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.
In the special case of MDP, there exist a large body of works on its sample complexity and sampling-based algorithms. For the tabular setting (finitely many state and actions), sample complexity of MDP with a sampling oracle has been studied in @cite_8 @cite_2 @cite_7 @cite_13 @cite_34 @cite_14 @cite_33 . Lower bounds for sample complexity have been studied in @cite_24 @cite_44 @cite_45 , where the first tight lower bound @math is obtained in @cite_24 . The first sample-optimal algorithm for finding an @math -optimal value is proposed in @cite_24 . @cite_12 gives the first algorithm that finds an @math -optimal policy using the optimal sample complexity @math for all values of @math . For solving MDP using @math linearly additive features, @cite_41 proved a lower bound of sample complexity that is @math . It also provided an algorithm that achieves this lower bound up to log factors, however, their analysis of the algorithm relies heavily on an extra anchor state'' assumption. In @cite_0 , a primal-dual method solving MDP with linear and bilinear representation of value functions and transition models is proposed for the undiscounted MDP. In @cite_18 , the sample complexity of contextual decision process is studied.
@cite_18 @cite_14 @cite_33 @cite_7 @cite_8 @cite_41 @cite_24 @cite_44 @cite_0 @cite_45 @cite_2 @cite_34 @cite_13 @cite_12
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Consider a Markov decision process (MDP) that admits a set of state-action features, which can linearly express the process's probabilistic transition model. We propose a parametric Q-learning algorithm that finds an approximate-optimal policy using a sample size proportional to the feature dimension @math and invariant with respect to the size of the state space. To further improve its sample efficiency, we exploit the monotonicity property and intrinsic noise structure of the Bellman operator, provided the existence of anchor state-actions that imply implicit non-negativity in the feature space. We augment the algorithm using techniques of variance reduction, monotonicity preservation, and confidence bounds. It is proved to find a policy which is @math -optimal from any initial state with high probability using @math sample transitions for arbitrarily large-scale MDP with a discount factor @math . A matching information-theoretical lower bound is proved, confirming the sample optimality of the proposed method with respect to all parameters (up to polylog factors). In this paper we consider the problem of computing an ϵ-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in O(1) time. Given such a DMDP with states , actions , discount factor γ∈(0,1), and rewards in range [0,1] we provide an algorithm which computes an ϵ-optimal policy with probability 1−δ where both the run time spent and number of sample taken is upper bounded by O[ | || | (1−γ)3ϵ2 log( | || | (1−γ)δϵ )log( 1 (1−γ)ϵ )] . For fixed values of ϵ, this improves upon the previous best known bounds by a factor of (1−γ)−1 and matches the sample complexity lower bounds proved in azar2013minimax up to logarithmic factors. We also extend our method to computing ϵ-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound. Consider the problem of approximating the optimal policy of a Markov decision process (MDP) by sampling state transitions. In contrast to existing reinforcement learning methods that are based on successive approximations to the nonlinear Bellman equation, we propose a Primal-Dual @math Learning method in light of the linear duality between the value and policy. The @math learning method is model-free and makes primal-dual updates to the policy and value vectors as new data are revealed. For infinite-horizon undiscounted Markov decision process with finite state space @math and finite action space @math , the @math learning method finds an @math -optimal policy using the following number of sample transitions @math where @math is an upper bound of mixing times across all policies and @math is a parameter characterizing the range of stationary distributions across policies. The @math learning method also applies to the computational problem of MDP where the transition probabilities and rewards are explicitly given as the input. In the case where each state transition can be sampled in @math time, the @math learning method gives a sublinear-time algorithm for solving the averaged-reward MDP. This paper addresses the problem of planning under uncertainty in large Markov Decision Processes (MDPs). Factored MDPs represent a complex state space using state variables and the transition model using a dynamic Bayesian network. This representation often allows an exponential reduction in the representation size of structured MDPs, but the complexity of exact solution algorithms for such MDPs can grow exponentially in the representation size. In this paper, we present two approximate solution algorithms that exploit structure in factored MDPs. Both use an approximate value function represented as a linear combination of basis functions, where each basis function involves only a small subset of the domain variables. A key contribution of this paper is that it shows how the basic operations of both algorithms can be performed efficiently in closed form, by exploiting both additive and context-specific structure in a factored MDP. A central element of our algorithms is a novel linear program decomposition technique, analogous to variable elimination in Bayesian networks, which reduces an exponentially large LP to a provably equivalent, polynomial-sized one. One algorithm uses approximate linear programming, and the second approximate dynamic programming. Our dynamic programming algorithm is novel in that it uses an approximation based on max-norm, a technique that more directly minimizes the terms that appear in error bounds for approximate MDP algorithms. We provide experimental results on problems with over 1040 states, demonstrating a promising indication of the scalability of our approach, and compare our algorithm to an existing state-of-the-art approach, showing, in some problems, exponential gains in computation time.