The impact of uncertainty on regularized learning in games (2506.13286v1)

Abstract: 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.

• The paper demonstrates that stochastic FTRL drives players' strategies to pure Nash equilibria in finite time through quantitative analysis.
• It reveals that random fluctuations collapse mixed strategy behaviors, particularly disrupting recurrence in zero-sum game settings.
• The findings imply that AI and multi-agent systems should favor conservative, pure strategies when operating under uncertain conditions.

The Impact of Uncertainty on Regularized Learning in Games

This paper examines how randomness and uncertainty influence learning dynamics in games, focusing specifically on the stochastic version of the Follow-The-Regularized-Leader (FTRL) methodology, where players' payoff observations and strategy updates are affected continuously by stochastic perturbations. This paper is approached through the lens of game theory, offering a nuanced view of the strategic adaptation involved in multi-agent systems. The authors explore the consequences of stochasticity in learning, providing robust theoretical insights backed by precise mathematical analysis.

Key Concepts and Findings

One of the central findings of the paper is summarized by the assertion that "uncertainty favors extremes." The research illustrates that regardless of the structure of the game or the level of noise, stochastic FTRL dynamics drive players' strategy profiles towards pure strategies over time. This is a crucial deviation from the behavior observed under deterministic settings, where mixed strategies and convergences outside pure Nash equilibria are possible.

The paper's results established several critical properties:

1. Finite Time Attraction to Pure Strategies: Every player in the game reaches an arbitrarily small neighborhood of a pure strategy in finite time, which is quantified by the hitting time $\tau_{\play,\varepsilon}$. The expected hitting time is bounded as $O(e^\lambda/\lambda)$ with respect to appropriately defined $\lambda$. This demonstrates the positioning effect of noise driving strategies towards pure choices.
2. Pure Nash Equilibria as Limit Points: The dynamics under uncertainty ensure that players will infinitely often revert to states close to pure strategies. Consequently, the only possible limit points of the stochastic FTRL dynamics in games, where learning behavior converges probabilistically, are pure Nash equilibria. Therefore, games lacking pure equilibria cannot exhibit convergence in stochastic FTRL settings.
3. Closed Sets and Attractiveness: The analysis shows that a span of pure strategies is stable and attractive if and only if it is closed under rational behavior. This finding aligns with deterministic analogs and underscores the profound impact of stochasticity, indicating that noise does not alter the club structure’s rational stability.
4. Disruption of Recurrence in Zero-Sum Settings: In settings where deterministic dynamics are typically recurrent (such as zero-sum games with interior equilibria), randomness alters this behavior, unexpectedly driving the dynamics to drift toward the boundary in average, contradicting the recurrence observed in deterministic systems.

Implications for AI and Future Research

The implications of this research extend into broader contexts such as AI and machine learning, especially in multi-agent domains. The potential fragility of mixed equilibria under stochastic conditions prompts reconsideration of strategic paradigms where uncertainty is prevalent. Within AI systems where strategic agents operate in a stochastic environment, more conservative approaches favor simpler, pure strategies.

Future directions might explore discrete analogs of the FTRL dynamics or consider extensions to games with continuous action spaces. Moreover, ongoing research could investigate applications where players exhibit learning over time through step-sizing regimes that progressively eliminate noise impacts.

Conclusion

In conclusion, this paper provides a profound understanding of how stochastic perturbations promote strategic convergence towards pure Nash equilibria. The theoretical modeling of these dynamics elaborates on the inherent favorability of extremes under conditions of uncertainty. As we move forward, the insights gained from this paper can guide innovations in AI strategies, machine learning algorithms, and multi-agent systems, where uncertainty remains a significant challenge. The implications call for further exploration and comprehensive models to predict behaviors in more complex environments, advancing both theory and application.