# Regret, equilibrium, and learning in games: A guided tour

arXiv: https://arxiv.org/abs/2608.09389 · HTML: https://mlsanity.com/paper/2608.09389/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 112 references that are in the map.
- First posted on arXiv 10 Aug 2026.

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

- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Fast Convergence of Regularized Learning in Games](https://mlsanity.com/paper/1507.00407/) (2015)
- [A Variational Inequality Perspective on Generative Adversarial Networks](https://mlsanity.com/paper/1802.10551/) (2018)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [Tsallis-INF: An Optimal Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1807.07623/) (2018)
- [On the convergence of single-call stochastic extra-gradient methods](https://mlsanity.com/paper/1908.08465/) (2019)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [Adaptive Learning in Continuous Games: Optimal Regret Bounds and Convergence to Nash Equilibrium](https://mlsanity.com/paper/2104.12761/) (2021)
- [No-regret learning and mixed Nash equilibria: They do not mix](https://mlsanity.com/paper/2010.09514/) (2020)
- [Explore Aggressively, Update Conservatively: Stochastic Extragradient Methods with Variable Stepsize Scaling](https://mlsanity.com/paper/2003.10162/) (2020)
- [Fighting Bandits with a New Kind of Smoothness](https://mlsanity.com/paper/1512.04152/) (2015)
- [Adaptive extra-gradient methods for min-max optimization and games](https://mlsanity.com/paper/2010.12100/) (2020)
- [Survival of the strictest: Stable and unstable equilibria under regularized learning with partial information](https://mlsanity.com/paper/2101.04667/) (2021)
- [A unified stochastic approximation framework for learning in games](https://mlsanity.com/paper/2206.03922/) (2022)
- [Smooth Calibration, Leaky Forecasts, Finite Recall, and Nash Dynamics](https://mlsanity.com/paper/2210.07152/) (2022)
- [Forecast Hedging and Calibration](https://mlsanity.com/paper/2210.07169/) (2022)
- [No-Regret Learning in Games with Noisy Feedback: Faster Rates and Adaptivity via Learning Rate Separation](https://mlsanity.com/paper/2206.06015/) (2022)
- [A continuous-time approach to online optimization](https://mlsanity.com/paper/1401.6956/) (2014)

## Abstract (arXiv)

This note aims to serve as an entry point to the literature on learning in games, a topic with significant theoretical appeal and a wide range of applications -- from machine learning and data science to economics and beyond. Our presentation is structured around two complementary viewpoints: We first consider a single agent -- the learner -- engaged in a sequential decision process in an unknown, non-stationary, and possibly adversarial environment. We then examine what happens when the environment is shaped by the decisions of several interacting agents, not necessarily aware of each other's actions or goals, and all seeking to improve their individual rewards. In this general context, we examine a family of regularized learning policies based on best-responding to the past history of play, up to a regularization penalty intended to encourage exploration and prevent over-commitment to suboptimal choices. In the single-agent setting, we present some basic regret bounds for regularized learning in adversarial multi-armed bandits; in the multi-agent setting, we describe an ergodic equilibrium convergence result for zero-sum games in the spirit of classical results on fictitious play, as well as a "folk theorem" linking strategic and dynamic notions of stability -- Nash equilibria and attracting points of regularized learning, respectively. We pay special attention to the information available to the players and, through a unified analysis framework, we study both oracle- and payoff-based (bandit) methods. Our goal is to provide a coherent and comprehensible -- albeit, by necessity, not comprehensive -- account of some recent ideas in the field, and to discuss their implications for the study of rationality.