Control Of Markov Decision Processes Research Articles

Risk-sensitive control of Markov decision processes: A moment-based approach with target distributions

In many revenue management applications risk-averse decision-making is crucial. In dynamic settings, however, it is challenging to find the right balance between maximizing expected rewards and minimizing various kinds of risk. In existing approaches utility functions, chance constraints, or (conditional) value at risk considerations are used to influence the distribution of rewards in a preferred way. Nevertheless, common techniques are not flexible enough and typically numerically complex. In our model, we exploit the fact that a distribution is characterized by its mean and higher moments. We present a multi-valued dynamic programming heuristic to compute risk-sensitive feedback policies that are able to directly control the moments of future rewards. Our approach is based on recursive formulations of higher moments and does not require an extension of the state space. Finally, we propose a self-tuning algorithm, which allows to identify feedback policies that approximate predetermined (risk-sensitive) target distributions. We illustrate the effectiveness and the flexibility of our approach for different dynamic pricing scenarios.

Analysis of averages over distributions of Markov processes

In problems of optimal control of Markov decision processes and optimal design of experiments, the occupation measure of a Markov process is designed in order to maximize a specific reward function. When the memory of such a process is too long, or the process is non-Markovian but mixing, it makes sense to approximate it by that of a shorter memory Markov process. This note provides a specific bound for the approximation error introduced in these schemes. The derived bound is then applied to the proposed solution of a recently introduced approach to optimal input design for nonlinear systems.

Policy Search for the Optimal Control of Markov Decision Processes: A Novel Particle-Based Iterative Scheme.

Classical approximate dynamic programming techniques based on state-space gridding become computationally impracticable for high-dimensional problems. Policy search techniques cope with this curse of dimensionality issue by searching for the optimal control policy in a restricted parameterized policy space. We here focus on the case of discrete action space and introduce a novel policy parametrization that adopts particles to describe the map from the state space to the action space, each particle representing a region of the state space that is mapped into a certain action. The locations and actions associated with the particles describing a policy can be tuned by means of a recently introduced policy gradient method with parameter-based exploration. The task of selecting an appropriately sized set of particles is here solved through an iterative policy building scheme that adds new particles to improve the policy performance and is also capable of removing redundant particles. Experiments demonstrate the scalability of the proposed approach as the dimensionality of the state-space grows.

Optimal Control of Markov Decision Processes With Linear Temporal Logic Constraints

In this paper, we develop a method to automatically generate a control policy for a dynamical system modeled as a Markov Decision Process (MDP). The control specification is given as a Linear Temporal Logic (LTL) formula over a set of propositions defined on the states of the MDP. Motivated by robotic applications requiring persistent tasks, such as environmental monitoring and data gathering, we synthesize a control policy that minimizes the expected cost between satisfying instances of a particular proposition over all policies that maximize the probability of satisfying the given LTL specification. Our approach is based on the definition of a novel optimization problem that extends the existing average cost per stage problem. We propose a sufficient condition for a policy to be optimal, and develop a dynamic programming algorithm that synthesizes a policy that is optimal for a set of LTL specifications.

Control of noisy differential-drive vehicles from time-bounded temporal logic specifications

We address the problem of controlling a noisy differential drive mobile robot such that the probability of satisfying a specification given as a bounded linear temporal logic formula over a set of properties at the regions in the environment is maximized. We assume that the vehicle can determine its precise initial position in a known map of the environment. However, motivated by practical limitations, we assume that the vehicle is equipped with noisy actuators and, during its motion in the environment, it can only measure the angular velocity of its wheels using limited accuracy incremental encoders. Assuming the duration of the motion is finite, we map the measurements to a Markov decision process (MDP). We use recent results in statistical model checking to obtain an MDP control policy that maximizes the probability of satisfaction. We translate this policy to a vehicle feedback control strategy and show that the probability that the vehicle satisfies the specification in the environment is bounded from below by the probability of satisfying the specification on the MDP. We illustrate our method with simulations and experimental results.

Policy Gradient Approach of Event‐Based Optimization and Its Online Implementation

AbstractIn the theory of event‐based optimization (EBO), the decision making is triggered by events, which is different from the traditional state‐based control in Markov decision processes (MDP). In this paper, we propose a policy gradient approach of EBO. First, an equation of performance gradient in the event‐based policy space is derived based on a fundamental quantity called Q‐factors of EBO. With the performance gradient, we can find the local optimum of EBO using the gradient‐based algorithm. Compared to the policy iteration approach in EBO, this policy gradient approach does not require restrictive conditions and it has a wider application scenario. The policy gradient approach is further implemented based on the online estimation of Q‐factors. This approach does not require the prior information about the system parameters, such as the transition probability. Finally, we use an EBO model to formulate the admission control problem and demonstrate the main idea of this paper. Such online algorithm provides an effective implementation of the EBO theory in practice.

A particle-based policy for the optimal control of Markov decision processes

When the state dimension is large, classical approximate dynamic programming techniques may become computationally unfeasible, since the complexity of the algorithm grows exponentially with the state space size (curse of dimensionality). Policy search techniques are able to overcome this problem because, instead of estimating the value function over the entire state space, they search for the optimal control policy in a restricted parameterized policy space. This paper presents a new policy parametrization that exploits a single point (particle) to represent an entire region of the state space and can be tuned through a recently introduced policy gradient method with parameter-based exploration. Experiments demonstrate the superior performance of the proposed approach in high dimensional environments.

Reinforcement learning algorithms with function approximation: Recent advances and applications

In recent years, the research on reinforcement learning (RL) has focused on function approximation in learning prediction and control of Markov decision processes (MDPs). The usage of function approximation techniques in RL will be essential to deal with MDPs with large or continuous state and action spaces. In this paper, a comprehensive survey is given on recent developments in RL algorithms with function approximation. From a theoretical point of view, the convergence and feature representation of RL algorithms are analyzed. From an empirical aspect, the performance of different RL algorithms was evaluated and compared in several benchmark learning prediction and learning control tasks. The applications of RL with function approximation are also discussed. At last, future works on RL with function approximation are suggested.

Reversible Markov Decision Processes with an Average-Reward Criterion

A wide variety of stochastic models have an important property known as reversibility. The analysis of reversible Markov chains is often significantly simpler than analysis of general Markov chains, particularly since there are often simple closed-form expressions for their invariant probability mass functions. In this paper we study the structure of optimal control policies for Markov decision processes with reversible dynamics. For optimal control of Markov decision processes, significant analytical and computational simplifications can arise as a result of reversibility. Here we present an analysis of a special class of reversible Markov decision processes, namely those that contain a cycle through their communication graph under any policy. These results provide encouraging evidence that similar simplifications hold for more general models, such as reversible processes without cycles and partition reversible processes. These results are demonstrated on a control-based variant of the Metropolis--Hastin...

Average control of Markov decision processes with Feller transition probabilities and general action spaces

This paper studies the average control problem of discrete-time Markov Decision Processes (MDPs for short) with general state space, Feller transition probabilities, and possibly non-compact control constraint sets A(x). Two hypotheses are considered: either the cost function c is strictly unbounded or the multifunctions Ar(x)={a∈A(x):c(x,a)≤r} are upper-semicontinuous and compact-valued for each real r. For these two cases we provide new results for the existence of a solution to the average-cost optimality equality and inequality using the vanishing discount approach. We also study the convergence of the policy iteration approach under these conditions. It should be pointed out that we do not make any assumptions regarding the convergence and the continuity of the limit function generated by the sequence of relative difference of the α-discounted value functions and the Poisson equations as often encountered in the literature.

Temporal Logic Motion Planning and Control With Probabilistic Satisfaction Guarantees

We describe a computational framework for automatic deployment of a robot with sensor and actuator noise from a temporal logic specification over a set of properties that are satisfied by the regions of a partitioned environment. We model the motion of the robot in the environment as a Markov decision process (MDP) and translate the motion specification to a formula of probabilistic computation tree logic (PCTL). As a result, the robot control problem is mapped to that of generating an MDP control policy from a PCTL formula. We present algorithms for the synthesis of such policies for different classes of PCTL formulas. We illustrate our method with simulation and experimental results.

Proto-value Functions: A Laplacian Framework for Learning Representation and Control in Markov Decision Processes

This paper introduces a novel spectral framework for solving Markov decision processes (MDPs) by jointly learning representations and optimal policies. The major components of the framework describ...

Policy Iteration for Decentralized Control of Markov Decision Processes

Coordination of distributed agents is required for problems arising in many areas, including multi-robot systems, networking and e-commerce. As a formal framework for such problems, we use the decentralized partially observable Markov decision process (DEC-POMDP). Though much work has been done on optimal dynamic programming algorithms for the single-agent version of the problem, optimal algorithms for the multiagent case have been elusive. The main contribution of this paper is an optimal policy iteration algorithm for solving DEC-POMDPs. The algorithm uses stochastic finite-state controllers to represent policies. The solution can include a correlation device, which allows agents to correlate their actions without communicating. This approach alternates between expanding the controller and performing value-preserving transformations, which modify the controller without sacrificing value. We present two efficient value-preserving transformations: one can reduce the size of the controller and the other can improve its value while keeping the size fixed. Empirical results demonstrate the usefulness of value-preserving transformations in increasing value while keeping controller size to a minimum. To broaden the applicability of the approach, we also present a heuristic version of the policy iteration algorithm, which sacrifices convergence to optimality. This algorithm further reduces the size of the controllers at each step by assuming that probability distributions over the other agents' actions are known. While this assumption may not hold in general, it helps produce higher quality solutions in our test problems.

Approximate Fixed Point Iteration with an Application to Infinite Horizon Markov Decision Processes

Many approximate iterative algorithms can be represented by the form $V_n=TV_{n-1}+U_n$, $n\geq1$, where $V_{n-1},U_n$, $n\geq1$, are elements of a seminormed linear space $(\mathcal{V},\|\cdot\|)$ and $T$ is a contractive operator. The objective is usually to calculate a fixed point $V^*$ of $T$. Here, the quantities $U_n$ may be interpreted as the errors obtained when the operator $T$ can be evaluated only approximately. As is well known, in the absence of such an approximation error, the algorithm converges to a fixed point with a geometric rate under general conditions. In this article the convergence properties of such algorithms in the presence of an approximation error $U_n$ are studied. It is shown that the error $\|V_n-V^*\|$ is dominated by $\|U_n\|$, so that convergence of $\|U_n\|$ to zero implies the convergence of $\|V_n-V^*\|$ to zero, at a rate determined by $\|U_n\|$. The results are naturally extended to a relative error of the form $\|U_n\|/\|V_{n-1}\|$, as well as to $J$-stage contractions. The utility of this general theory is then demonstrated by an extended application to the problem of model-based approximate and adaptive control of Markov decision processes. The theory is shown to permit a sharpening of known convergence rates under more general conditions. Additionally, bounds on regret for adaptive controls with forced exploration are calculated in terms of a stagewise exploration rate. This permits the determination of an optimal choice of exploration rate within the class of certainty-equivalence adaptive control policies.

Uniqueness and Stability of Optimal Policies of Finite State Markov Decision Processes

In this paper we consider infinite horizon discrete-time optimal control of Markov decision processes (MDPs) with finite state spaces and compact action sets. We restrict attention to unicost MDPs, which form a class that contains all the weakly communicating MDPs. The unicost MDPs are characterized as those MDPs for which there exists a solution to the single optimality equation. We address the problem of uniqueness and stability of minimizing Markov actions. Our main result asserts that when we endow the set of unicost MDPs with a certain natural metric, under which it is complete, then the class of MDPs with essentially unique and stable minimizing Markov actions contains the intersection of countably many open dense sets (hence is itself dense). Thus, the property of having essentially unique and stable minimizing Markov actions is generic for unicost MDPs.

Learning Representation and Control in Markov Decision Processes: New Frontiers

This paper describes a novel machine learning framework for solving sequential decision problems called Markov decision processes (MDPs) by iteratively computing low-dimensional representations and approximately optimal policies. A unified mathematical framework for learning representation and optimal control in MDPs is presented based on a class of singular operators called Laplacians, whose matrix representations have nonpositive off-diagonal elements and zero row sums. Exact solutions of discounted and average-reward MDPs are expressed in terms of a generalized spectral inverse of the Laplacian called the Drazin inverse . A generic algorithm called representation policy iteration (RPI) is presented which interleaves computing low-dimensional representations and approximately optimal policies. Two approaches for dimensionality reduction of MDPs are described based on geometric and reward-sensitive regularization, whereby low-dimensional representations are formed by diagonalization or dilation of Laplacian operators. Model-based and model-free variants of the RPI algorithm are presented; they are also compared experimentally on discrete and continuous MDPs. Some directions for future work are finally outlined.

Robust Control of Markov Decision Processes with Uncertain Transition Matrices

Optimal solutions to Markov decision problems may be very sensitive with respect to the state transition probabilities. In many practical problems, the estimation of these probabilities is far from accurate. Hence, estimation errors are limiting factors in applying Markov decision processes to real-world problems. We consider a robust control problem for a finite-state, finite-action Markov decision process, where uncertainty on the transition matrices is described in terms of possibly nonconvex sets. We show that perfect duality holds for this problem, and that as a consequence, it can be solved with a variant of the classical dynamic programming algorithm, the “robust dynamic programming” algorithm. We show that a particular choice of the uncertainty sets, involving likelihood regions or entropy bounds, leads to both a statistically accurate representation of uncertainty, and a complexity of the robust recursion that is almost the same as that of the classical recursion. Hence, robustness can be added at practically no extra computing cost. We derive similar results for other uncertainty sets, including one with a finite number of possible values for the transition matrices. We describe in a practical path planning example the benefits of using a robust strategy instead of the classical optimal strategy; even if the uncertainty level is only crudely guessed, the robust strategy yields a much better worst-case expected travel time.

AN RBF BASED NEURO-DYNAMIC APPROACH FOR THE CONTROL OF STOCHASTIC DYNAMIC SYSTEMS

This paper presents a neuro-dynamic programming methodology for the control of markov decision processes. The proposed method can be considered as a variant of the optimistic policy iteration, where radial basis function (RBF) networks are employed as a compact representation of the cost-to-go function and the Λ-LSPE is used for policy evaluation. We also emphasize the reformulation of the Bellman equation around the post-decision state in order to circumvent the calculation of the expectation. The proposed algorithm is applied to a retailer-inventory management problem.

The Complexity of Decentralized Control of Markov Decision Processes

We consider decentralized control of Markov decision processes and give complexity bounds on the worst-case running time for algorithms that find optimal solutions. Generalizations of both the fully observable case and the partially observable case that allow for decentralized control are described. For even two agents, the finite-horizon problems corresponding to both of these models are hard for nondeterministic exponential time. These complexity results illustrate a fundamental difference between centralized and decentralized control of Markov decision processes. In contrast to the problems involving centralized control, the problems we consider provably do not admit polynomial-time algorithms. Furthermore, assuming EXP ≠ NEXP, the problems require superexponential time to solve in the worst case.

Recursive adaptive control of Markov decision processes with the average reward criterion

We are concerned with Markov decision processes with Borel state and action spaces; the transition law and the reward function depend on anunknown parameter. In this framework, we study therecursive adaptive nonstationary value iteration policy, which is proved to be optimal under thesame conditions usually imposed to obtain the optimality of other well-knownnonrecursive adaptive policies. The results are illustrated by showing the existence of optimal adaptive policies for a class of additive-noise systems with unknown noise distribution.