We consider policy evaluation algorithms within the context of infinite-horizon dynamic programming problems with discounted cost. We focus on discrete-time dynamic systems with a large number of states, and we discuss two methods, which use simulation, temporal differences, and linear cost function approximation. The first method is a new gradient-like algorithm involving least-squares subproblems and a diminishing stepsize, which is based on the #-policy iteration method of Bertsekas and Ioffe. The second method is the LSTD(#) algorithm recently proposed by Boyan, which for # =0coincides with the linear least-squares temporal-difference algorithm of Bradtke and Barto. At present, there is only a convergence result by Bradtke and Barto for the LSTD(0) algorithm. Here, we strengthen this result by showing the convergence of LSTD(#), with probability 1, for every # [0, 1].

We address the problem of automatically constructing basis functions for linear approximation of the value function of a Markov Decision Process (MDP). Our work builds on results by Bertsekas and Castañon (1989) who proposed a method for automatically aggregating states to speed up value iteration. We propose to use neighborhood component analysis (Goldberger et al., 2005), a dimensionality reduction technique created for supervised learning, in order to map a high-dimensional state space to a lowdimensional space, based on the Bellman error, or on the temporal difference (TD) error. We then place basis function in the lower-dimensional space. These are added as new features for the linear function approximator. This approach is applied to a high-dimensional inventory control problem. 1.

We introduce the first algorithm for off-policy temporal-difference learning that is stable with linear function approximation. Off-policy learning is of interest because it forms the basis for popular reinforcement learning methods such as Q-learning, which has been known to diverge with linear function approximation, and because it is critical to the practical utility of multi-scale, multi-goal, learning frameworks such as options, HAMs, and MAXQ. Our new algorithm combines TD(λ) over state–action pairs with importance sampling ideas from our previous work. We prove that, given training under any ɛ-soft policy, the algorithm converges w.p.1 to a close approximation (as in Tsitsiklis and Van Roy, 1997; Tadic, 2001) to the action-value function for an arbitrary target policy. Variations of the algorithm designed to reduce variance introduce additional bias but are also guaranteed convergent. We also illustrate our method empirically on a small policy evaluation problem. Our current results are limited to episodic tasks with episodes of bounded length. 1 Although Q-learning remains the most popular of all reinforcement learning algorithms, it has been known since about 1996 that it is unsound with linear function approximation (see Gordon, 1995; Bertsekas and Tsitsiklis, 1996). The most telling counterexample, due to Baird (1995) is a seven-state Markov decision process with linearly independent feature vectors, for which an exact solution exists, yet 1 This is a re-typeset version of an article published in the Proceedings

The traditional Kalman filter can be viewed as a recursive stochastic algorithm that approximates an unknown function via a linear combination of prespecified basis functions given a sequence of noisy samples. In this paper, we generalize the algorithm to one that approximates the fixed point of an operator that is known to be a Euclidean norm contraction. Instead of noisy samples of the desired fixed point, the algorithm updates parameters based on noisy samples of functions generated by application of the operator, in the spirit of Robbins–Monro stochastic approximation. The algorithm is motivated by temporal–difference learning, and our developments lead to a possibly more efficient variant of temporal–difference learning. We establish convergence of the algorithm and explore efficiency gains through computational experiments involving optimal stopping and queueing problems.

We address the problem of computing the optimal Q-function in Markov decision problems with infinite state-space. We analyze the convergence properties of several variations of Q-learning when combined with function approximation, extending the analysis of TD-learning in (Tsitsiklis & Van Roy, 1996a) to stochastic control settings. We identify conditions under which such approximate methods converge with probability 1. We conclude with a brief discussion on the general applicability of our results and compare them with several related works. 1.

We introduce the first temporal-difference learning algorithm that is stable with linear function approximation and off-policy training, for any finite Markov decision process, behavior policy, and target policy, and whose complexity scales linearly in the number of parameters. We consider an i.i.d. policy-evaluation setting in which the data need not come from on-policy experience. The gradient temporal-difference (GTD) algorithm estimates the expected update vector of the TD(0) algorithm and performs stochastic gradient descent on its L2 norm. We prove that this algorithm is stable and convergent under the usual stochastic approximation conditions to the same least-squares solution as found by the LSTD, but without LSTD’s quadratic computational complexity. GTD is online and incremental, and does not involve multiplying by products of likelihood ratios as in importance-sampling methods. 1 Off-policy learning methods Off-policy methods have an important role to play in the larger ambitions of modern reinforcement learning. In general, updates to a statistic of a dynamical process are said to be “off-policy ” if their distribution does not match the dynamics of the process, particularly if the mismatch is due to the way actions are chosen. The prototypical example in reinforcement learning is the learning of the value function for one policy, the target policy, using data obtained while following another policy, the behavior policy. For example, the popular Q-learning algorithm (Watkins 1989) is an offpolicy temporal-difference algorithm in which the target policy is greedy with respect to estimated action values, and the behavior policy is something more exploratory, such as a corresponding ɛ-greedy policy. Off-policy methods are also critical to reinforcement-learning-based efforts to model human-level world knowledge and state representations as predictions of option outcomes (e.g.,

Kernel methods have attracted many research interests recently since by utilizing Mercer kernels, non-linear and non-parametric versions of conventional supervised or unsupervised learning algorithms can be implemented and usually better generalization abilities can be obtained. However, kernel methods in reinforcement learning have not been popularly studied in the literature. In this paper, we present a novel kernel-based least-squares temporal-difference (TD) learning algorithm called KLS-TD(λ), which can be viewed as the kernel version or nonlinear form of the previous linear LS-TD(λ) algorithms. By introducing kernel-based nonlinear mapping, the KLS-TD(λ) algorithm is superior to conventional linear TD(λ) algorithms in value function prediction or policy evaluation problems with nonlinear value functions. Furthermore, in KLS-TD(λ), the eligibility traces in kernel-based TD learning are derived to make use of data more efficiently, which is different from the recent work on Gaussian Processes in reinforcement learning. Experimental results on a typical value-function learning prediction problem of a Markov chain demonstrate the

Abstract. Reinforcement learning is commonly used with function approximation. However, very few positive results are known about the convergence of function approximation based RL control algorithms. In this paper we show that TD(0) and Sarsa(0) with linear function approximation is convergent for a simple class of problems, where the system is linear and the costs are quadratic (the LQ control problem). Furthermore, we show that for systems with Gaussian noise and non-completely observable states (the LQG problem), the mentioned RL algorithms are still convergent, if they are combined with Kalman filtering. 1.

Solving a semi-Markov decision process (SMDP) using value or policy iteration requires precise knowledge of the probabilistic model and suffers from the curse of dimensionality. To overcome these limitations, we present a reinforcement learning approach where one optimizes the SMDP performance criterion with respect to a family of parameterised policies. We propose an online algorithm that simultaneously estimates the gradient of the performance criterion and optimises it using stochastic approximation. We apply our algorithm to call admission control. Index Terms Stochastic processes; Semi-Markov decision process; Policy gradient; Two-time scale; Call admis-sion control.

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