arXiv subjects

Gabriele Farina

Publications and source records attributed to Gabriele Farina.

At least 55 records · Page 3Linked to original sources

arXiv2025

On Separation Between Best-Iterate, Random-Iterate, and Last-Iterate Convergence of Learning in Games

Non-ergodic convergence of learning dynamics in games is widely studied recently because of its importance in both theory and practice. Recent work (Cai et al., 2024) showed that a broad class of learning dynamics, including Optimistic Multiplicative Weights Update (OMWU), can exhibit arbitrarily slow last-iterate convergence even in simple $2 \times 2$ matrix games, despite many of these dynamics being known to converge asymptotically in the last iterate. It remains unclear, however, whether these algorithms achieve fast non-ergodic convergence under weaker criteria, such as best-iterate convergence. We show that for $2\times 2$ matrix games, OMWU achieves an $O(T^{-1/6})$ best-iterate convergence rate, in stark contrast to its slow last-iterate convergence in the same class of games. Furthermore, we establish a lower bound showing that OMWU does not achieve any polynomial random-iterate convergence rate, measured by the expected duality gaps across all iterates. This result challenges the conventional wisdom that random-iterate convergence is essentially equivalent to best-iterate convergence, with the former often used as a proxy for establishing the latter. Our analysis uncovers a new connection to dynamic regret and presents a novel two-phase approach to best-iterate convergence, which could be of independent interest.

cs.LG↗

arXiv2025

Expected Variational Inequalities

Variational inequalities (VIs) encompass many fundamental problems in diverse areas ranging from engineering to economics and machine learning. However, their considerable expressivity comes at the cost of computational intractability. In this paper, we introduce and analyze a natural relaxation -- which we refer to as expected variational inequalities (EVIs) -- where the goal is to find a distribution that satisfies the VI constraint in expectation. By adapting recent techniques from game theory, we show that, unlike VIs, EVIs can be solved in polynomial time under general (nonmonotone) operators. EVIs capture the seminal notion of correlated equilibria, but enjoy a greater reach beyond games. We also employ our framework to capture and generalize several existing disparate results, including from settings such as smooth games, and games with coupled constraints or nonconcave utilities.

cs.GT↗

arXiv2025

Efficient $Φ$-Regret Minimization with Low-Degree Swap Deviations in Extensive-Form Games

Recent breakthrough results by Dagan, Daskalakis, Fishelson and Golowich [2023] and Peng and Rubinstein [2023] established an efficient algorithm attaining at most $ε$ swap regret over extensive-form strategy spaces of dimension $N$ in $N^{\tilde O(1/ε)}$ rounds. On the other extreme, Farina and Pipis [2023] developed an efficient algorithm for minimizing the weaker notion of linear-swap regret in $\mathsf{poly}(N)/ε^2$ rounds. In this paper, we develop efficient parameterized algorithms for regimes between these two extremes. We introduce the set of $k$-mediator deviations, which generalize the untimed communication deviations recently introduced by Zhang, Farina and Sandholm [2024] to the case of having multiple mediators, and we develop algorithms for minimizing the regret with respect to this set of deviations in $N^{O(k)}/ε^2$ rounds. Moreover, by relating $k$-mediator deviations to low-degree polynomials, we show that regret minimization against degree-$k$ polynomial swap deviations is achievable in $N^{O(kd)^3}/ε^2$ rounds, where $d$ is the depth of the game, assuming a constant branching factor. For a fixed degree $k$, this is polynomial for Bayesian games and quasipolynomial more broadly when $d = \mathsf{polylog} N$ -- the usual balancedness assumption on the game tree.

cs.GT↗

arXiv2025

Fast Last-Iterate Convergence of Learning in Games Requires Forgetful Algorithms

Self-play via online learning is one of the premier ways to solve large-scale two-player zero-sum games, both in theory and practice. Particularly popular algorithms include optimistic multiplicative weights update (OMWU) and optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy $O(1/T)$ ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU offers several advantages including logarithmic dependence on the size of the payoff matrix and $\widetilde{O}(1/T)$ convergence to coarse correlated equilibria even in general-sum games. However, in terms of last-iterate convergence in two-player zero-sum games, an increasingly popular topic in this area, OGDA guarantees that the duality gap shrinks at a rate of $O(1/\sqrt{T})$, while the best existing last-iterate convergence for OMWU depends on some game-dependent constant that could be arbitrarily large. This begs the question: is this potentially slow last-iterate convergence an inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat surprisingly, we show that the former is true. More generally, we prove that a broad class of algorithms that do not forget the past quickly all suffer the same issue: for any arbitrarily small $δ>0$, there exists a $2\times 2$ matrix game such that the algorithm admits a constant duality gap even after $1/δ$ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.

cs.GT↗

arXiv2024

Efficient Learning and Computation of Linear Correlated Equilibrium in General Convex Games

We propose efficient no-regret learning dynamics and ellipsoid-based methods for computing linear correlated equilibria$\unicode{x2014}$a relaxation of correlated equilibria and a strengthening of coarse correlated equilibria$\unicode{x2014}$in general convex games. These are games where the number of pure strategies is potentially exponential in the natural representation of the game, such as extensive-form games. Our work identifies linear correlated equilibria as the tightest known notion of equilibrium that is computable in polynomial time and is efficiently learnable for general convex games. Our results are enabled by a generalization of the seminal framework of of Gordon et al. [2008] for $Φ$-regret minimization, providing extensions to this framework that can be used even when the set of deviations $Φ$ is intractable to separate/optimize over. Our polynomial-time algorithms are similarly enabled by extending the Ellipsoid-Against-Hope approach of Papadimitriou and Roughgarden [2008] and its generalization to games of non-polynomial type proposed by Farina and Pipis [2024a]. We provide an extension to these approaches when we do not have access to the separation oracles required by these works for the dual player.

cs.GT↗

arXiv2024

On the Optimality of Dilated Entropy and Lower Bounds for Online Learning in Extensive-Form Games

First-order methods (FOMs) are arguably the most scalable algorithms for equilibrium computation in large extensive-form games. To operationalize these methods, a distance-generating function, acting as a regularizer for the strategy space, must be chosen. The ratio between the strong convexity modulus and the diameter of the regularizer is a key parameter in the analysis of FOMs. A natural question is then: what is the optimal distance-generating function for extensive-form decision spaces? In this paper, we make a number of contributions, ultimately establishing that the weight-one dilated entropy (DilEnt) distance-generating function is optimal up to logarithmic factors. The DilEnt regularizer is notable due to its iterate-equivalence with Kernelized OMWU (KOMWU) -- the algorithm with state-of-the-art dependence on the game tree size in extensive-form games -- when used in conjunction with the online mirror descent (OMD) algorithm. However, the standard analysis for OMD is unable to establish such a result; the only current analysis is by appealing to the iterate equivalence to KOMWU. We close this gap by introducing a pair of primal-dual treeplex norms, which we contend form the natural analytic viewpoint for studying the strong convexity of DilEnt. Using these norm pairs, we recover the diameter-to-strong-convexity ratio that predicts the same performance as KOMWU. Along with a new regret lower bound for online learning in sequence-form strategy spaces, we show that this ratio is nearly optimal. Finally, we showcase our analytic techniques by refining the analysis of Clairvoyant OMD when paired with DilEnt, establishing an $\mathcal{O}(n \log |\mathcal{V}| \log T/T)$ approximation rate to coarse correlated equilibrium in $n$-player games, where $|\mathcal{V}|$ is the number of reduced normal-form strategies of the players, establishing the new state of the art.

cs.LG↗

arXiv2024

Optimal Correlated Equilibria in General-Sum Extensive-Form Games: Fixed-Parameter Algorithms, Hardness, and Two-Sided Column-Generation

We study the problem of finding optimal correlated equilibria of various sorts in extensive-form games: normal-form coarse correlated equilibrium (NFCCE), extensive-form coarse correlated equilibrium (EFCCE), and extensive-form correlated equilibrium (EFCE). We make two primary contributions. First, we introduce a new algorithm for computing optimal equilibria in all three notions. Its runtime depends exponentially only on a parameter related to the information structure of the game. We also prove a fundamental complexity gap: while our size bounds for NFCCE are similar to those achieved in the case of team games by Zhang et al., this is impossible to achieve for the other two concepts under standard complexity assumptions. Second, we propose a two-sided column generation approach for use when the runtime or memory usage of the previous algorithm is prohibitive. Our algorithm improves upon the one-sided approach of Farina et al. by means of a new decomposition of correlated strategies which allows players to re-optimize their sequence-form strategies with respect to correlation plans which were previously added to the support. Experiments show that our techniques outperform the prior state of the art for computing optimal general-sum correlated equilibria.

cs.GT↗

arXiv2024

Contested Logistics: A Game-Theoretic Approach

We introduce Contested Logistics Games, a variant of logistics problems that account for the presence of an adversary that can disrupt the movement of goods in selected areas. We model this as a large two-player zero-sum one-shot game played on a graph representation of the physical world, with the optimal logistics plans described by the (possibly randomized) Nash equilibria of this game. Our logistics model is fairly sophisticated, and is able to handle multiple modes of transport and goods, accounting for possible storage of goods in warehouses, as well as Leontief utilities based on demand satisfied. We prove computational hardness results related to equilibrium finding and propose a practical double-oracle solver based on solving a series of best-response mixed-integer linear programs. We experiment on both synthetic and real-world maps, demonstrating that our proposed method scales to reasonably large games. We also demonstrate the importance of explicitly modeling the capabilities of the adversary via ablation studies and comparisons with a naive logistics plan based on heuristics.

cs.GT↗

arXiv2024

LiteEFG: An Efficient Python Library for Solving Extensive-form Games

LiteEFG is an efficient library with easy-to-use Python bindings, which can solve multiplayer extensive-form games (EFGs). LiteEFG enables the user to express computation graphs in Python to define updates on the game tree structure. The graph is then executed by the C++ backend, leading to significant speedups compared to running the algorithm in Python. Moreover, in LiteEFG, the user needs to only specify the computation graph of the update rule in a decision node of the game, and LiteEFG will automatically distribute the update rule to each decision node and handle the structure of the imperfect-information game.

cs.GT↗

arXiv2024

A Lower Bound on Swap Regret in Extensive-Form Games

Recent simultaneous works by Peng and Rubinstein [2024] and Dagan et al. [2024] have demonstrated the existence of a no-swap-regret learning algorithm that can reach $ε$ average swap regret against an adversary in any extensive-form game within $m^{\tilde{\mathcal O}(1/ε)}$ rounds, where $m$ is the number of nodes in the game tree. However, the question of whether a $\mathrm{poly}(m, 1/ε)$-round algorithm could exist remained open. In this paper, we show a lower bound that precludes the existence of such an algorithm. In particular, we show that achieving average swap regret $ε$ against an oblivious adversary in general extensive-form games requires at least $\mathrm{exp}\left(Ω\left(\min\left\{m^{1/14}, ε^{-1/6}\right\}\right)\right)$ rounds.

cs.GT↗

arXiv2024

Computing Optimal Equilibria and Mechanisms via Learning in Zero-Sum Extensive-Form Games

We introduce a new approach for computing optimal equilibria via learning in games. It applies to extensive-form settings with any number of players, including mechanism design, information design, and solution concepts such as correlated, communication, and certification equilibria. We observe that optimal equilibria are minimax equilibrium strategies of a player in an extensive-form zero-sum game. This reformulation allows to apply techniques for learning in zero-sum games, yielding the first learning dynamics that converge to optimal equilibria, not only in empirical averages, but also in iterates. We demonstrate the practical scalability and flexibility of our approach by attaining state-of-the-art performance in benchmark tabular games, and by computing an optimal mechanism for a sequential auction design problem using deep reinforcement learning.

cs.GT↗

arXiv2024

The Update-Equivalence Framework for Decision-Time Planning

The process of revising (or constructing) a policy at execution time -- known as decision-time planning -- has been key to achieving superhuman performance in perfect-information games like chess and Go. A recent line of work has extended decision-time planning to imperfect-information games, leading to superhuman performance in poker. However, these methods involve solving subgames whose sizes grow quickly in the amount of non-public information, making them unhelpful when the amount of non-public information is large. Motivated by this issue, we introduce an alternative framework for decision-time planning that is not based on solving subgames, but rather on update equivalence. In this update-equivalence framework, decision-time planning algorithms replicate the updates of last-iterate algorithms, which need not rely on public information. This facilitates scalability to games with large amounts of non-public information. Using this framework, we derive a provably sound search algorithm for fully cooperative games based on mirror descent and a search algorithm for adversarial games based on magnetic mirror descent. We validate the performance of these algorithms in cooperative and adversarial domains, notably in Hanabi, the standard benchmark for search in fully cooperative imperfect-information games. Here, our mirror descent approach exceeds or matches the performance of public information-based search while using two orders of magnitude less search time. This is the first instance of a non-public-information-based algorithm outperforming public-information-based approaches in a domain they have historically dominated.

cs.AI↗

arXiv2024

Polynomial-Time Computation of Exact $Φ$-Equilibria in Polyhedral Games

It is a well-known fact that correlated equilibria can be computed in polynomial time in a large class of concisely represented games using the celebrated Ellipsoid Against Hope algorithm (Papadimitriou and Roughgarden, 2008; Jiang and Leyton-Brown, 2015). However, the landscape of efficiently computable equilibria in sequential (extensive-form) games remains unknown. The Ellipsoid Against Hope does not apply directly to these games, because they do not have the required "polynomial type" property. Despite this barrier, Huang and von Stengel (2008) altered the algorithm to compute exact extensive-form correlated equilibria. In this paper, we generalize the Ellipsoid Against Hope and develop a simple algorithmic framework for efficiently computing saddle-points in bilinear zero-sum games, even when one of the dimensions is exponentially large. Moreover, the framework only requires a "good-enough-response" oracle, which is a weakened notion of a best-response oracle. Using this machinery, we develop a general algorithmic framework for computing exact linear $Φ$-equilibria in any polyhedral game (under mild assumptions), including correlated equilibria in normal-form games, and extensive-form correlated equilibria in extensive-form games. This enables us to give the first polynomial-time algorithm for computing exact linear-deviation correlated equilibria in extensive-form games, thus resolving an open question by Farina and Pipis (2023). Furthermore, even for the cases for which a polynomial time algorithm for exact equilibria was already known, our framework provides a conceptually simpler solution.

cs.GT↗

arXiv2024

Mediator Interpretation and Faster Learning Algorithms for Linear Correlated Equilibria in General Extensive-Form Games

A recent paper by Farina & Pipis (2023) established the existence of uncoupled no-linear-swap regret dynamics with polynomial-time iterations in extensive-form games. The equilibrium points reached by these dynamics, known as linear correlated equilibria, are currently the tightest known relaxation of correlated equilibrium that can be learned in polynomial time in any finite extensive-form game. However, their properties remain vastly unexplored, and their computation is onerous. In this paper, we provide several contributions shedding light on the fundamental nature of linear-swap regret. First, we show a connection between linear deviations and a generalization of communication deviations in which the player can make queries to a "mediator" who replies with action recommendations, and, critically, the player is not constrained to match the timing of the game as would be the case for communication deviations. We coin this latter set the untimed communication (UTC) deviations. We show that the UTC deviations coincide precisely with the linear deviations, and therefore that any player minimizing UTC regret also minimizes linear-swap regret. We then leverage this connection to develop state-of-the-art no-regret algorithms for computing linear correlated equilibria, both in theory and in practice. In theory, our algorithms achieve polynomially better per-iteration runtimes; in practice, our algorithms represent the state of the art by several orders of magnitude.

cs.GT↗

arXiv2024

Team Belief DAG: Generalizing the Sequence Form to Team Games for Fast Computation of Correlated Team Max-Min Equilibria via Regret Minimization

A classic result in the theory of extensive-form games asserts that the set of strategies available to any perfect-recall player is strategically equivalent to a low-dimensional convex polytope, called the sequence-form polytope. Online convex optimization tools operating on this polytope are the current state-of-the-art for computing several notions of equilibria in games, and have been crucial in landmark applications of computational game theory. However, when optimizing over the joint strategy space of a team of players, one cannot use the sequence form to obtain a strategically-equivalent convex description of the strategy set of the team. In this paper, we provide new complexity results on the computation of optimal strategies for teams, and propose a new representation, coined team belief DAG (TB-DAG), that describes team strategies as a convex set. The TB-DAG enjoys state-of-the-art parameterized complexity bounds, while at the same time enjoying the advantages of efficient regret minimization techniques. We show that TB-DAG can be exponentially smaller and can be computed exponentially faster than all other known representations, and that the converse is never true. Experimentally, we show that the TB-DAG, when paired with learning techniques, yields state of the art on a wide variety of benchmark team games.

cs.GT↗

arXiv2023

Optimistic Policy Gradient in Multi-Player Markov Games with a Single Controller: Convergence Beyond the Minty Property

Policy gradient methods enjoy strong practical performance in numerous tasks in reinforcement learning. Their theoretical understanding in multiagent settings, however, remains limited, especially beyond two-player competitive and potential Markov games. In this paper, we develop a new framework to characterize optimistic policy gradient methods in multi-player Markov games with a single controller. Specifically, under the further assumption that the game exhibits an equilibrium collapse, in that the marginals of coarse correlated equilibria (CCE) induce Nash equilibria (NE), we show convergence to stationary $ε$-NE in $O(1/ε^2)$ iterations, where $O(\cdot)$ suppresses polynomial factors in the natural parameters of the game. Such an equilibrium collapse is well-known to manifest itself in two-player zero-sum Markov games, but also occurs even in a class of multi-player Markov games with separable interactions, as established by recent work. As a result, we bypass known complexity barriers for computing stationary NE when either of our assumptions fails. Our approach relies on a natural generalization of the classical Minty property that we introduce, which we anticipate to have further applications beyond Markov games.

cs.GT↗

arXiv2023

Efficient Learning in Polyhedral Games via Best Response Oracles

We study online learning and equilibrium computation in games with polyhedral decision sets, a property shared by both normal-form games and extensive-form games (EFGs), when the learning agent is restricted to using a best-response oracle. We show how to achieve constant regret in zero-sum games and $O(T^{1/4})$ regret in general-sum games while using only $O(\log t)$ best-response queries at a given iteration $t$, thus improving over the best prior result, which required $O(T)$ queries per iteration. Moreover, our framework yields the first last-iterate convergence guarantees for self-play with best-response oracles in zero-sum games. This convergence occurs at a linear rate, though with a condition-number dependence. We go on to show a $O(1/\sqrt{T})$ best-iterate convergence rate without such a dependence. Our results build on linear-rate convergence results for variants of the Frank-Wolfe (FW) algorithm for strongly convex and smooth minimization problems over polyhedral domains. These FW results depend on a condition number of the polytope, known as facial distance. In order to enable application to settings such as EFGs, we show two broad new results: 1) the facial distance for polytopes in standard form is at least $γ/\sqrt{k}$ where $γ$ is the minimum value of a nonzero coordinate of a vertex of the polytope and $k\leq n$ is the number of tight inequality constraints in the optimal face, and 2) the facial distance for polytopes of the form $\mathbf{A}\boldsymbol{x}=\boldsymbol{b},\mathbf{C}\boldsymbol{x}\leq\boldsymbol{d}, \boldsymbol{x}\geq \mathbf{0}$ where $\boldsymbol{x}\in\mathbb{R}^n$, $\mathbf{C}\geq\boldsymbol{0}$ is a nonzero integral matrix, and $\boldsymbol{d}\geq \boldsymbol{0}$, is at least $1/(\|\mathbf{C}\|_\infty\sqrt{n})$. This yields the first such results for several problems such as sequence-form polytopes, flow polytopes, and matching polytopes.

cs.GT↗

arXiv2023

Regularized Conventions: Equilibrium Computation as a Model of Pragmatic Reasoning

We present a model of pragmatic language understanding, where utterances are produced and understood by searching for regularized equilibria of signaling games. In this model (which we call ReCo, for Regularized Conventions), speakers and listeners search for contextually appropriate utterance--meaning mappings that are both close to game-theoretically optimal conventions and close to a shared, ''default'' semantics. By characterizing pragmatic communication as equilibrium search, we obtain principled sampling algorithms and formal guarantees about the trade-off between communicative success and naturalness. Across several datasets capturing real and idealized human judgments about pragmatic implicatures, ReCo matches or improves upon predictions made by best response and rational speech act models of language understanding.

cs.CL↗