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.

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.

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.

Social diversity and social preferences in mixed-motive reinforcement learning

Recent research on reinforcement learning in pure-conflict and pure-common interest games has emphasized the importance of population heterogeneity. In contrast, studies of reinforcement learning in mixed-motive games have primarily leveraged homogeneous approaches. Given the defining characteristic of mixed-motive games--the imperfect correlation of incentives between group members--we study the effect of population heterogeneity on mixed-motive reinforcement learning. We draw on interdependence theory from social psychology and imbue reinforcement learning agents with Social Value Orientation (SVO), a flexible formalization of preferences over group outcome distributions. We subsequently explore the effects of diversity in SVO on populations of reinforcement learning agents in two mixed-motive Markov games. We demonstrate that heterogeneity in SVO generates meaningful and complex behavioral variation among agents similar to that suggested by interdependence theory. Empirical results in these mixed-motive dilemmas suggest agents trained in heterogeneous populations develop particularly generalized, high-performing policies relative to those trained in homogeneous populations.

Social Diversity and Social Preferences in Mixed-Motive Reinforcement Learning

Recent research on reinforcement learning in pure-conflict and pure-common interest games has emphasized the importance of population heterogeneity. In contrast, studies of reinforcement learning in mixed-motive games have primarily leveraged homogeneous approaches. Given the defining characteristic of mixed-motive games--the imperfect correlation of incentives between group members--we study the effect of population heterogeneity on mixed-motive reinforcement learning. We draw on interdependence theory from social psychology and imbue reinforcement learning agents with Social Value Orientation (SVO), a flexible formalization of preferences over group outcome distributions. We subsequently explore the effects of diversity in SVO on populations of reinforcement learning agents in two mixed-motive Markov games. We demonstrate that heterogeneity in SVO generates meaningful and complex behavioral variation among agents similar to that suggested by interdependence theory. Empirical results in these mixed-motive dilemmas suggest agents trained in heterogeneous populations develop particularly generalized, high-performing policies relative to those trained in homogeneous populations.

Global Convergence to the Equilibrium of GANs using Variational Inequalities

In optimization, the negative gradient of a function denotes the direction of steepest descent. Furthermore, traveling in any direction orthogonal to the gradient maintains the value of the function. In this work, we show that these orthogonal directions that are ignored by gradient descent can be critical in equilibrium problems. Equilibrium problems have drawn heightened attention in machine learning due to the emergence of the Generative Adversarial Network (GAN). We use the framework of Variational Inequalities to analyze popular training algorithms for a fundamental GAN variant: the Wasserstein Linear-Quadratic GAN. We show that the steepest descent direction causes divergence from the equilibrium, and guaranteed convergence to the equilibrium is achieved through following a particular orthogonal direction. We call this successful technique Crossing-the-Curl, named for its mathematical derivation as well as its intuition: identify the game's axis of rotation and move "across" space in the direction towards smaller "curling".

Online Monotone Games

Algorithmic game theory (AGT) focuses on the design and analysis of algorithms for interacting agents, with interactions rigorously formalized within the framework of games. Results from AGT find applications in domains such as online bidding auctions for web advertisements and network routing protocols. Monotone games are games where agent strategies naturally converge to an equilibrium state. Previous results in AGT have been obtained for convex, socially-convex, or smooth games, but not monotone games. Our primary theoretical contributions are defining the monotone game setting and its extension to the online setting, a new notion of regret for this setting, and accompanying algorithms that achieve sub-linear regret. We demonstrate the utility of online monotone game theory on a variety of problem domains including variational inequalities, reinforcement learning, and generative adversarial networks.

Inverting Variational Autoencoders for Improved Generative Accuracy

Recent advances in semi-supervised learning with deep generative models have shown promise in generalizing from small labeled datasets ($\mathbf{x},\mathbf{y}$) to large unlabeled ones ($\mathbf{x}$). In the case where the codomain has known structure, a large unfeatured dataset ($\mathbf{y}$) is potentially available. We develop a parameter-efficient, deep semi-supervised generative model for the purpose of exploiting this untapped data source. Empirical results show improved performance in disentangling latent variable semantics as well as improved discriminative prediction on Martian spectroscopic and handwritten digit domains.

Generative Multi-Adversarial Networks

Generative adversarial networks (GANs) are a framework for producing a generative model by way of a two-player minimax game. In this paper, we propose the \emph{Generative Multi-Adversarial Network} (GMAN), a framework that extends GANs to multiple discriminators. In previous work, the successful training of GANs requires modifying the minimax objective to accelerate training early on. In contrast, GMAN can be reliably trained with the original, untampered objective. We explore a number of design perspectives with the discriminator role ranging from formidable adversary to forgiving teacher. Image generation tasks comparing the proposed framework to standard GANs demonstrate GMAN produces higher quality samples in a fraction of the iterations when measured by a pairwise GAM-type metric.