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Generative Models · Reasoning

Gabriel Poesia, David Broman, Nick Haber, Noah Goodman

How did humanity coax mathematics from the aether? We explore the Platonic view that mathematics can be discovered from its axioms---a game of conjecture and proof. We describe an agent that jointly learns to pose challenging problems for itself (conjecturing) and solve them (theorem proving). Given a mathematical domain axiomatized in dependent type theory, we first combine methods for constrained decoding and type-directed synthesis to sample valid conjectures from a language model. Our method guarantees well-formed conjectures by construction, even as we start with a randomly initialized model. We use the same model to represent a policy and value function for guiding proof search. Our agent targets generating hard but provable conjectures --- a moving target, since its own theorem proving ability also improves as it trains. We propose novel methods for hindsight relabeling on proof search trees to significantly improve the agent's sample efficiency in both tasks. Experiments on 3 axiomatic domains (propositional logic, arithmetic and group theory) demonstrate that our agent can bootstrap from only the axioms, self-improving in generating true and challenging conjectures and in finding proofs.

Deep Learning · Large Language Models

Federico Bianchi, Patrick John Chia, Mert Yuksekgonul, Jacopo Tagliabue, Dan Jurafsky, James Zou

Negotiation is the basis of social interactions; humans negotiate everything from the price of cars to how to share common resources. With rapidly growing interest in using large language models (LLMs) to act as agents on behalf of human users, such LLM agents would also need to be able to negotiate. In this paper, we study how well LLMs can negotiate with each other. We develop NegotiationArena: a flexible framework for evaluating and probing the negotiation abilities of LLM agents. We implemented three types of scenarios in NegotiationArena to assess LLM's behaviors in allocating shared resources (ultimatum games), aggregate resources (trading games) and buy/sell goods (price negotiations). Each scenario allows for multiple turns of flexible dialogues between LLM agents to allow for more complex negotiations. Interestingly, LLM agents can significantly boost their negotiation outcomes by employing certain behavioral tactics. For example, by pretending to be desolate and desperate, LLMs can improve their payoffs by 20% when negotiating against the standard GPT-4. We also quantify irrational negotiation behaviors exhibited by the LLM agents, many of which also appear in humans. Together, NegotiationArena offers a new environment to investigate LLM interactions, enabling new insights into LLM's theory of mind, irrationality, and reasoning abilities

Theory · Game Theory

Davide Legacci, Panayotis Mertikopoulos, Bary Pradelski

In view of the complexity of the dynamics of learning in games, we seek to decompose a game into simpler components where the dynamics' long-run behavior is well understood. A natural starting point for this is Helmholtz's theorem, which decomposes a vector field into a potential and an incompressible component. However, the geometry of game dynamics - and, in particular, the dynamics of exponential / multiplicative weights (EW) schemes - is not compatible with the Euclidean underpinnings of Helmholtz's theorem. This leads us to consider a specific Riemannian framework based on the so-called *Shahshahani metric*, and introduce the class of *incompressible games*, for which we establish the following results: First, in addition to being volume-preserving, the continuous-time EW dynamics in incompressible games admit a constant of motion and are *Poincaré recurrent* - i.e., almost every trajectory of play comes arbitrarily close to its starting point infinitely often. Second, we establish a deep connection with a well-known decomposition of games into a potential and harmonic component (where the players' objectives are aligned and anti-aligned respectively): a game is incompressible if and only if it is harmonic, implying in turn that the EW dynamics lead to Poincaré recurrence in harmonic games.

Theory · Optimization

Ana-Andreea Stoica, Vivian Y. Nastl, Moritz Hardt

Many applications of RCTs involve the presence of multiple treatment administrators---from field experiments to online advertising---that compete for the subjects' attention. In the face of competition, estimating a causal effect becomes difficult, as the position at which a subject sees a treatment influences their response, and thus the treatment effect. In this paper, we build a game-theoretic model of agents who wish to estimate causal effects in the presence of competition, through a bidding system and a utility function that minimizes estimation error. Our main technical result establishes an approximation with a tractable objective that maximizes the sample value obtained through strategically allocating budget on subjects. This allows us to find an equilibrium in our model: we show that the tractable objective has a pure Nash equilibrium, and that any Nash equilibrium is an approximate equilibrium for our general objective that minimizes estimation error under broad conditions. Conceptually, our work successfully combines elements from causal inference and game theory to shed light on the equilibrium behavior of experimentation under competition.

Theory · Game Theory

Fabian Fumagalli, Maximilian Muschalik, Patrick Kolpaczki, Eyke Hüllermeier, CITEC Barbara Hammer

The Shapley value (SV) is a prevalent approach of allocating credit to machine learning (ML) entities to understand black box ML models. Enriching such interpretations with higher-order interactions is inevitable for complex systems, where the Shapley Interaction Index (SII) is a direct axiomatic extension of the SV. While it is well-known that the SV yields an optimal approximation of any game via a weighted least square (WLS) objective, an extension of this result to SII has been a long-standing open problem, which even led to the proposal of an alternative index. In this work, we characterize higher-order SII as a solution to a WLS problem, which constructs an optimal approximation via SII and k-Shapley values (k-SII). We prove this representation for the SV and pairwise SII and give empirically validated conjectures for higher orders. As a result, we propose KernelSHAP-IQ, a direct extension of KernelSHAP for SII, and demonstrate state-of-the-art performance for feature interactions.

Theory · Online Learning and Bandits

Houshuang Chen, Yuchen He, Chihao Zhang

Learning with expert advice and multi-armed bandit are two classic online decision problems which differ on how the information is observed in each round of the game. We study a family of problems interpolating the two. For a vector $\mathbf{m}=(m_1,\dots,m_K)\in \mathbb N^K$, an instance of $\mathbf m$-MAB indicates that the arms are partitioned into $K$ groups and the $i$-th group contains $m_i$ arms. Once an arm is pulled, the losses of all arms in the same group are observed. We prove tight minimax regret bounds for $\mathbf m$-MAB and design an optimal PAC algorithm for its pure exploration version, $\mathbf m$-BAI, where the goal is to identify the arm with minimum loss with as few rounds as possible. We show that the minimax regret of $\mathbf m$-MAB is $\Theta\left(\sqrt{T\sum_{k=1}^K\log (m_k+1)}\right)$ and the minimum number of pulls for an $(\varepsilon,0.05)$-PAC algorithm of $\mathbf m$-BAI is $\Theta\left(\frac{1}{\varepsilon^2}\cdot \sum_{k=1}^K\log (m_k+1)\right)$. Both our upper bounds and lower bounds for $\mathbf m$-MAB can be extended to a more general setting, namely the bandit with graph feedback, in terms of the *clique cover* and related graph parameters. As consequences, we obtained tight minimax regret bounds for several families of feedback graphs.

Theory · Game Theory

Fuzhong Zhou, Chenyu Zhang, Xu Chen, Xuan Di

We propose a discrete time graphon game formulation on continuous state and action spaces using a representative player to study stochastic games with heterogeneous interaction among agents. This formulation admits both conceptual and mathematical advantages, compared to a widely adopted formulation using a continuum of players. We prove the existence and uniqueness of the graphon equilibrium with mild assumptions, and show that this equilibrium can be used to construct an approximate solution for the finite player game, which is challenging to analyze and solve due to curse of dimensionality. An online oracle-free learning algorithm is developed to solve the equilibrium numerically, and sample complexity analysis is provided for its convergence.

Theory · Reinforcement Learning and Planning

Weiye Zhao, Feihan Li, Yifan Sun, Rui Chen, Tianhao Wei, Changliu Liu

In recent years, trust region on-policy reinforcement learning has achieved impressive results in addressing complex control tasks and gaming scenarios. However, contemporary state-of-the-art algorithms within this category primarily emphasize improvement in expected performance, lacking the ability to control over the worst-case performance outcomes. To address this limitation, we introduce a novel objective function, optimizing which leads to guaranteed monotonic improvement in the lower probability bound of performance with high confidence. Building upon this groundbreaking theoretical advancement, we further introduce a practical solution called Absolute Policy Optimization (APO). Our experiments demonstrate the effectiveness of our approach across challenging continuous control benchmark tasks and extend its applicability to mastering Atari games. Our findings reveal that APO as well as its efficient variation Proximal Absolute Policy Optimization (PAPO) significantly outperforms state-of-the-art policy gradient algorithms, resulting in substantial improvements in worst-case performance, as well as expected performance.

Theory · Game Theory

Youzhi Zhang, Bo An, Daniel Zeng

Many works recently have focused on computing optimal solutions for the ex ante coordination of a team for solving sequential adversarial team games, where a team of players coordinate against an opponent (or a team of players) in a zero-sum extensive-form game. However, it is challenging to directly compute such an optimal solution because the team’s coordinated strategy space is exponential in the size of the game tree due to the asymmetric information of team members. Column Generation (CG) algorithms have been proposed to overcome this challenge by iteratively expanding the team’s coordinated strategy space via a Best Response Oracle (BRO). More recently, more compact representations (particularly, the Team Belief Directed Acyclic Graph (TB-DAG)) of the team’s coordinated strategy space have been proposed, but the TB-DAG-based algorithms only outperform the CG-based algorithms in games with a small TB-DAG. Unfortunately, it is inefficient to directly apply CG to the TB-DAG because the size of the TB-DAG is still exponential in the size of the game tree and then makes the BRO unscalable. To this end, we develop our novel TB-DAG CG (DCG) algorithm framework by computing a coordinated best response in the original game first and then transforming this strategy into the TB-DAG form. To further improve the scalability, we propose a more suitable BRO for DCG to reduce the cost of the transformation at each iteration. We theoretically show that our algorithm converges exponentially faster than the state-of-the-art CG algorithms, and experimental results show that our algorithm is at least two orders of magnitude faster than the state-of-the-art baselines.

Theory · Game Theory

Young Wu, Jeremy McMahan, Yiding Chen, Yudong Chen, Jerry Zhu, Qiaomin Xie

We study the game modification problem, where a benevolent game designer or a malevolent adversary modifies the reward function of a zero-sum Markov game so that a target deterministic or stochastic policy profile becomes the unique Markov perfect Nash equilibrium and has a value within a target range, in a way that minimizes the modification cost. We characterize the set of policy profiles that can be installed as the unique equilibrium of a game and establish sufficient and necessary conditions for successful installation. We propose an efficient algorithm that solves a convex optimization problem with linear constraints and then performs random perturbation to obtain a modification plan with a near-optimal cost.

Reinforcement Learning · Multi-agent

Jianhong Wang, Yang Li, Yuan Zhang, Wei Pan, Samuel Kaski

Ad hoc teamwork poses a challenging problem, requiring the design of an agent to collaborate with teammates without prior coordination or joint training. Open ad hoc teamwork (OAHT) further complicates this challenge by considering environments with a changing number of teammates, referred to as open teams. One promising solution in practice to this problem is leveraging the generalizability of graph neural networks to handle an unrestricted number of agents with various agent-types, named graph-based policy learning (GPL). However, its joint Q-value representation over a coordination graph lacks convincing explanations. In this paper, we establish a new theory to understand the representation of the joint Q-value for OAHT and its learning paradigm, through the lens of cooperative game theory. Building on our theory, we propose a novel algorithm named CIAO, based on GPL's framework, with additional provable implementation tricks that can facilitate learning. The demos of experimental results are available on https://sites.google.com/view/ciao2024, and the code of experiments is published on https://github.com/hsvgbkhgbv/CIAO.

Theory · Online Learning and Bandits

Taira Tsuchiya, Shinji Ito, Junya Honda

Partial monitoring is a generic framework of online decision-making problems with limited feedback. To make decisions from such limited feedback, it is necessary to find an appropriate distribution for exploration. Recently, a powerful approach for this purpose, exploration by optimization (ExO), was proposed, which achieves optimal bounds in adversarial environments with follow-the-regularized-leader for a wide range of online decision-making problems. However, a naive application of ExO in stochastic environments significantly degrades regret bounds. To resolve this issue in locally observable games, we first establish a new framework and analysis for ExO with a hybrid regularizer. This development allows us to significantly improve existing regret bounds of best-of-both-worlds (BOBW) algorithms, which achieves nearly optimal bounds both in stochastic and adversarial environments. In particular, we derive a stochastic regret bound of $O(\sum_{a \neq a^*} k^2 m^2 \log T / \Delta_a)$, where $k$, $m$, and $T$ are the numbers of actions, observations and rounds, $a^*$ is an optimal action, and $\Delta_a$ is the suboptimality gap for action $a$. This bound is roughly $\Theta(k^2 \log T)$ times smaller than existing BOBW bounds. In addition, for globally observable games, we provide a new BOBW algorithm with the first $O(\log T)$ stochastic bound.

Theory · Online Learning and Bandits

Jon Schneider, Kiran Vodrahalli

We study the problem of full-information online learning in the ``bounded recall'' setting popular in the study of repeated games. An online learning algorithm $\mathcal{A}$ is $M$-*bounded-recall* if its output at time $t$ can be written as a function of the $M$ previous rewards (and not e.g. any other internal state of $\mathcal{A}$). We first demonstrate that a natural approach to constructing bounded-recall algorithms from mean-based no-regret learning algorithms (e.g., running Hedge over the last $M$ rounds) fails, and that any such algorithm incurs constant regret per round. We then construct a stationary bounded-recall algorithm that achieves a per-round regret of $\Theta(1/\sqrt{M})$, which we complement with a tight lower bound. Finally, we show that unlike the perfect recall setting, any low regret bound bounded-recall algorithm must be aware of the ordering of the past $M$ losses -- any bounded-recall algorithm which plays a symmetric function of the past $M$ losses must incur constant regret per round.

Theory · Online Learning and Bandits

Antoine Scheid, Daniil Tiapkin, Etienne Boursier, Aymeric Capitaine, Eric Moulines, Michael Jordan, El-Mahdi El-Mhamdi, Alain Oliviero Durmus

This work considers a repeated principal-agent bandit game, where the principal can only interact with her environment through the agent. The principal and the agent have misaligned objectives and the choice of action is only left to the agent. However, the principal can influence the agent's decisions by offering incentives which add up to his rewards. The principal aims to iteratively learn an incentive policy to maximize her own total utility. This framework extends usual bandit problems and is motivated by several practical applications, such as healthcare or ecological taxation, where traditionally used mechanism design theories often overlook the learning aspect of the problem. We present nearly optimal (with respect to a horizon $T$) learning algorithms for the principal's regret in both multi-armed and linear contextual settings. Finally, we support our theoretical guarantees through numerical experiments.

Reinforcement Learning · Multi-agent

Johan Peralez, Aurélien Delage, Olivier Buffet, Jilles Dibangoye

A recent theory shows that a multi-player decentralized partially observable Markov decision process can be transformed into an equivalent single-player game, enabling the application of Bellman's principle of optimality to solve the single-player game by breaking it down into single-stage subgames. However, this approach entangles the decision variables of all players at each single-stage subgame, resulting in backups with a double-exponential complexity. This paper demonstrates how to disentangle these decision variables while maintaining optimality under hierarchical information sharing, a prominent management style in our society. To achieve this, we apply the principle of optimality to solve any single-stage subgame by breaking it down further into smaller subgames, enabling us to make single-player decisions at a time. Our approach reveals that extensive-form games always exist with solutions to a single-stage subgame, significantly reducing time complexity. Our experimental results show that the algorithms leveraging these findings can scale up to much larger multi-player games without compromising optimality.

Theory · Game Theory

Boning Li, Zhixuan Fang, Longbo Huang

Effective action abstraction is crucial in tackling challenges associated with large action spaces in Imperfect Information Extensive-Form Games (IIEFGs). However, due to the vast state space and computational complexity in IIEFGs, existing methods often rely on fixed abstractions, resulting in sub-optimal performance. In response, we introduce RL-CFR, a novel reinforcement learning (RL) approach for dynamic action abstraction. RL-CFR builds upon our innovative Markov Decision Process (MDP) formulation, with states corresponding to public information and actions represented as feature vectors indicating specific action abstractions. The reward is defined as the expected payoff difference between the selected and default action abstractions. RL-CFR constructs a game tree with RL-guided action abstractions and utilizes counterfactual regret minimization (CFR) for strategy derivation. Impressively, it can be trained from scratch, achieving higher expected payoff without increased CFR solving time. In experiments on Heads-up No-limit Texas Hold'em, RL-CFR outperforms ReBeL's replication and Slumbot, demonstrating significant win-rate margins of $64\pm 11$ and $84\pm 17$ mbb/hand, respectively.

Theory · Reinforcement Learning and Planning

Jiawei Huang, Niao He, Andreas Krause

We study the sample complexity of reinforcement learning (RL) in Mean-Field Games (MFGs) with model-based function approximation that requires strategic exploration to find a Nash Equilibrium policy. We introduce the Partial Model-Based Eluder Dimension (P-MBED), a more effective notion to characterize the model class complexity. Notably, P-MBED measures the complexity of the single-agent model class converted from the given mean-field model class, and potentially, can be exponentially lower than the MBED proposed by Huang et al. (2024). We contribute a model elimination algorithm featuring a novel exploration strategy and establish sample complexity results polynomial w.r.t. P-MBED. Crucially, our results reveal that, under the basic realizability and Lipschitz continuity assumptions, *learning Nash Equilibrium in MFGs is no more statistically challenging than solving a logarithmic number of single-agent RL problems*. We further extend our results to Multi-Type MFGs, generalizing from conventional MFGs and involving multiple types of agents. This extension implies statistical tractability of a broader class of Markov Games through the efficacy of mean-field approximation. Finally, inspired by our theoretical algorithm, we present a heuristic approach with improved computational efficiency and empirically demonstrate its effectiveness.

Theory · Game Theory

Safwan Hossain, Tonghan Wang, Tao Lin, Yiling Chen, David Parkes, Haifeng Xu

We consider *multiple senders* with informational advantage signaling to convince a single self-interested actor to take certain actions. Generalizing the seminal *Bayesian Persuasion* framework, such settings are ubiquitous in computational economics, multi-agent learning, and machine learning with multiple objectives. The core solution concept here is the Nash equilibrium of senders' signaling policies. Theoretically, we prove that finding an equilibrium in general is PPAD-Hard; in fact, even computing a sender's best response is NP-Hard. Given these intrinsic difficulties, we turn to finding local Nash equilibria. We propose a novel differentiable neural network to approximate this game's non-linear and discontinuous utilities. Complementing this with the extra-gradient algorithm, we discover local equilibria that Pareto dominates full-revelation equilibria and those found by existing neural networks. Broadly, our theoretical and empirical contributions are of interest to a large class of economic problems.

Theory · Game Theory

Safwan Hossain, Yiling Chen

We model real-world data markets, where sellers post fixed prices and buyers are free to purchase from any set of sellers, as a simultaneous game. A key component here is the negative externality buyers induce on one another due to data purchases. Starting with a simple setting where buyers know their valuations a priori, we characterize both the existence and welfare properties of the pure Nash equilibrium in the presence of such externality. While the outcomes are bleak without any intervention, mirroring the limitations of current data markets, we prove that for a standard class of externality functions, platforms intervening through a transaction cost can lead to a pure equilibrium with strong welfare guarantees. We next consider a more realistic setting where buyers learn their valuations over time through market interactions. Our intervention is feasible here as well, and we consider learning algorithms to achieve low regret concerning both individual and cumulative utility metrics. Lastly, we analyze the promises of this intervention under a much richer externality model.

Theory · Online Learning and Bandits

Maxime Heuillet, Ola Ahmad, Audrey Durand

The partial monitoring (PM) framework provides a theoretical formulation of sequential learning problems with incomplete feedback. At each round, a learning agent plays an action while the environment simultaneously chooses an outcome. The agent then observes a feedback signal that is only partially informative about the (unobserved) outcome. The agent leverages the received feedback signals to select actions that minimize the (unobserved) cumulative loss. In contextual PM, the outcomes depend on some side information that is observable by the agent before selecting the action. In this paper, we consider the contextual and non-contextual PM settings with stochastic outcomes. We introduce a new class of PM strategies based on the randomization of deterministic confidence bounds. We also extend regret guarantees to settings where existing stochastic strategies are not applicable. Our experiments show that the proposed RandCBP and RandCBPside* strategies have competitive performance against state-of-the-art baselines in multiple PM games. To illustrate how the PM framework can benefit real world applications, we design a use case on the real-world problem of monitoring the error rate of any deployed classification system.