An Oracle-Guided Approach to Constrained Policy Synthesis Under Uncertainty

<p>Partially Observable Markov Decision Processes (POMDPs) is a modeling framework that allow decision makers to make decisions in uncertain environments. This framework is particularly suited for sequential decision making problems for which the uncertainties are revealed in stages. Constrained POMDPs (CPOMDPs) extend this framework by also accounting for various restrictions in the environment. This PhD thesis focuses on POMDPs, CPOMDPs and its extensions, proposes novel solution methods and studies applications of POMDPs in disease screening. First, we focus on developing efficient versions of the well-known algorithms for POMDPs. POMDPs are notoriously difficult to solve optimally, and exact solution algorithms typically fail to solve the instances with more than a few core states. In this regard, we consider scalable algorithms solve the POMDP problems. Specifically, we propose distributed and parallel versions of the exact solution methods such as Monahan's algorithm and the incremental pruning algorithm as well as the heuristics such as Lovejoy's upper bound and lower bound methods. Next, we study the CPOMDPs and propose linear programming-based solution methods as a flexible approach for those. We perform a detailed numerical study with five different problem instances to showcase the viability of the proposed solution approach for CPOMDPs. Lastly, we propose multi-objective CPOMDP models for the breast cancer screening problem. These CPOMDP models extend the previously proposed POMDP models for breast cancer screening by considering different screening modalities and simultaneously optimizing for expected total quality-adjusted life years maximization and life-time risk minimization objectives.</p>

Resolving structural uncertainty in natural resources management using POMDP approaches

Aiming to reduce energy consumption, minimize transmission latency, and optimize the long-term performance of Wireless Sensor Networks (WSN), this paper proposes an improved routing model of Directed Diffusion. We formulated the routing issue as a partially observable Markov decision process (POMDP), such that the optimal nodes selection policy yielded by maximizing the expected reward. Since the solution of POMDP is ready-made, the attention is concentrated only on the formulation of the routing issue into a POMDP model, more specifically, the definition of the tuple which contains states, actions, observations, transition probabilities, observation probabilities, and reward function. Our experimental study shows that the POMDP-based Directed Diffusion routing outperforms other routing techniques in terms of three widely used metrics including average dissipated energy, average delay, and average lifetime.

Ambiguous partially observable Markov decision processes: Structural results and applications

In this article, we study the problem of controlling a partially observed Markov decision process (POMDP) to either aid or hinder the estimation of its state trajectory. We encode the estimation objectives via the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">smoother entropy</i> , which is the conditional entropy of the state trajectory given measurements and controls. Consideration of the smoother entropy contrasts with previous approaches that instead resort to marginal (or instantaneous) state entropies due to tractability concerns. By establishing novel expressions for the smoother entropy in terms of the POMDP belief state, we show that both the problems of minimizing and maximizing the smoother entropy in POMDPs can surprisingly be reformulated as belief-state Markov decision processes with concave cost and value functions. The significance of these reformulations is that they render the smoother entropy a tractable optimization objective, with structural properties amenable to the use of standard POMDP solution techniques for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">both</i> active estimation and obfuscation. Simulations illustrate that optimization of the smoother entropy leads to superior trajectory estimation and obfuscation compared to alternative approaches.

The framework of partially observable Markov decision processes (POMDPs) offers a standard approach to model uncertainty in many robot tasks. Traditionally, POMDPs are formulated with optimality objectives. In this article, we study a different formulation of POMDPs with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Boolean objectives</i> . For robotic domains that require a correctness guarantee of accomplishing tasks, Boolean objectives are natural formulations. We investigate the problem of POMDPs with a common Boolean objective: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">safe reachability</i> , requiring that the robot eventually reaches a goal state with a probability above a threshold while keeping the probability of visiting unsafe states below a different threshold. Our approach builds upon the previous work that represents POMDPs with Boolean objectives using symbolic constraints. We employ a satisfiability modulo theories (SMTs) solver to efficiently search for solutions, i.e., policies or conditional plans that specify the action to take contingent on every possible event. A full policy or conditional plan is generally expensive to compute. To improve computational efficiency, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">partial conditional plans</i> that cover sampled events to approximate a full conditional plan. Our approach constructs a partial conditional plan parameterized by a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">replanning probability</i> . We prove that the failure rate of the constructed partial conditional plan is bounded by the replanning probability. Our approach allows users to specify an appropriate bound on the replanning probability to balance efficiency and correctness. Moreover, we update this bound properly to quickly detect whether the current partial conditional plan meets the bound and avoid unnecessary computation. In addition, to further improve the efficiency, we cache partial conditional plans for sampled belief states and reuse these cached plans if possible. We validate our approach in several robotic domains. The results show that our approach outperforms a previous policy synthesis approach for POMDPs with safe-reachability objectives in these domains. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article was motivated by two observations. On the one hand, in robotics applications where uncertainty in sensing and actions is present, the solution to the classical partially observable Markov decision process (POMDP) formulation is expensive to compute in general. On the other hand, in certain practical scenarios, formulations other than the classical POMDP make a lot of sense and can provide flexibility in balancing efficiency and correctness. This article considers a modified POMDP formulation that includes a Boolean objective, namely safe reachability. This article uses the notion of a partial conditional plan. Rather than explicitly enumerating all possible observations to construct a full conditional plan, this work samples a subset of all observations to ensure bounded replanning probability. Our theoretical and empirical results show that the failure rate of the constructed partial conditional plan is bounded by the replanning probability. Moreover, these partial conditional plans can be cached to further improve the performance. Our results suggest that for domains where replanning is easy, increasing the replanning probability bound usually leads to better scalability, and for domains where replanning is difficult or impossible in some states, we can decrease the bound and allocate more computation time to achieve a higher success rate. Hence, in certain cases, the practitioner can take advantage of their knowledge of the problem domain to scale to larger problems. Preliminary physical experiments suggest that this approach is applicable to real-world robotic domains, but it requires a discrete representation of the workspace. How to deal with continuous workspace directly is an interesting future direction.

Partially observable Markov decision processes (POMDPs) have recently become popular among many AI researchers because they serve as a natural model for planning under uncertainty. Value iteration ...

Iteratively solving a set of linear programs (LPs) is a common strategy for solving various decision-making problems in Artificial Intelligence, such as planning in multi-objective or partially observable Markov Decision Processes (MDPs). A prevalent feature is that the solutions to these LPs become increasingly similar as the solving algorithm converges, because the solution computed by the algorithm approaches the fixed point of a Bellman backup operator. In this paper, we propose to speed up the solving process of these LPs by bootstrapping based on similar LPs solved previously. We use these LPs to initialize a subset of relevant LP constraints, before iteratively generating the remaining constraints. The resulting algorithm is the first to consider such information sharing across iterations. We evaluate our approach on planning in Multi-Objective MDPs (MOMDPs) and Partially Observable MDPs (POMDPs), showing that it solves fewer LPs than the state of the art, which leads to a significant speed-up. Moreover, for MOMDPs we show that our method scales better in both the number of states and the number of objectives, which is vital for multi-objective planning.

Partially observable Markov decision processes (POMDPs) are models for sequential decision-making under uncertainty and incomplete information. Machine learning methods typically train recurrent neural networks (RNN) as effective representations of POMDP policies that can efficiently process sequential data. However, it is hard to verify whether the POMDP driven by such RNN-based policies satisfies safety constraints, for instance, given by temporal logic specifications. We propose a novel method that combines techniques from machine learning with the field of formal methods: training an RNN-based policy and then automatically extracting a so-called finite-state controller (FSC) from the RNN. Such FSCs offer a convenient way to verify temporal logic constraints. Implemented on a POMDP, they induce a Markov chain, and probabilistic verification methods can efficiently check whether this induced Markov chain satisfies a temporal logic specification. Using such methods, if the Markov chain does not satisfy the specification, a byproduct of verification is diagnostic information about the states in the POMDP that are critical for the specification. The method exploits this diagnostic information to either adjust the complexity of the extracted FSC or improve the policy by performing focused retraining of the RNN. The method synthesizes policies that satisfy temporal logic specifications for POMDPs with up to millions of states, which are three orders of magnitude larger than comparable approaches.

In recent years, Markov decision processes (MDPs) and partially observable Markov decision processes (POMDPs) have found important applications to medical decision making in the context of prevention, screening, and treatment of diseases. In this chapter, we provide a review of state-of-the-art models and methods that have been applied to chronic diseases. We provide a tutorial about how to formulate and solve these important problems emphasizing some of the challenges specific to chronic diseases such as diabetes, heart disease, and cancer. Then, we illustrate important considerations for model formulation and solution methods through two examples. The first example is an MDP model for optimal control of drug treatment decisions for managing the risk of heart disease and stroke in patients with type 2 diabetes. The second example is a POMDP model for optimal design of biomarker-based screening policies in the context of prostate cancer. We end the chapter with a discussion of the challenges of using MDPs and POMDPs for medical contexts and describe some important future directions for research.

Motion planning with imperfect state information is a crucial capability for autonomous robots to operate reliably in uncertain and dynamic environments. Partially observable Markov decision processes (POMDPs) provide a principled general framework for planning under uncertainty. Using probabilistic sampling, point-based POMDP solvers have drastically improved the speed of POMDP planning, enabling us to handle moderately complex robotic tasks. However, robot motion planning tasks with long time horizons remains a severe obstacle for even the fastest point-based POMDP solvers today. This paper proposes Milestone Guided Sampling (MiGS), a new point-based POMDP solver, which exploits state space information to reduce effective planning horizons. MiGS samples a set of points, called milestones, from a robot’s state space and constructs a simplified representation of the state space from the sampled milestones. It then uses this representation of the state space to guide sampling in the belief space and tries to capture the essential features of the belief space with a small number of sampled points. Preliminary results are very promising. We tested MiGS in simulation on several difficult POMDPs that model distinct robotic tasks with long time horizons in both 2-D and 3-D environments. These POMDPs are impossible to solve with the fastest point-based solvers today, but MiGS solved them in a few minutes.

Profit Sharing is one of the reinforcement learning methods. An agent, as a learner, selects an action with a state-action value and receives rewards when it reaches a goal state. Then it distributes receiving rewards to state-action values. This paper discusses how to set the initial value of a state-action value. A distribution function ƒ(x) is called as the reinforcement function. On Profit Sharing, an agent learns a policy by distributing rewards with the reinforcement function. On Markov Decision Processes (MDPs), the reinforcement function ƒ(x) = 1/Lx is useful, and on Partially Observable Markov Decision Processes (POMDPs), ƒ(x) = 1/Lw is useful, where L is the sufficient number of rules at each state, and W is the length of an episode. If episodes are always long, the value of the reinforcement function is little. So the differences of rule values become little, and the agent learns little by using the roulette selection as an action selection. This problem is called as Learning Speed Problem. If the value of the reinforcement function for an action is very higher than its state-action value, an agent will not select other action. There is a problem when its action is not a optimal action. This problem is called as Past Experiences Problem. This paper shows that both Learning Speed Problem and Past Experiences Problem are caused by the bad setting between the initial values of a state-action values and the function values of a reinforcement function. We propose how to set the initial values of a state-action values at each state. The experiment shows that an agent can learn correctly even if the length of episode is large. And shows the effectiveness on both MDPs and POMDPs. Our proposed method focuses on the initialization of state-action values and does not limit reinforcement functions. So it can apply to any reinforcement function.

The effectiveness of stochastic power management relies on the accurate system and workload model and effective policy optimization. Workload modeling is a machine learning procedure that finds the intrinsic pattern of the incoming tasks based on the observed workload attributes. Markov Decision Process (MDP) based model has been widely adopted for stochastic power management because it delivers provable optimal policy. Given a sequence of observed workload attributes, the hidden Markov model (HMM) of the workload is trained. If the observed workload attributes and states in the workload model do not have one-to-one correspondence, the MDP becomes a Partially Observable Markov Decision Process (POMDP). This paper presents a framework of modeling and optimization for stochastic power management using HMM and POMDP. The proposed technique discovers the HMM of the workload by maximizing the likelihood of the observed attribute sequence. The POMDP optimization is formulated and solved as a quadraticly constrained linear programming (QCLP). Compared with traditional optimization technique, which is based on value iteration, the QCLP based optimization provides superior policy by enabling stochastic control.

Offline reinforcement learning (RL) is valuable in settings where online interactions with an environment are impractical. While such settings are often partially-observable, existing offline RL methods typically focus on fully-observable Markov decision processes (MDPs) rather than partially-observable MDPs (POMDPs). To help close that gap, we present an offline RL algorithm for POMDPs that leverages expert policies from simpler, fully-observable versions of environments in an asymmetric learning setting. We provide theoretical grounding for how overlap between MDPs and POMDPs can be exploited to improve learning in the partially-observable setting, and our experiments empirically demonstrate that our method significantly improves performance compared to existing state-of-the-art MDP offline RL algorithms.

In this paper, we consider the problem of controlling a partially observed Markov decision process (POMDP) in order to actively estimate its state trajectory over a fixed horizon with minimal uncertainty. We pose a novel active smoothing problem in which the objective is to directly minimise the smoother entropy, that is, the conditional entropy of the (joint) state trajectory distribution of concern in fixed-interval Bayesian smoothing. Our formulation contrasts with prior active approaches that minimise the sum of conditional entropies of the (marginal) state estimates provided by Bayesian filters. By establishing a novel form of the smoother entropy in terms of the POMDP belief (or information) state, we show that our active smoothing problem can be reformulated as a (fully observed) Markov decision process with a value function that is concave in the belief state. The concavity of the value function is of particular importance since it enables the approximate solution of our active smoothing problem using piecewise-linear function approximations in conjunction with standard POMDP solvers. We illustrate the approximate solution of our active smoothing problem in simulation and compare its performance to alternative approaches based on minimising marginal state estimate uncertainties.