Consider a two-player zero-sum stochastic game where the transition function can be embedded in a given feature space. We propose a two-player Q-learning algorithm for approximating the Nash equilibrium strategy via sampling. The algorithm is shown to find an @math -optimal strategy using sample size linear to the number of features. To further improve its sample efficiency, we develop an accelerated algorithm by adopting techniques such as variance reduction, monotonicity preservation and two-sided strategy approximation. We prove that the algorithm is guaranteed to find an @math -optimal strategy using no more than @math samples with high probability, where @math is the number of features and @math is a discount factor. The sample, time and space complexities of the algorithm are independent of original dimensions of the game.
As for general stochastic games, the minimax Q-learning algorithm and the friend-and-foe Q-learning algorithm is introduced in @cite_37 and @cite_15 , respectively. The Nash Q-learning algorithm is proposed for zero-sum games in @cite_4 and for general-sum games in @cite_40 @cite_23 . Also in @cite_21 , the error of approximate Q-learning is estimated. In @cite_9 , finite-sample analysis of multi-agent reinforcement learning is provided. To our best knowledge, there is no known algorithm that solves 2-TBSG using features with sample complexity analysis.
@cite_37 @cite_4 @cite_9 @cite_21 @cite_40 @cite_23 @cite_15
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The single-agent multi-armed bandit problem can be solved by an agent that learns the values of each action using reinforcement learning. However, the multi-agent version of the problem, the iterated normal form game, presents a more complex challenge, since the rewards available to each agent depend on the strategies of the others. We consider the behavior of value-based learning agents in this situation, and show that such agents cannot generally play at a Nash equilibrium, although if smooth best responses are used, a Nash distribution can be reached. We introduce a particular value-based learning algorithm, which we call individual Q-learning, and use stochastic approximation to study the asymptotic behavior, showing that strategies will converge to Nash distribution almost surely in 2-player zero-sum games and 2-player partnership games. Player-dependent learning rates are then considered, and it is shown that this extension converges in some games for which many algorithms, including the basic algorithm initially considered, fail to converge. This paper provides an analysis of error propagation in Approximate Dynamic Programming applied to zero-sum two-player Stochastic Games. We provide a novel and unified error propagation analysis in Lp-norm of three well-known algorithms adapted to Stochastic Games (namely Approximate Value Iteration, Approximate Policy Iteration and Approximate Generalized Policy Iteration). We show that we can achieve a stationary policy which is 2γe+e′ (1-γ)2 -optimal, where e is the value function approximation error and e′ is the approximate greedy operator error. In addition, we provide a practical algorithm (AGPI-Q) to solve infinite horizon γ-discounted two-player zero-sum Stochastic Games in a batch setting. It is an extension of the Fitted-Q algorithm (which solves Markov Decisions Processes from data) and can be non-parametric. Finally, we demonstrate experimentally the performance of AGPI-Q on a simultaneous two-player game, namely Alesia. We extend Q-learning to a noncooperative multiagent context, using the framework of general-sum stochastic games. A learning agent maintains Q-functions over joint actions, and performs updates based on assuming Nash equilibrium behavior over the current Q-values. This learning protocol provably converges given certain restrictions on the stage games (defined by Q-values) that arise during learning. Experiments with a pair of two-player grid games suggest that such restrictions on the game structure are not necessarily required. Stage games encountered during learning in both grid environments violate the conditions. However, learning consistently converges in the first grid game, which has a unique equilibrium Q-function, but sometimes fails to converge in the second, which has three different equilibrium Q-functions. In a comparison of offline learning performance in both games, we find agents are more likely to reach a joint optimal path with Nash Q-learning than with a single-agent Q-learning method. When at least one agent adopts Nash Q-learning, the performance of both agents is better than using single-agent Q-learning. We have also implemented an online version of Nash Q-learning that balances exploration with exploitation, yielding improved performance. We consider in this chapter a class of two-player nonzero-sum stochastic games with incomplete information, which is inspired by recent applications of game theory in network security. We develop fully distributed reinforcement learning algorithms, which require for each player a minimal amount of information regarding the other player. At each time, each player can be in an active mode or in a sleep mode. If †This material is based upon work supported in part by the U.S. Air Force Office of Scientific Research (AFOSR) under grant number AFOSR MURI FA9550-09-1-0249. D R A F T October 2, 2011, 1:59am D R A F T 2 HYBRID LEARNING IN STOCHASTIC GAMES AND ITS APPLICATION IN NETWORK SECURITY a player is in an active mode, she updates her strategy and estimates of unknown quantities using a specific pure or hybrid learning pattern. The players’ intelligence and rationality are captured by the weighted linear combination of different learning patterns. We use stochastic approximation techniques to show that, under appropriate conditions, the pure or hybrid learning schemes with random updates can be studied using their deterministic ordinary differential equation (ODE) counterparts. Convergence to state-independent equilibria is analyzed for special classes of games, namely, games with two actions, and potential games. Results are applied to network security games between an intruder and an administrator, where the noncooperative behaviors are well characterized by the features of distributed hybrid learning.