Finite Sample Analysis of the GTD Policy Evaluation Algorithms in Markov Setting

In reinforcement learning (RL) , one of the key components is policy evaluation, which aims to estimate the value function (i.e., expected long-term accumulated reward) of a policy. With a good policy evaluation method, the RL algorithms will estimate the value function more accurately and find a better policy. When the state space is large or continuous \emph{Gradient-based Temporal Difference(GTD)} policy evaluation algorithms with linear function approximation are widely used. Considering that the collection of the evaluation data is both time and reward consuming, a clear understanding of the finite sample performance of the policy evaluation algorithms is very important to reinforcement learning. Under the assumption that data are i.i.d. generated, previous work provided the finite sample analysis of the GTD algorithms with constant step size by converting them into convex-concave saddle point problems. However, it is well-known that, the data are generated from Markov processes rather than i.i.d. in RL problems.. In this paper, in the realistic Markov setting, we derive the finite sample bounds for the general convex-concave saddle point problems, and hence for the GTD algorithms. We have the following discussions based on our bounds. (1) With variants of step size, GTD algorithms converge. (2) The convergence rate is determined by the step size, with the mixing time of the Markov process as the coefficient. The faster the Markov processes mix, the faster the convergence. (3) We explain that the experience replay trick is effective by improving the mixing property of the Markov process. To the best of our knowledge, our analysis is the first to provide finite sample bounds for the GTD algorithms in Markov setting

Finite-sample Analysis of Greedy-GQ with Linear Function Approximation under Markovian Noise

Greedy-GQ is an off-policy two timescale algorithm for optimal control in reinforcement learning. This paper develops the first finite-sample analysis for the Greedy-GQ algorithm with linear function approximation under Markovian noise. Our finite-sample analysis provides theoretical justification for choosing stepsizes for this two timescale algorithm for faster convergence in practice, and suggests a trade-off between the convergence rate and the quality of the obtained policy. Our paper extends the finite-sample analyses of two timescale reinforcement learning algorithms from policy evaluation to optimal control, which is of more practical interest. Specifically, in contrast to existing finite-sample analyses for two timescale methods, e.g., GTD, GTD2 and TDC, where their objective functions are convex, the objective function of the Greedy-GQ algorithm is non-convex. Moreover, the Greedy-GQ algorithm is also not a linear two-timescale stochastic approximation algorithm. Our techniques in this paper provide a general framework for finite-sample analysis of non-convex value-based reinforcement learning algorithms for optimal control.Comment: UAI 202

On a convergent off -policy temporal difference learning algorithm in on-line learning environment

In this paper we provide a rigorous convergence analysis of a "off"-policy temporal difference learning algorithm with linear function approximation and per time-step linear computational complexity in "online" learning environment. The algorithm considered here is TDC with importance weighting introduced by Maei et al. We support our theoretical results by providing suitable empirical results for standard off-policy counterexamples.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1503.0910

Finite-Sample Analysis of Proximal Gradient TD Algorithms

In this paper, we analyze the convergence rate of the gradient temporal difference learning (GTD) family of algorithms. Previous analyses of this class of algorithms use ODE techniques to prove asymptotic convergence, and to the best of our knowledge, no finite-sample analysis has been done. Moreover, there has been not much work on finite-sample analysis for convergent off-policy reinforcement learning algorithms. In this paper, we formulate GTD methods as stochastic gradient algorithms w.r.t.~a primal-dual saddle-point objective function, and then conduct a saddle-point error analysis to obtain finite-sample bounds on their performance. Two revised algorithms are also proposed, namely projected GTD2 and GTD2-MP, which offer improved convergence guarantees and acceleration, respectively. The results of our theoretical analysis show that the GTD family of algorithms are indeed comparable to the existing LSTD methods in off-policy learning scenarios.Comment: 31st Conference on Uncertainty in Artificial Intelligence (UAI). arXiv admin note: substantial text overlap with arXiv:2006.0397

A Greedy Approach to Adapting the Trace Parameter for Temporal Difference Learning

One of the main obstacles to broad application of reinforcement learning methods is the parameter sensitivity of our core learning algorithms. In many large-scale applications, online computation and function approximation represent key strategies in scaling up reinforcement learning algorithms. In this setting, we have effective and reasonably well understood algorithms for adapting the learning-rate parameter, online during learning. Such meta-learning approaches can improve robustness of learning and enable specialization to current task, improving learning speed. For temporal-difference learning algorithms which we study here, there is yet another parameter, $\lambda$, that similarly impacts learning speed and stability in practice. Unfortunately, unlike the learning-rate parameter, $\lambda$ parametrizes the objective function that temporal-difference methods optimize. Different choices of $\lambda$ produce different fixed-point solutions, and thus adapting $\lambda$ online and characterizing the optimization is substantially more complex than adapting the learning-rate parameter. There are no meta-learning method for $\lambda$ that can achieve (1) incremental updating, (2) compatibility with function approximation, and (3) maintain stability of learning under both on and off-policy sampling. In this paper we contribute a novel objective function for optimizing $\lambda$ as a function of state rather than time. We derive a new incremental, linear complexity $\lambda$-adaption algorithm that does not require offline batch updating or access to a model of the world, and present a suite of experiments illustrating the practicality of our new algorithm in three different settings. Taken together, our contributions represent a concrete step towards black-box application of temporal-difference learning methods in real world problems

Investigating practical linear temporal difference learning

Off-policy reinforcement learning has many applications including: learning from demonstration, learning multiple goal seeking policies in parallel, and representing predictive knowledge. Recently there has been an proliferation of new policy-evaluation algorithms that fill a longstanding algorithmic void in reinforcement learning: combining robustness to off-policy sampling, function approximation, linear complexity, and temporal difference (TD) updates. This paper contains two main contributions. First, we derive two new hybrid TD policy-evaluation algorithms, which fill a gap in this collection of algorithms. Second, we perform an empirical comparison to elicit which of these new linear TD methods should be preferred in different situations, and make concrete suggestions about practical use.Comment: Autonomous Agents and Multi-agent Systems, 201

Direct Gradient Temporal Difference Learning

Off-policy learning enables a reinforcement learning (RL) agent to reason counterfactually about policies that are not executed and is one of the most important ideas in RL. It, however, can lead to instability when combined with function approximation and bootstrapping, two arguably indispensable ingredients for large-scale reinforcement learning. This is the notorious deadly triad. Gradient Temporal Difference (GTD) is one powerful tool to solve the deadly triad. Its success results from solving a doubling sampling issue indirectly with weight duplication or Fenchel duality. In this paper, we instead propose a direct method to solve the double sampling issue by simply using two samples in a Markovian data stream with an increasing gap. The resulting algorithm is as computationally efficient as GTD but gets rid of GTD's extra weights. The only price we pay is a logarithmically increasing memory as time progresses. We provide both asymptotic and finite sample analysis, where the convergence rate is on-par with the canonical on-policy temporal difference learning. Key to our analysis is a novel refined discretization of limiting ODEs.Comment: Submitted to JMLR in Apr 202

On the Global Convergence of Actor-Critic: A Case for Linear Quadratic Regulator with Ergodic Cost

Despite the empirical success of the actor-critic algorithm, its theoretical understanding lags behind. In a broader context, actor-critic can be viewed as an online alternating update algorithm for bilevel optimization, whose convergence is known to be fragile. To understand the instability of actor-critic, we focus on its application to linear quadratic regulators, a simple yet fundamental setting of reinforcement learning. We establish a nonasymptotic convergence analysis of actor-critic in this setting. In particular, we prove that actor-critic finds a globally optimal pair of actor (policy) and critic (action-value function) at a linear rate of convergence. Our analysis may serve as a preliminary step towards a complete theoretical understanding of bilevel optimization with nonconvex subproblems, which is NP-hard in the worst case and is often solved using heuristics.Comment: 41 page

Concentration bounds for temporal difference learning with linear function approximation: The case of batch data and uniform sampling

We propose a stochastic approximation (SA) based method with randomization of samples for policy evaluation using the least squares temporal difference (LSTD) algorithm. Our proposed scheme is equivalent to running regular temporal difference learning with linear function approximation, albeit with samples picked uniformly from a given dataset. Our method results in an $O(d)$ improvement in complexity in comparison to LSTD, where $d$ is the dimension of the data. We provide non-asymptotic bounds for our proposed method, both in high probability and in expectation, under the assumption that the matrix underlying the LSTD solution is positive definite. The latter assumption can be easily satisfied for the pathwise LSTD variant proposed in [23]. Moreover, we also establish that using our method in place of LSTD does not impact the rate of convergence of the approximate value function to the true value function. These rate results coupled with the low computational complexity of our method make it attractive for implementation in big data settings, where $d$ is large. A similar low-complexity alternative for least squares regression is well-known as the stochastic gradient descent (SGD) algorithm. We provide finite-time bounds for SGD. We demonstrate the practicality of our method as an efficient alternative for pathwise LSTD empirically by combining it with the least squares policy iteration (LSPI) algorithm in a traffic signal control application. We also conduct another set of experiments that combines the SA based low-complexity variant for least squares regression with the LinUCB algorithm for contextual bandits, using the large scale news recommendation dataset from Yahoo

Distributed Policy Evaluation Under Multiple Behavior Strategies

We apply diffusion strategies to develop a fully-distributed cooperative reinforcement learning algorithm in which agents in a network communicate only with their immediate neighbors to improve predictions about their environment. The algorithm can also be applied to off-policy learning, meaning that the agents can predict the response to a behavior different from the actual policies they are following. The proposed distributed strategy is efficient, with linear complexity in both computation time and memory footprint. We provide a mean-square-error performance analysis and establish convergence under constant step-size updates, which endow the network with continuous learning capabilities. The results show a clear gain from cooperation: when the individual agents can estimate the solution, cooperation increases stability and reduces bias and variance of the prediction error; but, more importantly, the network is able to approach the optimal solution even when none of the individual agents can (e.g., when the individual behavior policies restrict each agent to sample a small portion of the state space).Comment: 36 pages, 4 figures, accepted for publication on IEEE Transactions on Automatic Contro