arXiv subjects

Ian Gemp

Publications and source records attributed to Ian Gemp.

At least 37 records · Page 2Linked to original sources

arXiv2024

Feature Likelihood Divergence: Evaluating the Generalization of Generative Models Using Samples

The past few years have seen impressive progress in the development of deep generative models capable of producing high-dimensional, complex, and photo-realistic data. However, current methods for evaluating such models remain incomplete: standard likelihood-based metrics do not always apply and rarely correlate with perceptual fidelity, while sample-based metrics, such as FID, are insensitive to overfitting, i.e., inability to generalize beyond the training set. To address these limitations, we propose a new metric called the Feature Likelihood Divergence (FLD), a parametric sample-based metric that uses density estimation to provide a comprehensive trichotomic evaluation accounting for novelty (i.e., different from the training samples), fidelity, and diversity of generated samples. We empirically demonstrate the ability of FLD to identify overfitting problem cases, even when previously proposed metrics fail. We also extensively evaluate FLD on various image datasets and model classes, demonstrating its ability to match intuitions of previous metrics like FID while offering a more comprehensive evaluation of generative models. Code is available at https://github.com/marcojira/fld.

cs.LG↗

arXiv2024

Visualizing 2x2 Normal-Form Games: twoxtwogame LaTeX Package

Normal-form games with two players, each with two strategies, are the most studied class of games. These so-called 2x2 games are used to model a variety of strategic interactions. They appear in game theory, economics, and artificial intelligence research. However, there lacks tools for describing and visualizing such games. This work introduces a LaTeX package for visualizing 2x2 games. This work has two goals: first, to provide high-quality tools and vector graphic visualizations, suitable for scientific publications. And second, to help promote standardization of names and representations of 2x2 games. The LaTeX package, twoxtwogame, is maintained on GitHub and mirrored on CTAN, and is available under a permissive Apache 2 license.

cs.GT↗

arXiv2024

NfgTransformer: Equivariant Representation Learning for Normal-form Games

Normal-form games (NFGs) are the fundamental model of strategic interaction. We study their representation using neural networks. We describe the inherent equivariance of NFGs -- any permutation of strategies describes an equivalent game -- as well as the challenges this poses for representation learning. We then propose the NfgTransformer architecture that leverages this equivariance, leading to state-of-the-art performance in a range of game-theoretic tasks including equilibrium-solving, deviation gain estimation and ranking, with a common approach to NFG representation. We show that the resulting model is interpretable and versatile, paving the way towards deep learning systems capable of game-theoretic reasoning when interacting with humans and with each other.

cs.GT↗

arXiv2024

Approximating the Core via Iterative Coalition Sampling

The core is a central solution concept in cooperative game theory, defined as the set of feasible allocations or payments such that no subset of agents has incentive to break away and form their own subgroup or coalition. However, it has long been known that the core (and approximations, such as the least-core) are hard to compute. This limits our ability to analyze cooperative games in general, and to fully embrace cooperative game theory contributions in domains such as explainable AI (XAI), where the core can complement the Shapley values to identify influential features or instances supporting predictions by black-box models. We propose novel iterative algorithms for computing variants of the core, which avoid the computational bottleneck of many other approaches; namely solving large linear programs. As such, they scale better to very large problems as we demonstrate across different classes of cooperative games, including weighted voting games, induced subgraph games, and marginal contribution networks. We also explore our algorithms in the context of XAI, providing further evidence of the power of the core for such applications.

cs.GT↗

arXiv2023

The Symmetric Generalized Eigenvalue Problem as a Nash Equilibrium

The symmetric generalized eigenvalue problem (SGEP) is a fundamental concept in numerical linear algebra. It captures the solution of many classical machine learning problems such as canonical correlation analysis, independent components analysis, partial least squares, linear discriminant analysis, principal components and others. Despite this, most general solvers are prohibitively expensive when dealing with streaming data sets (i.e., minibatches) and research has instead concentrated on finding efficient solutions to specific problem instances. In this work, we develop a game-theoretic formulation of the top-$k$ SGEP whose Nash equilibrium is the set of generalized eigenvectors. We also present a parallelizable algorithm with guaranteed asymptotic convergence to the Nash. Current state-of-the-art methods require $O(d^2k)$ runtime complexity per iteration which is prohibitively expensive when the number of dimensions ($d$) is large. We show how to modify this parallel approach to achieve $O(dk)$ runtime complexity. Empirically we demonstrate that this resulting algorithm is able to solve a variety of SGEP problem instances including a large-scale analysis of neural network activations.

cs.LG↗

arXiv2023

Equilibrium-Invariant Embedding, Metric Space, and Fundamental Set of $2\times2$ Normal-Form Games

Equilibrium solution concepts of normal-form games, such as Nash equilibria, correlated equilibria, and coarse correlated equilibria, describe the joint strategy profiles from which no player has incentive to unilaterally deviate. They are widely studied in game theory, economics, and multiagent systems. Equilibrium concepts are invariant under certain transforms of the payoffs. We define an equilibrium-inspired distance metric for the space of all normal-form games and uncover a distance-preserving equilibrium-invariant embedding. Furthermore, we propose an additional transform which defines a better-response-invariant distance metric and embedding. To demonstrate these metric spaces we study $2\times2$ games. The equilibrium-invariant embedding of $2\times2$ games has an efficient two variable parameterization (a reduction from eight), where each variable geometrically describes an angle on a unit circle. Interesting properties can be spatially inferred from the embedding, including: equilibrium support, cycles, competition, coordination, distances, best-responses, and symmetries. The best-response-invariant embedding of $2\times2$ games, after considering symmetries, rediscovers a set of 15 games, and their respective equivalence classes. We propose that this set of game classes is fundamental and captures all possible interesting strategic interactions in $2\times2$ games. We introduce a directed graph representation and name for each class. Finally, we leverage the tools developed for $2\times2$ games to develop game theoretic visualizations of large normal-form and extensive-form games that aim to fingerprint the strategic interactions that occur within.

cs.GT↗

arXiv2023

Turbocharging Solution Concepts: Solving NEs, CEs and CCEs with Neural Equilibrium Solvers

Solution concepts such as Nash Equilibria, Correlated Equilibria, and Coarse Correlated Equilibria are useful components for many multiagent machine learning algorithms. Unfortunately, solving a normal-form game could take prohibitive or non-deterministic time to converge, and could fail. We introduce the Neural Equilibrium Solver which utilizes a special equivariant neural network architecture to approximately solve the space of all games of fixed shape, buying speed and determinism. We define a flexible equilibrium selection framework, that is capable of uniquely selecting an equilibrium that minimizes relative entropy, or maximizes welfare. The network is trained without needing to generate any supervised training data. We show remarkable zero-shot generalization to larger games. We argue that such a network is a powerful component for many possible multiagent algorithms.

cs.LG↗

arXiv2023

Generative Adversarial Equilibrium Solvers

We introduce the use of generative adversarial learning to compute equilibria in general game-theoretic settings, specifically the generalized Nash equilibrium (GNE) in pseudo-games, and its specific instantiation as the competitive equilibrium (CE) in Arrow-Debreu competitive economies. Pseudo-games are a generalization of games in which players' actions affect not only the payoffs of other players but also their feasible action spaces. Although the computation of GNE and CE is intractable in the worst-case, i.e., PPAD-hard, in practice, many applications only require solutions with high accuracy in expectation over a distribution of problem instances. We introduce Generative Adversarial Equilibrium Solvers (GAES): a family of generative adversarial neural networks that can learn GNE and CE from only a sample of problem instances. We provide computational and sample complexity bounds, and apply the framework to finding Nash equilibria in normal-form games, CE in Arrow-Debreu competitive economies, and GNE in an environmental economic model of the Kyoto mechanism.

cs.GT↗

arXiv2022

AlphaSnake: Policy Iteration on a Nondeterministic NP-hard Markov Decision Process

Reinforcement learning has recently been used to approach well-known NP-hard combinatorial problems in graph theory. Among these problems, Hamiltonian cycle problems are exceptionally difficult to analyze, even when restricted to individual instances of structurally complex graphs. In this paper, we use Monte Carlo Tree Search (MCTS), the search algorithm behind many state-of-the-art reinforcement learning algorithms such as AlphaZero, to create autonomous agents that learn to play the game of Snake, a game centered on properties of Hamiltonian cycles on grid graphs. The game of Snake can be formulated as a single-player discounted Markov Decision Process (MDP) where the agent must behave optimally in a stochastic environment. Determining the optimal policy for Snake, defined as the policy that maximizes the probability of winning - or win rate - with higher priority and minimizes the expected number of time steps to win with lower priority, is conjectured to be NP-hard. Performance-wise, compared to prior work in the Snake game, our algorithm is the first to achieve a win rate over $0.5$ (a uniform random policy achieves a win rate $< 2.57 \times 10^{-15}$), demonstrating the versatility of AlphaZero in approaching NP-hard environments.

cs.AI↗

arXiv2022

Game Theoretic Rating in N-player general-sum games with Equilibria

Rating strategies in a game is an important area of research in game theory and artificial intelligence, and can be applied to any real-world competitive or cooperative setting. Traditionally, only transitive dependencies between strategies have been used to rate strategies (e.g. Elo), however recent work has expanded ratings to utilize game theoretic solutions to better rate strategies in non-transitive games. This work generalizes these ideas and proposes novel algorithms suitable for N-player, general-sum rating of strategies in normal-form games according to the payoff rating system. This enables well-established solution concepts, such as equilibria, to be leveraged to efficiently rate strategies in games with complex strategic interactions, which arise in multiagent training and real-world interactions between many agents. We empirically validate our methods on real world normal-form data (Premier League) and multiagent reinforcement learning agent evaluation.

cs.GT↗

arXiv2022

Developing, Evaluating and Scaling Learning Agents in Multi-Agent Environments

The Game Theory & Multi-Agent team at DeepMind studies several aspects of multi-agent learning ranging from computing approximations to fundamental concepts in game theory to simulating social dilemmas in rich spatial environments and training 3-d humanoids in difficult team coordination tasks. A signature aim of our group is to use the resources and expertise made available to us at DeepMind in deep reinforcement learning to explore multi-agent systems in complex environments and use these benchmarks to advance our understanding. Here, we summarise the recent work of our team and present a taxonomy that we feel highlights many important open challenges in multi-agent research.

cs.MA↗

arXiv2022

Stochastic Parallelizable Eigengap Dilation for Large Graph Clustering

Large graphs commonly appear in social networks, knowledge graphs, recommender systems, life sciences, and decision making problems. Summarizing large graphs by their high level properties is helpful in solving problems in these settings. In spectral clustering, we aim to identify clusters of nodes where most edges fall within clusters and only few edges fall between clusters. This task is important for many downstream applications and exploratory analysis. A core step of spectral clustering is performing an eigendecomposition of the corresponding graph Laplacian matrix (or equivalently, a singular value decomposition, SVD, of the incidence matrix). The convergence of iterative singular value decomposition approaches depends on the eigengaps of the spectrum of the given matrix, i.e., the difference between consecutive eigenvalues. For a graph Laplacian corresponding to a well-clustered graph, the eigenvalues will be non-negative but very small (much less than $1$) slowing convergence. This paper introduces a parallelizable approach to dilating the spectrum in order to accelerate SVD solvers and in turn, spectral clustering. This is accomplished via polynomial approximations to matrix operations that favorably transform the spectrum of a matrix without changing its eigenvectors. Experiments demonstrate that this approach significantly accelerates convergence, and we explain how this transformation can be parallelized and stochastically approximated to scale with available compute.

stat.ML↗

arXiv2022

EigenGame Unloaded: When playing games is better than optimizing

We build on the recently proposed EigenGame that views eigendecomposition as a competitive game. EigenGame's updates are biased if computed using minibatches of data, which hinders convergence and more sophisticated parallelism in the stochastic setting. In this work, we propose an unbiased stochastic update that is asymptotically equivalent to EigenGame, enjoys greater parallelism allowing computation on datasets of larger sample sizes, and outperforms EigenGame in experiments. We present applications to finding the principal components of massive datasets and performing spectral clustering of graphs. We analyze and discuss our proposed update in the context of EigenGame and the shift in perspective from optimization to games.

stat.ML↗

arXiv2022

D3C: Reducing the Price of Anarchy in Multi-Agent Learning

In multiagent systems, the complex interaction of fixed incentives can lead agents to outcomes that are poor (inefficient) not only for the group, but also for each individual. Price of anarchy is a technical, game-theoretic definition that quantifies the inefficiency arising in these scenarios -- it compares the welfare that can be achieved through perfect coordination against that achieved by self-interested agents at a Nash equilibrium. We derive a differentiable, upper bound on a price of anarchy that agents can cheaply estimate during learning. Equipped with this estimator, agents can adjust their incentives in a way that improves the efficiency incurred at a Nash equilibrium. Agents do so by learning to mix their reward (equiv. negative loss) with that of other agents by following the gradient of our derived upper bound. We refer to this approach as D3C. In the case where agent incentives are differentiable, D3C resembles the celebrated Win-Stay, Lose-Shift strategy from behavioral game theory, thereby establishing a connection between the global goal of maximum welfare and an established agent-centric learning rule. In the non-differentiable setting, as is common in multiagent reinforcement learning, we show the upper bound can be reduced via evolutionary strategies, until a compromise is reached in a distributed fashion. We demonstrate that D3C improves outcomes for each agent and the group as a whole on several social dilemmas including a traffic network exhibiting Braess's paradox, a prisoner's dilemma, and several multiagent domains.

cs.MA↗

arXiv2022

Sample-based Approximation of Nash in Large Many-Player Games via Gradient Descent

Nash equilibrium is a central concept in game theory. Several Nash solvers exist, yet none scale to normal-form games with many actions and many players, especially those with payoff tensors too big to be stored in memory. In this work, we propose an approach that iteratively improves an approximation to a Nash equilibrium through joint play. It accomplishes this by tracing a previously established homotopy that defines a continuum of equilibria for the game regularized with decaying levels of entropy. This continuum asymptotically approaches the limiting logit equilibrium, proven by McKelvey and Palfrey (1995) to be unique in almost all games, thereby partially circumventing the well-known equilibrium selection problem of many-player games. To encourage iterates to remain near this path, we efficiently minimize average deviation incentive via stochastic gradient descent, intelligently sampling entries in the payoff tensor as needed. Monte Carlo estimates of the stochastic gradient from joint play are biased due to the appearance of a nonlinear max operator in the objective, so we introduce additional innovations to the algorithm to alleviate gradient bias. The descent process can also be viewed as repeatedly constructing and reacting to a polymatrix approximation to the game. In these ways, our proposed approach, average deviation incentive descent with adaptive sampling (ADIDAS), is most similar to three classical approaches, namely homotopy-type, Lyapunov, and iterative polymatrix solvers. The lack of local convergence guarantees for biased gradient descent prevents guaranteed convergence to Nash, however, we demonstrate through extensive experiments the ability of this approach to approximate a unique Nash in normal-form games with as many as seven players and twenty one actions (several billion outcomes) that are orders of magnitude larger than those possible with prior algorithms.

cs.GT↗

arXiv2022

Learning to Play No-Press Diplomacy with Best Response Policy Iteration

Recent advances in deep reinforcement learning (RL) have led to considerable progress in many 2-player zero-sum games, such as Go, Poker and Starcraft. The purely adversarial nature of such games allows for conceptually simple and principled application of RL methods. However real-world settings are many-agent, and agent interactions are complex mixtures of common-interest and competitive aspects. We consider Diplomacy, a 7-player board game designed to accentuate dilemmas resulting from many-agent interactions. It also features a large combinatorial action space and simultaneous moves, which are challenging for RL algorithms. We propose a simple yet effective approximate best response operator, designed to handle large combinatorial action spaces and simultaneous moves. We also introduce a family of policy iteration methods that approximate fictitious play. With these methods, we successfully apply RL to Diplomacy: we show that our agents convincingly outperform the previous state-of-the-art, and game theoretic equilibrium analysis shows that the new process yields consistent improvements.

cs.LG↗

arXiv2021

EigenGame: PCA as a Nash Equilibrium

We present a novel view on principal component analysis (PCA) as a competitive game in which each approximate eigenvector is controlled by a player whose goal is to maximize their own utility function. We analyze the properties of this PCA game and the behavior of its gradient based updates. The resulting algorithm -- which combines elements from Oja's rule with a generalized Gram-Schmidt orthogonalization -- is naturally decentralized and hence parallelizable through message passing. We demonstrate the scalability of the algorithm with experiments on large image datasets and neural network activations. We discuss how this new view of PCA as a differentiable game can lead to further algorithmic developments and insights.

cs.LG↗

arXiv2020

Proximal Gradient Temporal Difference Learning: Stable Reinforcement Learning with Polynomial Sample Complexity

In this paper, we introduce proximal gradient temporal difference learning, which provides a principled way of designing and analyzing true stochastic gradient temporal difference learning algorithms. We show how gradient TD (GTD) reinforcement learning methods can be formally derived, not by starting from their original objective functions, as previously attempted, but rather from a primal-dual saddle-point objective function. We also conduct a saddle-point error analysis to obtain finite-sample bounds on their performance. Previous analyses of this class of algorithms use stochastic approximation techniques to prove asymptotic convergence, and do not provide any finite-sample analysis. We also propose an accelerated algorithm, called GTD2-MP, that uses proximal ``mirror maps'' to yield an improved convergence rate. The results of our theoretical analysis imply that the GTD family of algorithms are comparable and may indeed be preferred over existing least squares TD methods for off-policy learning, due to their linear complexity. We provide experimental results showing the improved performance of our accelerated gradient TD methods.

cs.LG↗