Reinforcement learning is a simple, and yet, comprehensive theory of learning that simultaneously models the adaptive behavior of artificial agents, such as robots and autonomous software programs, as well as attempts to explain the emergent behavior of biological systems. It also gives rise to computational ideas that provide a powerful tool to solve problems involving sequential prediction and decision making. Temporal difference learning is the most widely used method to solve reinforcement learning problems, with a rich history dating back more than three decades. For these and many other reasons, devel-

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 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".

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.

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.

The input vocabulary and their learned representations are crucial to the performance of neural NLP models. Using the full vocabulary results in less explainable and more memory intensive models, with the embedding layer often constituting the majority of model parameters. It is thus common to use a smaller vocabulary to lower memory requirements and construct more interpertable models.We propose a vocabulary selection method that views words as members of a team trying to maximize the model's performance. We apply power indices from cooperative game theory, including the Shapley value and Banzhaf index, that measure the relative importance of individual team members in accomplishing a joint task. We approximately compute these indices to identify the most influential words.Our empirical evaluation examines multiple NLP tasks, including sentence and document classification, question answering and textual entailment. We compare to baselines that select words based on frequency, TF-IDF and regression coefficients under L1 regularization, and show that this game-theoretic vocabulary selection outperforms all baselines on a range of different tasks and datasets. * Work done during an internship at DeepMind.