Ambiguous partially observable Markov decision processes: Structural results and applications

This paper considers the optimization problem for partially observable Markov decision processes (POMDPs) with the continuous state, observation, and action spaces. POMDPs with the discrete spaces have emerged as a promising approach to the decision systems with imperfect state information. However, in recent applications of POMDPs, there are many problems that have continuous states, observations, and actions. For such problems, due to the infinite dimensionality of the belief space, the existing studies usually discretize the continuous spaces with the sufficient or nonsufficient statistics, which may cause the curse of dimensionality and performance degradation. In this paper, based on the sensitivity analysis of the performance criteria, we have developed a simulation-based policy iteration algorithm to find the local optimal observation-based policy for POMDPs with the continuous spaces. The proposed algorithm needs none of the specific assumptions and prior information, and has a low computational complexity. One numerical example of the complicated multiple-input multiple-output beamforming problem shows that the algorithm has a significant performance improvement.

Many robotic tasks, such as mobile manipulation, often require interaction with unstructured environments and are subject to imperfect sensing and actuation. This brings substantial uncertainty into the problems. Reasoningunder this uncertainty can provide higher level of robustness but is computationally significantly more challenging. More specifically, sequential decision making under motion and sensing uncertainty can be formulated in a principled form as a Partially Observable Markov Decision Process (POMDP). Solving POMDPs exactly is computationally intractable due to their exponential complexity with the number of states and the depth of the planning horizon, so called, the curse of dimensionality and the curse ofhistory, respectively. In this work, we propose a novel search-based robust planning framework that sample-efficiently finds a solution with theoretical suboptimalitybounds by leveraging multiple heuristics that are designed using domain knowledge. The main contributions of this work can be summarized as follows: 1) it works with generative models, in the absence of mathematical models, similarly to the reinforcement learning paradigm, but 2) it still exhibits high sample-efficiency by bootstrapping with the domain knowledge in the form of heuristics, and 3) it is empowered with effective guidance of the search toward the goal through the systematic employment of multipleheuristics. Its solution can be returned in an anytime fashion, and converges to the bounded suboptimality with theoretical guarantees. We validate the proposed frameworks through simulation and robot experiments,which includes 2D rover navigation, PR2 parts assembly, and full size mobile manipulator truck unloading tasks. The truck unloading task is especially interesting, where the custom-built robot is to unload up to several thousands of boxes from a truck. It is a challenging real-world manipulation problem in continuous space with excessively high uncertainty, and we demonstrate the efficacy of our approach in such a domain.

Making principled decisions in the presence of uncertainty is often facilitated by Partially Observable Markov Decision Processes (POMDPs). Despite tremendous advances in POMDP solvers, finding good policies with large action spaces remains difficult. To alleviate this difficulty, this paper presents an on-line approximate solver, called Quantile-Based Action Selector (QBASE). It uses quantile-statistics to adaptively evaluate a small subset of the action space without sacrificing the quality of the generated decision strategies by much. Experiments on four different robotics tasks with up to 10,000 actions indicate that QBASE can generate substantially better strategies than a state-of-the-art method.

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.

Resolving structural uncertainty in natural resources management using POMDP 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) provide a general framework for AI planning, but they lack the structure for representing real world planning problems in a convenient and efficient way. Representations built on logic allow for problems to be specified in a compact and transparent manner. Moreover, decision making algorithms can assume and exploit structure found in the state space, actions, observations, and success criteria, and can solve with relative efficiency problems with large state spaces. In recent years researchers have sought to combine the benefits of logic with the expressiveness of POMDPs. In this paper, we show how to build upon and extend the results in this fusing of logic and decision theory. In particular, we present a compact representation of POMDPs and a method to update beliefs after actions and observations. The key contribution is our compact representation of belief states and of the operations used to update them. We then use heuristic search to find optimal plans that maximize expected total reward given an initial belief state

&lt;p&gt;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.&lt;/p&gt;

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.

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 ...

This paper proposes a novel direct policy search (DPS) method with model selection for partially observed Markov decision processes (POMDPs). DPSs have been standard for learning POMDPs due to their computational efficiency and natural ability to maximize total rewards. An important open challenge for the best use of DPS methods is model selection, i.e., determination of the proper dimensionality of hidden states and complexity of policy functions, to mitigate overfitting in highly-flexible model representations of POMDPs. This paper bridges Bayesian inference and reward maximization and derives marginalized weighted log-likelihood~(MWL) for POMDPs which takes both advantages of Bayesian model selection and DPS. Then we propose factorized asymptotic Bayesian policy search (FABPS) to explore the model and the policy which maximizes MWL by expanding recently-developed factorized asymptotic Bayesian inference. Experimental results show that FABPS outperforms state-of-the-art model selection methods for POMDPs, with respect both to model selection and to expected total rewards.

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.

Partially Observable Markov Decision Processes (POMDPs) are a powerful framework for planning under uncertainty. They allow to model state uncertainty as a belief probability distribution. Approximate solvers based on Monte Carlo sampling show great success to relax the computational demand and perform online planning. However, scaling to complex realistic domains with many actions and long planning horizons is still a major challenge, and a key point to achieve good performance is guiding the action-selection process with domain-dependent policy heuristics which are tailored for the specific application domain. We propose to learn high-quality heuristics from POMDP traces of executions generated by any solver. We convert the belief-action pairs to a logical semantics, and exploit data- and time-efficient Inductive Logic Programming (ILP) to generate interpretable belief-based policy specifications, which are then used as online heuristics. We evaluate thoroughly our methodology on two notoriously challenging POMDP problems, involving large action spaces and long planning horizons, namely, rocksample and pocman. Considering different state-of-the-art online POMDP solvers, including POMCP, DESPOT and AdaOPS, we show that learned heuristics expressed in Answer Set Programming (ASP) yield performance superior to neural networks and similar to optimal handcrafted task-specific heuristics within lower computational time. Moreover, they well generalize to more challenging scenarios not experienced in the training phase (e.g., increasing rocks and grid size in rocksample, incrementing the size of the map and the aggressivity of ghosts in pocman).

Foraging mammals exhibit a familiar yet poorly characterized phenomenon, ‘alternation’, a pause to sniff in the air preceded by the animal rearing on its hind legs or raising its head. Rodents spontaneously alternate in the presence of airflow, suggesting that alternation serves an important role during plume-tracking. To test this hypothesis, we combine fully resolved simulations of turbulent odor transport and Bellman optimization methods for decision-making under partial observability. We show that an agent trained to minimize search time in a realistic odor plume exhibits extensive alternation together with the characteristic cast-and-surge behavior observed in insects. Alternation is linked with casting and occurs more frequently far downwind of the source, where the likelihood of detecting airborne cues is higher relative to ground cues. Casting and alternation emerge as complementary tools for effective exploration with sparse cues. A model based on marginal value theory captures the interplay between casting, surging, and alternation.

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.