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Reinforcement Learning · Inverse

Junyi Liao, Zihan Zhu, Ethan Fang, Zhuoran Yang, Vahid Tarokh

Estimating the unknown reward functions driving agents' behavior is a central challenge in inverse games and reinforcement learning. This paper introduces a unified framework for reward function recovery in two-player zero-sum matrix games and Markov games with entropy regularization. Given observed player strategies and actions, we aim to reconstruct the underlying reward functions. This task is challenging due to the inherent ambiguity of inverse problems, the non-uniqueness of feasible rewards, and limited observational data coverage. To address these challenges, we establish reward function identifiability using the quantal response equilibrium (QRE) under linear assumptions. Building on this theoretical foundation, we propose an algorithm to learn reward from observed actions, designed to capture all plausible reward parameters by constructing confidence sets. Our algorithm works in both static and dynamic settings and is adaptable to incorporate other methods, such as Maximum Likelihood Estimation (MLE). We provide strong theoretical guarantees for the reliability and sample-efficiency of our algorithm. Empirical results demonstrate the framework’s effectiveness in accurately recovering reward functions across various scenarios, offering new insights into decision-making in competitive environments.

Theory · Online Learning and Bandits

Quan Nguyen, Nishant Mehta, Cristóbal Guzmán

The minimax sample complexity of group distributionally robust optimization (GDRO) has been determined up to a $\log(K)$ factor, where $K$ is the number of groups. In this work, we venture beyond the minimax perspective via a novel notion of sparsity that we call $(\lambda, \beta)$-sparsity. In short, this condition means that at any parameter $\theta$, there is a set of at most $\beta$ groups whose risks at $\theta$ are all at least $\lambda$ larger than the risks of the other groups. To find an $\epsilon$-optimal $\theta$, we show via a novel algorithm and analysis that the $\epsilon$-dependent term in the sample complexity can swap a linear dependence on $K$ for a linear dependence on the potentially much smaller $\beta$. This improvement leverages recent progress in sleeping bandits, showing a fundamental connection between the two-player zero-sum game optimization framework for GDRO and per-action regret bounds in sleeping bandits. We next show an adaptive algorithm which, up to logarithmic factors, obtains a sample complexity bound that adapts to the best $(\lambda, \beta)$-sparsity condition that holds. We also show how to obtain a dimension-free semi-adaptive sample complexity bound with a computationally efficient method.Finally, we demonstrate the practicality of the $(\lambda, \beta)$-sparsity condition and the improved sample efficiency of our algorithms on both synthetic and real-life datasets.

Theory · Game Theory

Bo Li, Fangxiao WANG, Xing Shiji

We study the fair scheduling of jobs among groups of (unrelated) machines and focus on the maximin share (MMS) fairness at the group level. The problem was first introduced by Li et al. [NeurIPS 2023], where each group consists of a number of identical machines (or identical up to different speeds), and the cost of a group is determined by the minimum makespan on completing all jobs assigned to it. It is left as an open problem when the machines within each group are unrelated. In this paper, we first resolve this problem and design a polynomial-time algorithm that computes a 2-approximate MMS allocation via linear programming techniques. We complement this result with a hard instance, showing that no algorithm can be better than $(2-\frac{1}{n})$-approximate MMS, where $n$ is the number of machines. Thus the approximation ratio 2 is asymptotically tight. When the groups consist of identical machines, we improve the approximation ratio to $\frac{4}{3}$.

Theory · Game Theory

Chris Dong, Jannik Peters

Multiwinner voting is the study of electing a fixed-size committee given individual agents' preferences over candidates. Most research in this field has been limited to a static setting, with only one election over a fixed set of candidates. However, this approach overlooks the dynamic nature of applications, where candidate sets are subject to change.We extend the study of proportionality in multiwinner voting to dynamic settings, allowing candidates to join or leave the election and demanding that each chosen committee satisfies proportionality without differing too much from the previously selected committee. We consider approval preferences, ranked preferences, and the proportional clustering setting. In these settings, we either give algorithms making few changes or show that such algorithms cannot exist for various proportionality axioms. In particular, we show that such algorithms cannot exist for ranked preferences and provide amortized and exact algorithms for several proportionality notions in the other two settings.

Theory · Game Theory

Ohad Einav, Nir Rosenfeld

Machine learning models play a key role for service providers looking to gain market share in consumer markets. However, traditional learning approaches do not take into account the existence of additional providers, who compete with each other for consumers. Our work aims to study learning in this market setting, as it affects providers, consumers, and the market itself. We begin by analyzing such markets through the lens of the learning objective, and show that accuracy cannot be the only consideration. We then propose a method for classification under competition, so that a learner can maximize market share in the presence of competitors. We show that our approach benefits the providers as well as the consumers, and find that the timing of market entry and model updates can be crucial. We display the effectiveness of our approach across a range of domains, from simple distributions to noisy datasets, and show that the market as a whole remains stable by converging quickly to an equilibrium.

Theory · Optimization

Brian Zhang, Ioannis Anagnostides, Emanuel Tewolde, Ratip Emin Berker, Gabriele Farina, Vincent Conitzer, Tuomas Sandholm

*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.

Theory · Game Theory

Ermis Soumalias, Jakob Heiss, Jakob Weissteiner, Sven Seuken

We study the design of *iterative combinatorial auctions (ICAs)*.The main challenge in this domain is that the bundle space grows exponentially in the number of items. To address this, recent work has proposed machine learning (ML)-based preference elicitation algorithms that aim to elicit only the most critical information from bidders to maximize efficiency.However, while the SOTA ML-based algorithms elicit bidders' preferences via *value queries*, ICAs that are used in practice elicit information via *demand queries*. In this paper, we introduce a novel ML algorithm that provably makes use of the full information from both value and demand queries, and we show via experiments that combining both query types results in significantly better learning performance in practice. Building on these insights, we present MLHCA, a new ML-powered auction that uses value and demand queries. MLHCA significantly outperforms the previous SOTA, reducing efficiency loss by up to a factor 10, with up to 58% fewer queries. Thus, MLHCA achieves large efficiency improvements while also reducing bidders' cognitive load, establishing a new benchmark for both practicability and efficiency. Our code is available at https://github.com/marketdesignresearch/MLHCA.

Brian Zhang, Ioannis Anagnostides, Emanuel Tewolde, Ratip Emin Berker, Gabriele Farina, Vincent Conitzer, Tuomas Sandholm

*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.

Theory · Game Theory

Yuan Deng, Amin Karbasi, Vahab Mirrokni, Renato Leme, Grigorios Velegkas, Song Zuo

We study the problem of procurement auctions, in which an auctioneer seeks to acquire services from a group of strategic sellers with private costs. The quality of the services is measured through some submodular function that is known to the auctioneer. Our goal is to design computationally efficient procurement auctions that (approximately) maximize the difference between the quality of the acquired services and the total cost of the sellers, in a way that is incentive compatible (IC) and individual rational (IR) for the sellers, and generates non-negative surplus (NAS) for the auctioneer. {Our contribution is twofold: \textbf{i)} we provide an improved analysis of existing algorithms for non-positive submodular function maximization and \textbf{ii)} we design computationally efficient frameworks that transform submodular function optimization algorithms to mechanisms that are IC and IR for the sellers, NAS for the auctioneer, and approximation-preserving.} Our frameworks are general and work both in the offline setting where the auctioneer can observe the bids and the services of all the sellers simultaneously, and in the online setting where the sellers arrive in an adversarial order and the auctioneer has to make an irrevocable decision whether to purchase their service or not. We further investigate whether it is possible to convert state-of-art submodular optimization algorithms into descending auctions. We focus on the adversarial setting, meaning that the schedule of the descending prices is determined by an adversary. We show that a submodular optimization algorithm satisfying bi-criteria $(1/2,1)$-approximation in welfare can be effectively converted to a descending auction in this setting. We further establish a connection between descending auctions and online submodular optimization. Finally, we demonstrate the practical applications of our frameworks by instantiating them with different state-of-the-art submodular optimization algorithms and comparing their welfare performance through empirical experiments on publicly available datasets that consist of thousands of sellers.

Theory · Game Theory

Yi Feng, Kaito Fujii, EFSTRATIOS PANTELEIMON SKOULAKIS, Xiao Wang, Volkan Cevher

Since Polyak's pioneering work, heavy ball (HB) momentum has been widely studied in minimization. However, its role in min-max games remains largely unexplored. As a key component of practical min-max algorithms like Adam, this gap limits their effectiveness. In this paper, we present a continuous-time analysis for HB with simultaneous and alternating update schemes in min-max games. Locally, we prove *smaller* momentum enhances algorithmic stability by enabling local convergence across a wider range of step sizes, with alternating updates generally converging faster. Globally, we study the implicit regularization of HB, and find *smaller* momentum guides algorithms trajectories towards shallower slope regions of the loss landscapes, with alternating updates amplifying this effect. Surprisingly, all these phenomena differ from those observed in minimization, where *larger* momentum yields similar effects. Our results reveal fundamental differences between HB in min-max games and minimization, and numerical experiments further validate our theoretical results.

Theory · Game Theory

Francesco Bacchiocchi, Jiarui Gan, Matteo Castiglioni, Alberto Marchesi, Nicola Gatti

Principal-agent problems model scenarios where a principal aims at incentivizing an agent to take costly, unobservable actions through the provision of payments. Such interactions are ubiquitous in several real-world applications, ranging from blockchain to the delegation of machine learning tasks. In this paper, we initiate the study of hidden-action principal-agent problems under approximate best responses, in which the agent may select any action that is not too much suboptimal given the principal's payment scheme (a.k.a. contract). Our main result is a polynomial-time algorithm to compute an optimal contract under approximate best responses. This is perhaps surprising, as computing an optimal commitment under approximate best responses is known to be computationally intractable in Stackelberg games. We also investigate the learnability of contracts under approximate best responses, by providing a no-regret learning algorithm for a natural application scenario where the principal does not know anything about the environment.

Reinforcement Learning · Multi-agent

Tin Dizdarevic, Ravi Hammond, Tobias Gessler, Anisoara Calinescu, Jonathan Cook, Matteo Gallici, Andrei Lupu, Jakob Foerster

Achieving seamless coordination between AI agents and humans is crucial for real-world applications, yet it remains a significant open challenge. Hanabi is a cooperative card game featuring imperfect information, constrained communication, theory of mind requirements, and coordinated action -- making it an ideal testbed for human-AI coordination. However, its use for human-AI interaction has been limited by the challenges of human evaluation. In this work, we introduce the Ad-Hoc Human-AI Coordination Challenge (AH2AC2) to overcome the constraints of costly and difficult-to-reproduce human evaluations. We develop \textit{human proxy agents} on a large-scale human dataset that serve as robust, cheap, and reproducible human-like evaluation partners in AH2AC2. To encourage the development of data-efficient methods, we open-source a dataset of 3,079 games, deliberately limiting the amount of available human gameplay data. We present baseline results for both two- and three- player Hanabi scenarios. To ensure fair evaluation, we host the proxy agents through a controlled evaluation system rather than releasing them publicly. The code is available at \href{https://github.com/FLAIROx/ah2ac2}{https://github.com/FLAIROx/ah2ac2}.

Theory · Online Learning and Bandits

Aymeric Capitaine, Etienne Boursier, Eric Moulines, Michael Jordan, Alain Oliviero Durmus

The framework of uncoupled online learning in multiplayer games has made significant progress in recent years. In particular, the development of time-varying games has considerably expanded its modeling capabilities. However, current regret bounds quickly become vacuous when the game undergoes significant variations over time, even when these variations are easy to predict. Intuitively, the ability of players to forecast future payoffs should lead to tighter guarantees, yet existing approaches fail to incorporate this aspect. This work aims to fill this gap by introducing a novel prediction-aware framework for time-varying games, where agents can forecast future payoffs and adapt their strategies accordingly. In this framework, payoffs depend on an underlying state of nature that agents predict in an online manner. To leverage these predictions, we propose the POMWU algorithm, a contextual extension of the optimistic Multiplicative Weight Update algorithm, for which we establish theoretical guarantees on social welfare and convergence to equilibrium. Our results demonstrate that, under bounded prediction errors, the proposed framework achieves performance comparable to the static setting. Finally, we empirically demonstrate the effectiveness of POMWU in a traffic routing experiment.

Theory · Online Learning and Bandits

Aditya Gangrade, Aldo Pacchiano, Clay Scott, Venkatesh Saligrama

We study the 'feasible action search' (FAS) problem for linear bandits, wherein a learner attempts to discover a feasible point for a set of linear constraints $\Phi_* a \ge 0,$ without knowledge of the matrix $\Phi_* \in \mathbb{R}^{m \times d}$. A FAS learner selects a sequence of actions $a_t,$ and uses observations of the form $\Phi_* a_t + \mathrm{noise}$ to either find a point with nearly optimal 'safety margin', or detect that the constraints are infeasible, where the safety margin of an action measures its (signed) distance from the constraint boundary. While of interest in its own right, the FAS problem also directly addresses a key deficiency in the extant theory of 'safe linear bandits' (SLBs), by discovering a safe initialisation for low-regret SLB methods.We propose and analyse a novel efficient FAS-learner. Our method, FAST, is based on Thompson Sampling. It applies a _coupled_ random perturbation to an estimate of $\Phi_*,$ and plays a maximin point of a game induced by this perturbed matrix. We prove that FAST stops in $\tilde{O}(d^3/\varepsilon^2 M_*^2)$ steps, and incurs $\tilde{O}(d^3/|M_*|)$ safety costs, to either correctly detect infeasibility, or output a point that is at least $(1-\varepsilon) M_*$-safe, where $M_*$ is the _optimal safety margin_ of $\Phi_*$. Further, instantiating prior SLB methods with the output of FAS yields the first SLB methods that incur $\tilde{O}(\sqrt{d^3 T/M_*^2})$ regret and $O(1)$ risk without a priori knowledge of a safe action. The main technical novelty lies in the extension of Thompson Sampling to this multiobjective setting, for which we both propose a coupled noise design, and provide an analysis that avoids convexity considerations.

Theory · Game Theory

Linjian Meng, Wubing Chen, Wenbin Li, Tianpei Yang, Youzhi Zhang, Yang Gao

Nash equilibrium (NE) plays an important role in game theory. How to efficiently compute an NE in NFGs is challenging due to its complexity and non-convex optimization property. Machine Learning (ML), the cornerstone of modern artificial intelligence, has demonstrated remarkable empirical performance across various applications including non-convex optimization. To leverage non-convex stochastic optimization techniques from ML for approximating an NE, various loss functions have been proposed. Among these, only one loss function is unbiased, allowing for unbiased estimation under the sampled play. Unfortunately, this loss function suffers from high variance, which degrades the convergence rate. To improve the convergence rate by mitigating the high variance associated with the existing unbiased loss function, we propose a novel surrogate loss function named Nash Advantage Loss (NAL). NAL is theoretically proved unbiased and exhibits significantly lower variance than the existing unbiased loss function. Experimental results demonstrate that the algorithm minimizing NAL achieves a significantly faster empirical convergence rates compared to other algorithms, while also reducing the variance of estimated loss value by several orders of magnitude.

Theory · Game Theory

Han Shao, Shuo Xie, Kunhe Yang

Strategic classification addresses a learning problem where a decision-maker implements a classifier over agents who may manipulate their features in order to receive favorable predictions. In the standard model of online strategic classification, in each round, the decision-maker implements and publicly reveals a classifier, after which agents perfectly best respond based on this knowledge. However, in practice, whether to disclose the classifier is often debated---some decision-makers believe that hiding the classifier can prevent misclassification errors caused by manipulation. In this paper, we formally examine how limiting the agents' access to the current classifier affects the decision-maker's performance. Specifically, we consider an extended online strategic classification setting where agents lack direct knowledge about the current classifier and instead manipulate based on a weighted average of historically implemented classifiers. Our main result shows that in this setting, the decision-maker incurs $(1-\gamma)^{-1}$ or $k_{\text{in}}$ times more mistakes compared to the full-knowledge setting, where $k_{\text{in}}$ is the maximum in-degree of the manipulation graph (representing how many distinct feature vectors can be manipulated to appear as a single one), and $\gamma$ is the discount factor indicating agents' memory of past classifiers. Our results demonstrate how withholding access to the classifier can backfire and degrade the decision-maker's performance in online strategic classification.

Theory · Reinforcement Learning and Planning

Tong Yang, Bo Dai, Lin Xiao, Yuejie Chi

Multi-agent reinforcement learning (MARL) lies at the heart of a plethora of applications involving the interaction of a group of agents in a shared unknown environment. A prominent framework for studying MARL is Markov games, with the goal of finding various notions of equilibria in a sample-efficient manner, such as the Nash equilibrium (NE) and the coarse correlated equilibrium (CCE). However, existing sample-efficient approaches either require tailored uncertainty estimation under function approximation, or careful coordination of the players. In this paper, we propose a novel model-based algorithm, called VMG, that incentivizes exploration via biasing the empiricalestimate of the model parameters towards those with a higher collective best-response values of all the players when fixing the other players' policies, thus encouraging the policy to deviate from its current equilibrium for more exploration. VMG is oblivious to different forms of function approximation, and permits simultaneous and uncoupled policy updates of all players. Theoretically, we also establish that VMG achieves a near-optimal regret for finding both the NEs of two-player zero-sum Markov games and CCEs of multi-player general-sum Markov games under linear function approximation in an online environment, which nearly match their counterparts with sophisticated uncertainty quantification.

Theory · Online Learning and Bandits

Junyan Liu, Lillian Ratliff

We study the repeated principal-agent bandit game, where the principal indirectly explores an unknown environment by incentivizing an agent to play arms. Unlike prior work that assumes a greedy agent with full knowledge of reward means, we consider a self-interested learning agent who iteratively updates reward estimates and may explore arbitrarily with some probability.As a warm-up, we first consider a self-interested learning agent without exploration.We propose algorithms for both i.i.d. and linear reward settings with bandit feedback in a finite horizon $T$, achieving regret bounds of $\widetilde{O}(\sqrt{T})$ and $\widetilde{O}(T^{\frac{2}{3}})$, respectively. Specifically, these algorithms rely on a novel elimination framework coupled with new search algorithms which accommodate the uncertainty from the agent's learning behavior. We then extend the framework to handle an exploratory learning agent and develop an algorithm to achieve a $\widetilde{O}(T^{\frac{2}{3}})$ regret bound in i.i.d. reward setup by enhancing the robustness of our elimination framework to the potential agent exploration. Finally, when our agent model reduces to that in (Dogan et al., 2023a), we propose an algorithm based on our robust framework, which achieves a $\widetilde{O}(\sqrt{T})$ regret bound, significantly improving upon their $\widetilde{O}(T^{\frac{11}{12}})$ bound.

Theory · Game Theory

Yizhou Zhang, Yian Ma, Eric Mazumdar

We consider the problem of learning to exploit learning algorithms through repeated interactions in games. Specifically, we focus on the case of repeated two player, finite-action games, in which an optimizer aims to steer a no-regret learner to a Stackelberg equilibrium without knowledge of its payoffs. We first show that this is impossible if the optimizer only knows that the learner is using an algorithm from the general class of no-regret algorithms. This suggests that the optimizer requires more information about the learner's objectives or algorithm to successfully exploit them. Building on this intuition, we reduce the problem for the optimizer to that of recovering the learner's payoff structure. We demonstrate the effectiveness of this approach if the learner’s algorithm is drawn from a smaller class by analyzing two examples: one where the learner uses an ascent algorithm, and another where the learner uses stochastic mirror ascent with known regularizer and step sizes.

Theory · Game Theory

Pierre-Louis Cauvin, Davide Legacci, Panayotis Mertikopoulos

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 show that randomness disrupts this behavior, causing the stochastic dynamics to drift toward the boundary on average.