Title: Robust Equilibria in Continuous Games: From Strategic to Dynamic Robustness
Venue: NeurIPS 2025
Accepted: yes
Keywords: game theory

Nash equilibrium

robust equilibrium

regularized learning
Abstract: In this paper, we examine the robustness of Nash equilibria in continuous games, under both strategic and dynamic uncertainty. Starting with the former, we introduce the notion of a robust equilibrium as those equilibria that remain invariant to small—but otherwise arbitrary—perturbations to the game’s payoff structure, and we provide a crisp geometric characterization thereof. Subsequently, we turn to the question of dynamic robustness, and we examine which equilibria may arise as stable limit points of the dynamics of “follow the regularized leader” (FTRL) in the presence of randomness and uncertainty. Despite their very distinct origins, we establish a structural correspondence between these two notions of robustness: strategic robustness implies dynamic robustness, and, conversely, the requirement of strategic robustness cannot be relaxed if dynamic robustness is to be maintained. Finally, we examine the rate of convergence to robust equilibria as a function of the underlying regularizer, and we show that entropically regularized learning converges at a geometric rate in games with affinely constrained action spaces.


Peer reviews:
comments: SUMMARY:
This paper focuses on continuous games which exhibit some level of uncertainty and the stability of equilibria in these games. The authors provide the concept of a robust equilibrium to define and describe equilibria that remain valid under sufficiently small perturbations of the payoff structure. The paper provides several theoretical results, such as a geometric characterization of robust equilibria and the relation of strategic and dynamic robustness with respect to FTRL dynamics.

STRENGTHS AND WEAKNESSES:
Overall, I enjoyed reading the paper and did not spot any major weaknesses. Here are the details:

Strengths:
- The paper has a clear structure and extensively discusses related work which enables readers to access the contributions of the present work
- The authors consistently discuss limitations and provide context for their theoretical results
- The results seem to close a gap in the existing literature
- The theoretical results appear rigorous and assumptions are clearly stated
- Examples complement the theory

Suggestions:
- Would it be possible to include an actual simulation of the examples? I understand that the character of this paper is theoretical, but an exemplary computational simulation could provide further context
- I think there is a typo between lines 139 and 140: "for all $x_i$" instead of "for all $x$"

QUESTIONS:
1. Would it be possible to include an actual simulation of the examples?
2. How could the theoretical findings be leveraged to advance practical learning applications?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-06

comments: SUMMARY:
This paper looks at the dynamics of the Follow The Regularized Leader (FTRL) algorithm in continuous games of finite dimension, in both deterministic and stochastic settings. It relies on an assumption of a robust equilibrium in order to ensure convergence under any regularizer. Especially, the rate is characterized depending on the regularizer, and FTRL is shown not to converge to non-equilibrium. The paper is purely theoretical.

STRENGTHS AND WEAKNESSES:
Strengths:

- The paper is well written and very clear.

- The statements are precise and rigorous. All properties are proven properly.

Weakness:

- The robust equilibrium assumption is very strong: it restricts the solutions to extreme points with gradients in the interior of the polar cone. As the main theorems assume that the primal iterates stay in a neighborhood of this extreme point, the results are relatively straightforward: in a neighborhood, all gradients are oriented "toward" this extreme point, and the trajectory of the dual iterates is easily characterized.

To support the idea of the restrictiveness of the assumption, in the specific case of mixed policies of a finite games, robustness is (if I'm not mistaken) equivalent to being a strict pure Nash equilibrium.

QUESTIONS:
Could the assumptions on being a non-equilibrium for the non-convergence in Proposition 1 be relaxed? Proposition 3 shows that being non-robust is not enough, but perhaps a more precise characterization depending on the regularizer is possible? This could justify the very restrictive assumption of robustness.

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 4
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-13

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper defines two notions of stability for local Nash equilibrium in continuous games: strategic robustness and dynamic robustness. For strategic robustness, the author refers to a Nash equilibrium that remains unchanged under perturbation in the underlying game's payoff structure in terms of the vector fields. For dynamic robustness, the authors refer to an equilibrium that attracts the FTRL dynamics with initial conditions in a neighborhood. \n\nThe main results include a characterization of the strategic robustness equilibrium (Thm 1) and a proof that FTRL converges to the strategic robustness equilibrium (Thm 2).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n* The problem studied in this paper is interesting, and the writting is clear and easy to follow.\n* The observation that small perturbation on the payoff value of games will destroy all the equilibrium (Example 3.1 & 3.2) is interesting.\n* Theorem 1 provides an interesting characterization of the geometry of robust equilibrium in terms of TC and PC.\n\nWeaknesses:\n\n* It is not clear which kinds of games could admit a robust equilibrium. Given that the existence of a robust equilibrium is a precondition for the convergence result in Thm 2, the lack of discussion on the existence of a robust equilibrium makes the value of this result less clear. For example, it seems that all the interior equilibria will not be robust according to Thm 1, which implies that a robust equilibrium may not exist in many interesting unconstrained games, such as GANs.\n\n* It is not very well-motivated to consider the constant step size FTRL dynamics, given that the vanishing step size is a benchmark in the theoretical analysis of the convergence properties of FTRL. Especially, it is not clear whether the result of Theorem 2 also applies to the vanishing step size case.\n\nQUESTIONS:\n* Can the author provide a characterization of the existence of a robust Nash equilibrium as defined in Definition 1? For example, in which kind of important and interesting games does the robust Nash equilibrium necessarily exist? \n\n  **(This is my main concern. I would be happy to increase my evaluation score if this issue is addressed.)**\n\n* Does the dynamic robustness result still hold when considering the average-iterate behavior of FTRL dynamics?\n\n* Is it possible to define robustness for mix Nash equilibrium?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper shows connections between two notions of robustness in games. The first is strategic robustness which requires that an equilibrium of a game continues to be an equilibrium even if the payoffs of the game are perturbed by a small amount. The second is dynamic robustness, which requires that learning dynamics converge to this equilibrium even if the initial point is perturbed by a small amount. \n\nThe second notion of robustness is interesting in dynamics where the step-sizes don’t decay. So the paper considers dynamic robustness with FTRL as learning dynamics and show that their results apply to FTRL with any choice of regularizer. \n\nThe paper provides a geometric interpretation of strategic robust equilibria. The paper shows that strategic robustness is a sufficient condition for dynamic robustness relative to FTRL for two feedback structures: unbiased estimate of loss gradients and simply the loss received. \n\nToward arguing that strategic robustness is also necessary, they show that strategic robustness is necessary for some regularizer in an example they construct.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: The paper makes a nice connection between two forms of robustness of equilibria. This also seems to be a connection between two forms of equilibrium selection --- based on payoffs and based on convergence of dynamics. The paper is well-written and the main arguments are easy to follow.\n\nWeaknesses: There could be more intuition provided about the proof for Theorem 2 which is the main theorem showing sufficiency of strategic robustness for dynamic robustness. Namely, describing the main technical ideas/tools to analyze convergence of FTRL and how strategic robustness enters this would be helpful.\n\nQUESTIONS:\nDo you think this connection will extend to other dynamics, such as gradient descent?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-03"}