# Scale-Invariant Fast Convergence in Games

arXiv: https://arxiv.org/abs/2602.11857 · HTML: https://mlsanity.com/paper/2602.11857/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 31 references that are in the map.
- First posted on arXiv 12 Feb 2026.

## Builds on (most cited first, 17 of 17)

- [DeepStack: Expert-Level Artificial Intelligence in No-Limit Poker](https://mlsanity.com/paper/1701.01724/) (2017)
- [Nash Learning from Human Feedback](https://mlsanity.com/paper/2312.00886/) (2023)
- [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 Minimaximalist Approach to Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2401.04056/) (2024)
- [Mastering the Game of Stratego with Model-Free Multiagent Reinforcement Learning](https://mlsanity.com/paper/2206.15378/) (2022)
- [Scale-Free Online Learning](https://mlsanity.com/paper/1601.01974/) (2016)
- [Hedging in games: Faster convergence of external and swap regrets](https://mlsanity.com/paper/2006.04953/) (2020)
- [Impossible Tuning Made Possible: A New Expert Algorithm and Its Applications](https://mlsanity.com/paper/2102.01046/) (2021)
- [Second-order Quantile Methods for Experts and Combinatorial Games](https://mlsanity.com/paper/1502.08009/) (2015)
- [Near-Optimal No-Regret Learning for Correlated Equilibria in Multi-Player General-Sum Games](https://mlsanity.com/paper/2111.06008/) (2021)
- [Lipschitz Adaptivity with Multiple Learning Rates in Online Learning](https://mlsanity.com/paper/1902.10797/) (2019)
- [Uncoupled Learning Dynamics with O(log T) Swap Regret in Multiplayer Games](https://mlsanity.com/paper/2204.11417/) (2022)
- [Extensive-Form Game Solving via Blackwell Approachability on Treeplexes](https://mlsanity.com/paper/2403.04680/) (2024)
- [Corrupted Learning Dynamics in Games](https://mlsanity.com/paper/2412.07120/) (2024)
- [Scale-Invariant Regret Matching and Online Learning with Optimal Convergence: Bridging Theory and Practice in Zero-Sum Games](https://mlsanity.com/paper/2510.04407/) (2025)
- [Tight Regret Upper and Lower Bounds for Optimistic Hedge in Two-Player Zero-Sum Games](https://mlsanity.com/paper/2510.11691/) (2025)

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

- [Sublogarithmic Swap Regret in Multiplayer General-Sum Games via Hybrid Regularization](https://mlsanity.com/paper/2608.04149/) (2026)

## Abstract (arXiv)

Scale-invariance in games has recently emerged as a widely valued desirable property. Yet, almost all fast convergence guarantees in learning in games require prior knowledge of the utility scale. To address this, we develop learning dynamics that achieve fast convergence while being both scale-free, requiring no prior information about utilities, and scale-invariant, remaining unchanged under positive rescaling of utilities. For two-player zero-sum games, we obtain scale-free and scale-invariant dynamics with external regret bounded by tildeO(Adiff), where Adiff is the payoff range, which implies an tildeO(Adiff / T) convergence rate to Nash equilibrium after T rounds. For multiplayer general-sum games with n players and m actions, we obtain scale-free and scale-invariant dynamics with swap regret bounded by O(Umax log T), where Umax is the range of the utilities, ignoring the dependence on the number of players and actions. This yields an O(Umax log T / T) convergence rate to correlated equilibrium. Our learning dynamics are based on optimistic follow-the-regularized-leader with an adaptive learning rate that incorporates the squared path length of the opponents' gradient vectors, together with a new stopping-time analysis that exploits negative terms in regret bounds without scale-dependent tuning. For general-sum games, scale-free learning is enabled also by a technique called doubling clipping, which clips observed gradients based on past observations.