Stable and unstable equilibria under regularized learning with partial information](https://mlsanity.com/paper/2101.04667/) (2021)
- [First-order optimization algorithms via inertial systems with Hessian driven damping](https://mlsanity.com/paper/1907.10536/) (2019)
- [Learning Nash Equilibria in Monotone Games](https://mlsanity.com/paper/1904.01882/) (2019)
- [Finite composite games: Equilibria and dynamics](https://mlsanity.com/paper/1503.07935/) (2015)
- [Convergence of the Iterates in Mirror Descent Methods](https://mlsanity.com/paper/1805.01526/) (2018)
- [Learning with minimal information in continuous games](https://mlsanity.com/paper/1806.11506/) (2018)
- [On Passivity, Reinforcement Learning and Higher-Order Learning in Multi-Agent Finite Games](https://mlsanity.com/paper/1808.04464/) (2018)
- [Continuous-time Discounted Mirror-Descent Dynamics in Monotone Concave Games](https://mlsanity.com/paper/1912.03460/) (2019)
- [RLC Circuits based Distributed Mirror Descent Method](https://mlsanity.com/paper/1911.06273/) (2019)
- [Continuous-Time Convergence Rates in Potential and Monotone Games](https://mlsanity.com/paper/2011.10682/) (2020)

## Built on by (most cited first, 1 of 1)

- [Accelerated regularized learning in finite N-person games](https://mlsanity.com/paper/2412.20365/) (2024)

## Abstract (arXiv)

In this paper, we propose a second-order extension of the continuous-time game-theoretic mirror descent (MD) dynamics, referred to as MD2, which provably converges to mere (but not necessarily strict) variationally stable states (VSS) without using common auxiliary techniques such as time-averaging or discounting. We show that MD2 enjoys no-regret as well as an exponential rate of convergence towards strong VSS upon a slight modification. MD2 can also be used to derive many novel continuous-time primal-space dynamics. We then use stochastic approximation techniques to provide a convergence guarantee of discrete-time MD2 with noisy observations towards interior mere VSS. Selected simulations are provided to illustrate our results.