Carlos Martin

Carlos Martin, Tuomas Sandholm

We study the problem of computing an approximate Nash equilibrium of continuous-action game without access to gradients. Such game access is common in reinforcement learning settings, where the environment is typically treated as a black box. To tackle this problem, we apply zeroth-order optimization techniques that combine smoothed gradient estimators with equilibrium-finding dynamics. We model players' strategies using artificial neural networks. In particular, we use randomized policy networks to model mixed strategies. These take noise in addition to an observation as input and can flexibly represent arbitrary observation-dependent, continuous-action distributions. Being able to model such mixed strategies is crucial for tackling continuous-action games that lack pure-strategy equilibria. We evaluate the performance of our method using an approximation of the Nash convergence metric from game theory, which measures how much players can benefit from unilaterally changing their strategy. We apply our method to continuous Colonel Blotto games, single-item and multi-item auctions, and a visibility game. The experiments show that our method can quickly find high-quality approximate equilibria. Furthermore, they show that the dimensionality of the input noise is crucial for performance. To our knowledge, this paper is the first to solve general continuous-action games with unrestricted mixed strategies and without any gradient information.

Carlos Martin, Tuomas Sandholm

There has been substantial progress on finding game-theoretic equilibria. Most of that work has focused on games with finite, discrete action spaces. However, many games involving space, time, money, and other fine-grained quantities have continuous action spaces (or are best modeled as having such). We study the problem of finding an approximate Nash equilibrium of games with continuous action sets. The standard measure of closeness to Nash equilibrium is exploitability, which measures how much players can benefit from unilaterally changing their strategy. We propose two new methods that minimize an approximation of exploitability with respect to the strategy profile. The first method uses a learned best-response function, which takes the current strategy profile as input and outputs candidate best responses for each player. The strategy profile and best-response functions are trained simultaneously, with the former trying to minimize exploitability while the latter tries to maximize it. The second method maintains an ensemble of candidate best responses for each player. In each iteration, the best-performing elements of each ensemble are used to update the current strategy profile. The strategy profile and ensembles are simultaneously trained to minimize and maximize the approximate exploitability, respectively. We evaluate our methods on various continuous games and GAN training, showing that they outperform prior methods.

Carlos Martin, Tuomas Sandholm

Search and planning algorithms have been a cornerstone of artificial intelligence since the field's inception. Giving reinforcement learning agents the ability to plan during execution time has resulted in significant performance improvements in various domains. However, in real-world environments, the model with respect to which the agent plans has been constrained to be grounded in the real environment itself, as opposed to a more abstract model which allows for planning over compound actions and behaviors. We propose a new method, called PiZero, that gives an agent the ability to plan in an abstract search space that the agent learns during training, which is completely decoupled from the real environment. Unlike prior approaches, this enables the agent to perform high-level planning at arbitrary timescales and reason in terms of compound or temporally-extended actions, which can be useful in environments where large numbers of base-level micro-actions are needed to perform relevant macro-actions. In addition, our method is more general than comparable prior methods because it seamlessly handles settings with continuous action spaces, combinatorial action spaces, and partial observability. We evaluate our method on multiple domains, including the traveling salesman problem, Sokoban, 2048, the facility location problem, and Pacman. Experimentally, it outperforms comparable prior methods without assuming access to an environment simulator at execution time.

Carlos Martin, Tuomas Sandholm

Planning at execution time has been shown to dramatically improve performance for agents in both single-agent and multi-agent settings. A well-known family of approaches to planning at execution time are AlphaZero and its variants, which use Monte Carlo Tree Search together with a neural network that guides the search by predicting state values and action probabilities. AlphaZero trains these networks by minimizing a planning loss that makes the value prediction match the episode return, and the policy prediction at the root of the search tree match the output of the full tree expansion. AlphaZero has been applied to both single-agent environments (such as Sokoban) and multi-agent environments (such as chess and Go) with great success. In this paper, we explore an intriguing question: In single-agent environments, can we outperform AlphaZero by directly maximizing the episode score instead of minimizing this planning loss, while leaving the MCTS algorithm and neural architecture unchanged? To directly maximize the episode score, we use evolution strategies, a family of algorithms for zeroth-order blackbox optimization. Our experiments indicate that, across multiple environments, directly maximizing the episode score outperforms minimizing the planning loss.

Carlos Martin, Tuomas Sandholm

We investigate the increasingly important and common game-solving setting where we do not have an explicit description of the game but only oracle access to it through gameplay, such as in financial or military simulations and computer games. During a limited-duration learning phase, the algorithm can control the actions of both players in order to try to learn the game and how to play it well. After that, the algorithm has to produce a strategy that has low exploitability. Our motivation is to quickly learn strategies that have low exploitability in situations where evaluating the payoffs of a queried strategy profile is costly. For the stochastic game setting, we propose using the distribution of state-action value functions induced by a belief distribution over possible environments. We compare the performance of various exploration strategies for this task, including generalizations of Thompson sampling and Bayes-UCB to this new setting. These two consistently outperform other strategies.