We introduce a new algorithm based on linear programming that approximates the differential value function of an average-cost Markov decision process via a linear combination of pre-selected basis functions. The algorithm carries out a form of cost shaping and minimizes a version of Bellman error. We establish an error bound that scales gracefully with the number of states without imposing the (strong) Lyapunov condition required by its counter- part in [6]. We propose a path-following method that automates selection of important algorithm parameters which represent coun- terparts to the "state-relevance weights" studied in [6]. Over the past few years, there has been a growing interest in linear programming (LP) approaches to approximate dynamic programming (DP).

We consider approximate value iteration with a parameterized approximator in which the state space is partitioned and the optimal cost-to-go function over each partition is approximated by a constant. We establish performance loss bounds for policies derived from approximations associated with fixed points. These bounds identify benefits to having projection weights equal to the invariant distribution of the resulting policy. Such projection weighting leads to the same fixed points as TD(0). Our analysis also leads to the first performance loss bound for approximate value iteration with an average cost objective.

Temporally extended actions have proven useful for reinforcement learning, but their duration also makes them valuable for efficient planning. The options framework provides a concrete way to implement and reason about temporally extended actions. Existing literature has demonstrated the value of planning with options empirically, but there is a lack of theoretical analysis formalizing when planning with options is more efficient than planning with primitive actions. We provide a general analysis of the convergence rate of a popular Approximate Value Iteration (AVI) algorithm called Fitted Value Iteration (FVI) with options. Our analysis reveals that longer duration options and a pessimistic estimate of the value function both lead to faster convergence. Furthermore, options can improve convergence even when they are suboptimal and sparsely distributed throughout the state-space. Next we consider the problem of generating useful options for planning based on a subset of landmark states. This suggests a new algorithm, Landmark-based AVI (LAVI), that represents the value function only at the landmark states. We analyze both FVI and LAVI using the proposed landmark-based options and compare the two algorithms. Our experimental results in three different domains demonstrate the key properties from the analysis. Our theoretical and experimental results demonstrate that options can play an important role in AVI by decreasing approximation error and inducing fast convergence.

Approximate value iteration (AVI) is a widely used technique in reinforcement learning. Most AVI methods do not take full advantage of the sequential relationship between samples within a trajectory in deriving value estimates, due to the challenges in dealing with the inherent bias and variance in the $n$-step returns. We propose a bounding method which uses a negatively biased but relatively low variance estimator generated from a complex return to provide a lower bound on the observed value of a traditional one-step return estimator. In addition, we develop a new Bounded FQI algorithm, which efficiently incorporates the bounding method into an AVI framework. Experiments show that our method produces more accurate value estimates than existing approaches, resulting in improved policies.

Approximate dynamic programming (ADP) has proven itself in a wide range of applications spanning large-scale transportation problems, health care, revenue management, and energy systems. The design of effective ADP algorithms has many dimensions, but one crucial factor is the stepsize rule used to update a value function approximation. Many operations research applications are computationally intensive, and it is important to obtain good results quickly. Furthermore, the most popular stepsize formulas use tunable parameters and can produce very poor results if tuned improperly. We derive a new stepsize rule that optimizes the prediction error in order to improve the short-term performance of an ADP algorithm. With only one, relatively insensitive tunable parameter, the new rule adapts to the level of noise in the problem and produces faster convergence in numerical experiments.

We consider the discrete-time infinite-horizon optimal control problem formalized by Markov Decision Processes (Puterman, 1994; Bertsekas and Tsitsiklis, 1996). We revisit the work of Bertsekas and Ioffe (1996), that introduced λ Policy Iteration, a family of algorithms parameterized by λ that generalizes the standard algorithms Value Iteration and Policy Iteration, and has some deep connections with the Temporal Differences algorithm TD(λ) described by Sutton and Barto (1998). We deepen the original theory developped by the authors by providing convergence rate bounds which generalize standard bounds for Value Iteration described for instance by Puterman (1994). Then, the main contribution of this paper is to develop the theory of this algorithm when it is used in an approximate form and show that this is sound. Doing so, we extend and unify the separate analyses developped by Munos for Approximate Value Iteration (Munos, 2007) and Approximate Policy Iteration (Munos, 2003). Eventually, we revisit the use of this algorithm in the training of a Tetris playing controller as originally done by Bertsekas and Ioffe (1996). We provide an original performance bound that can be applied to such an undiscounted control problem. Our empirical results are different from those of Bertsekas and Ioffe (which were originally qualified as "paradoxical" and "intriguing"), and much more conform to what one would expect from a learning experiment. We discuss the possible reason for such a difference.

Most algorithms for solving Markov decision processes rely on a discount factor, which ensures their convergence. It is generally assumed that using an artificially low discount factor will improve the convergence rate, while sacrificing the solution quality.We however demonstrate that using an artificially low discount factor may significantly improve the solution quality, when used in approximate dynamic programming. We propose two explanations of this phenomenon. The first justification followsdirectly from the standard approximation error bounds: using a lower discount factor may decrease the approximation error bounds. However, we also show that these bounds are loose, thus their decrease does not entirely justify the improved solution quality. We thus propose another justification: when the rewards are received only sporadically (as in the case of Tetris), we can derive tighter bounds, which support a significant improvement in the solution quality with a decreased discount factor.

In this paper we propose a novel algorithm, factored value iteration (FVI), for the approximate solution of factored Markov decision processes (fMDPs). The traditional approximate value iteration algorithm is modified in two ways. For one, the least-squares projection operator is modified so that it does not increase max-norm, and thus preserves convergence. The other modification is that we uniformly sample polynomially many samples from the (exponentially large) state space. This way, the complexity of our algorithm becomes polynomial in the size of the fMDP description length. We prove that the algorithm is convergent. We also derive an upper bound on the difference between our approximate solution and the optimal one, and also on the error introduced by sampling. We analyze various projection operators with respect to their computation complexity and their convergence when combined with approximate value iteration.

We consider approximate value iteration with a parameterized approximator in which the state space is partitioned and the optimal cost-to-go function over each partition is approximated by a constant. We establish performance loss bounds for policies derived from approximations associated with fixed points. These bounds identify benefits to having projection weights equal to the invariant distribution of the resulting policy. Such projection weighting leads to the same fixed points as TD(0). Our analysis also leads to the first performance loss bound for approximate value iteration with an average cost objective.

We consider approximate value iteration with a parameterized approximator in which the state space is partitioned and the optimal cost-to-go function over each partition is approximated by a constant. We establish performance loss bounds for policies derived from approximations associated with fixed points. These bounds identify benefits to having projection weights equal to the invariant distribution of the resulting policy. Such projection weighting leads to the same fixed points as TD(0). Our analysis also leads to the first performance loss bound for approximate value iteration with an average cost objective.