Learning near-optimal policies with Bellman-residual minimization based fitted policy iteration and a single sample path

We consider the problem of finding a near-optimal policy in continuous space, discounted Markovian Decision Problems given the trajectory of some behaviour policy. We study the policy iteration algorithm where in successive iterations the action-value functions of the intermediate policies are obtained by picking a function from some fixed function set (chosen by the user) that minimizes an unbiased finite-sample approximation to a novel loss function that upper-bounds the unmodified Bellman-residual criterion. The main result is a finite-sample, high-probability bound on the performance of the resulting policy that depends on the mixing rate of the trajectory, the capacity of the function set as measured by a novel capacity concept that we call the VC-crossing dimension, the approximation power of the function set and the discounted-average concentrability of the future-state distribution. To the best of our knowledge this is the first theoretical reinforcement learning result for off-policy control learning over continuous state-spaces using a single trajectory

Is the Bellman residual a bad proxy?

This paper aims at theoretically and empirically comparing two standard optimization criteria for Reinforcement Learning: i) maximization of the mean value and ii) minimization of the Bellman residual. For that purpose, we place ourselves in the framework of policy search algorithms, that are usually designed to maximize the mean value, and derive a method that minimizes the residual $\|T_* v_\pi - v_\pi\|_{1,\nu}$ over policies. A theoretical analysis shows how good this proxy is to policy optimization, and notably that it is better than its value-based counterpart. We also propose experiments on randomly generated generic Markov decision processes, specifically designed for studying the influence of the involved concentrability coefficient. They show that the Bellman residual is generally a bad proxy to policy optimization and that directly maximizing the mean value is much better, despite the current lack of deep theoretical analysis. This might seem obvious, as directly addressing the problem of interest is usually better, but given the prevalence of (projected) Bellman residual minimization in value-based reinforcement learning, we believe that this question is worth to be considered.Comment: Final NIPS 2017 version (title, among other things, changed

Q-learning with Nearest Neighbors

We consider model-free reinforcement learning for infinite-horizon discounted Markov Decision Processes (MDPs) with a continuous state space and unknown transition kernel, when only a single sample path under an arbitrary policy of the system is available. We consider the Nearest Neighbor Q-Learning (NNQL) algorithm to learn the optimal Q function using nearest neighbor regression method. As the main contribution, we provide tight finite sample analysis of the convergence rate. In particular, for MDPs with a $d$-dimensional state space and the discounted factor $\gamma \in (0,1)$, given an arbitrary sample path with "covering time" $L$, we establish that the algorithm is guaranteed to output an $\varepsilon$-accurate estimate of the optimal Q-function using $\tilde{O}\big(L/(\varepsilon^3(1-\gamma)^7)\big)$ samples. For instance, for a well-behaved MDP, the covering time of the sample path under the purely random policy scales as $\tilde{O}\big(1/\varepsilon^d\big),$ so the sample complexity scales as $\tilde{O}\big(1/\varepsilon^{d+3}\big).$ Indeed, we establish a lower bound that argues that the dependence of $\tilde{\Omega}\big(1/\varepsilon^{d+2}\big)$ is necessary.Comment: Accepted to NIPS 201

Batch Policy Learning under Constraints

When learning policies for real-world domains, two important questions arise: (i) how to efficiently use pre-collected off-policy, non-optimal behavior data; and (ii) how to mediate among different competing objectives and constraints. We thus study the problem of batch policy learning under multiple constraints, and offer a systematic solution. We first propose a flexible meta-algorithm that admits any batch reinforcement learning and online learning procedure as subroutines. We then present a specific algorithmic instantiation and provide performance guarantees for the main objective and all constraints. To certify constraint satisfaction, we propose a new and simple method for off-policy policy evaluation (OPE) and derive PAC-style bounds. Our algorithm achieves strong empirical results in different domains, including in a challenging problem of simulated car driving subject to multiple constraints such as lane keeping and smooth driving. We also show experimentally that our OPE method outperforms other popular OPE techniques on a standalone basis, especially in a high-dimensional setting

On the Use of Non-Stationary Policies for Stationary Infinite-Horizon Markov Decision Processes

We consider infinite-horizon stationary $\gamma$-discounted Markov Decision Processes, for which it is known that there exists a stationary optimal policy. Using Value and Policy Iteration with some error $\epsilon$ at each iteration, it is well-known that one can compute stationary policies that are $\frac{2\gamma}{(1-\gamma)^2}\epsilon$-optimal. After arguing that this guarantee is tight, we develop variations of Value and Policy Iteration for computing non-stationary policies that can be up to $\frac{2\gamma}{1-\gamma}\epsilon$-optimal, which constitutes a significant improvement in the usual situation when $\gamma$ is close to 1. Surprisingly, this shows that the problem of "computing near-optimal non-stationary policies" is much simpler than that of "computing near-optimal stationary policies"

Should one compute the Temporal Difference fix point or minimize the Bellman Residual? The unified oblique projection view

We investigate projection methods, for evaluating a linear approximation of the value function of a policy in a Markov Decision Process context. We consider two popular approaches, the one-step Temporal Difference fix-point computation (TD(0)) and the Bellman Residual (BR) minimization. We describe examples, where each method outperforms the other. We highlight a simple relation between the objective function they minimize, and show that while BR enjoys a performance guarantee, TD(0) does not in general. We then propose a unified view in terms of oblique projections of the Bellman equation, which substantially simplifies and extends the characterization of (schoknecht,2002) and the recent analysis of (Yu & Bertsekas, 2008). Eventually, we describe some simulations that suggest that if the TD(0) solution is usually slightly better than the BR solution, its inherent numerical instability makes it very bad in some cases, and thus worse on average

Finite-Sample Analysis of Bellman Residual Minimization

International audienceWe consider the Bellman residual minimization approach for solving discounted Markov decision problems, where we assume that a generative model of the dynamics and rewards is available. At each policy iteration step, an approximation of the value function for the current policy is obtained by minimizing an empirical Bellman residual defined on a set of n states drawn i.i.d. from a distribution, the immediate rewards, and the next states sampled from the model. Our main result is a generalization bound for the Bellman residual in linear approximation spaces. In particular, we prove that the empirical Bellman residual approaches the true (quadratic) Bellman residual with a rate of order O(1/sqrt((n)). This result implies that minimizing the empirical residual is indeed a sound approach for the minimization of the true Bellman residual which guarantees a good approximation of the value function for each policy. Finally, we derive performance bounds for the resulting approximate policy iteration algorithm in terms of the number of samples n and a measure of how well the function space is able to approximate the sequence of value functions.