# The impact of uncertainty on regularized learning in games

arXiv: https://arxiv.org/abs/2506.13286 · HTML: https://mlsanity.com/paper/2506.13286/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 66 references that are in the map.
- First posted on arXiv 16 Jun 2025.

## Builds on (most cited first, 19 of 19)

- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [More Adaptive Algorithms for Adversarial Bandits](https://mlsanity.com/paper/1801.03265/) (2018)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [The limits of min-max optimization algorithms: convergence to spurious non-critical sets](https://mlsanity.com/paper/2006.09065/) (2020)
- [No-regret learning and mixed Nash equilibria: They do not mix](https://mlsanity.com/paper/2010.09514/) (2020)
- [Hessian Riemannian gradient flows in convex programming](https://mlsanity.com/paper/1811.10331/) (2018)
- [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)
- [Riemannian game dynamics](https://mlsanity.com/paper/1603.09173/) (2016)
- [A continuous-time approach to online optimization](https://mlsanity.com/paper/1401.6956/) (2014)
- [On the robustness of learning in games with stochastically perturbed payoff observations](https://mlsanity.com/paper/1412.6565/) (2014)
- [On the convergence of gradient-like flows with noisy gradient input](https://mlsanity.com/paper/1611.06730/) (2016)
- [Stochastic mirror descent dynamics and their convergence in monotone variational inequalities](https://mlsanity.com/paper/1710.01551/) (2017)
- [Decomposition of games: some strategic considerations](https://mlsanity.com/paper/1901.06048/) (2019)
- [The equivalence of dynamic and strategic stability under regularized learning in games](https://mlsanity.com/paper/2311.02407/) (2023)
- [A geometric decomposition of finite games: Convergence vs. recurrence under exponential weights](https://mlsanity.com/paper/2405.07224/) (2024)
- [No-regret learning in harmonic games: Extrapolation in the face of conflicting interests](https://mlsanity.com/paper/2412.20203/) (2024)
- [Imitation Dynamics with Payoff Shocks](https://mlsanity.com/paper/1412.7842/) (2014)

## Built on by (most cited first, 2 of 2)

- [Multi-agent learning under uncertainty: Recurrence vs. concentration](https://mlsanity.com/paper/2512.08132/) (2025)
- [Robust equilibria in continuous games: From strategic to dynamic robustness](https://mlsanity.com/paper/2512.08138/) (2025)

## Abstract (arXiv)

In this paper, we investigate how randomness and uncertainty influence learning in games. Specifically, we examine a perturbed variant of the dynamics of "follow-the-regularized-leader" (FTRL), where the players' payoff observations and strategy updates are continually impacted by random shocks. Our findings reveal that, in a fairly precise sense, "uncertainty favors extremes": in any game, regardless of the noise level, every player's trajectory of play reaches an arbitrarily small neighborhood of a pure strategy in finite time (which we estimate). Moreover, even if the player does not ultimately settle at this strategy, they return arbitrarily close to some (possibly different) pure strategy infinitely often. This prompts the question of which sets of pure strategies emerge as robust predictions of learning under uncertainty. We show that (a) the only possible limits of the FTRL dynamics under uncertainty are pure Nash equilibria; and (b) a span of pure strategies is stable and attracting if and only if it is closed under better replies. Finally, we turn to games where the deterministic dynamics are recurrent - such as zero-sum games with interior equilibria - and we show that randomness disrupts this behavior, causing the stochastic dynamics to drift toward the boundary on average.