Optimal Cost Almost-sure Reachability in POMDPs

We consider partially observable Markov decision processes (POMDPs) with a set of target states and every transition is associated with an integer cost. The optimization objective we study asks to minimize the expected total cost till the target set is reached, while ensuring that the target set is reached almost-surely (with probability 1). We show that for integer costs approximating the optimal cost is undecidable. For positive costs, our results are as follows: (i) we establish matching lower and upper bounds for the optimal cost and the bound is double exponential; (ii) we show that the problem of approximating the optimal cost is decidable and present approximation algorithms developing on the existing algorithms for POMDPs with finite-horizon objectives. While the worst-case running time of our algorithm is double exponential, we also present efficient stopping criteria for the algorithm and show experimentally that it performs well in many examples of interest.Comment: Full Version of Optimal Cost Almost-sure Reachability in POMDPs, AAAI 2015. arXiv admin note: text overlap with arXiv:1207.4166 by other author

Sensor Synthesis for POMDPs with Reachability Objectives

Partially observable Markov decision processes (POMDPs) are widely used in probabilistic planning problems in which an agent interacts with an environment using noisy and imprecise sensors. We study a setting in which the sensors are only partially defined and the goal is to synthesize "weakest" additional sensors, such that in the resulting POMDP, there is a small-memory policy for the agent that almost-surely (with probability~1) satisfies a reachability objective. We show that the problem is NP-complete, and present a symbolic algorithm by encoding the problem into SAT instances. We illustrate trade-offs between the amount of memory of the policy and the number of additional sensors on a simple example. We have implemented our approach and consider three classical POMDP examples from the literature, and show that in all the examples the number of sensors can be significantly decreased (as compared to the existing solutions in the literature) without increasing the complexity of the policies.Comment: arXiv admin note: text overlap with arXiv:1511.0845

IST Austria Technical Report

We consider partially observable Markov decision processes (POMDPs) with a set of target states and every transition is associated with an integer cost. The optimization objective we study asks to minimize the expected total cost till the target set is reached, while ensuring that the target set is reached almost-surely (with probability 1). We show that for integer costs approximating the optimal cost is undecidable. For positive costs, our results are as follows: (i) we establish matching lower and upper bounds for the optimal cost and the bound is double exponential; (ii) we show that the problem of approximating the optimal cost is decidable and present approximation algorithms developing on the existing algorithms for POMDPs with finite-horizon objectives. While the worst-case running time of our algorithm is double exponential, we also present efficient stopping criteria for the algorithm and show experimentally that it performs well in many examples of interest

Parameter-Independent Strategies for pMDPs via POMDPs

Markov Decision Processes (MDPs) are a popular class of models suitable for solving control decision problems in probabilistic reactive systems. We consider parametric MDPs (pMDPs) that include parameters in some of the transition probabilities to account for stochastic uncertainties of the environment such as noise or input disturbances. We study pMDPs with reachability objectives where the parameter values are unknown and impossible to measure directly during execution, but there is a probability distribution known over the parameter values. We study for the first time computing parameter-independent strategies that are expectation optimal, i.e., optimize the expected reachability probability under the probability distribution over the parameters. We present an encoding of our problem to partially observable MDPs (POMDPs), i.e., a reduction of our problem to computing optimal strategies in POMDPs. We evaluate our method experimentally on several benchmarks: a motivating (repeated) learner model; a series of benchmarks of varying configurations of a robot moving on a grid; and a consensus protocol.Comment: Extended version of a QEST 2018 pape

Stochastic Shortest Path with Energy Constraints in POMDPs

We consider partially observable Markov decision processes (POMDPs) with a set of target states and positive integer costs associated with every transition. The traditional optimization objective (stochastic shortest path) asks to minimize the expected total cost until the target set is reached. We extend the traditional framework of POMDPs to model energy consumption, which represents a hard constraint. The energy levels may increase and decrease with transitions, and the hard constraint requires that the energy level must remain positive in all steps till the target is reached. First, we present a novel algorithm for solving POMDPs with energy levels, developing on existing POMDP solvers and using RTDP as its main method. Our second contribution is related to policy representation. For larger POMDP instances the policies computed by existing solvers are too large to be understandable. We present an automated procedure based on machine learning techniques that automatically extracts important decisions of the policy allowing us to compute succinct human readable policies. Finally, we show experimentally that our algorithm performs well and computes succinct policies on a number of POMDP instances from the literature that were naturally enhanced with energy levels.Comment: Technical report accompanying a paper published in proceedings of AAMAS 201

Stochastic Control via Entropy Compression

We consider an agent trying to bring a system to an acceptable state by repeated probabilistic action. Several recent works on algorithmizations of the Lovasz Local Lemma (LLL) can be seen as establishing sufficient conditions for the agent to succeed. Here we study whether such stochastic control is also possible in a noisy environment, where both the process of state-observation and the process of state-evolution are subject to adversarial perturbation (noise). The introduction of noise causes the tools developed for LLL algorithmization to break down since the key LLL ingredient, the sparsity of the causality (dependence) relationship, no longer holds. To overcome this challenge we develop a new analysis where entropy plays a central role, both to measure the rate at which progress towards an acceptable state is made and the rate at which noise undoes this progress. The end result is a sufficient condition that allows a smooth tradeoff between the intensity of the noise and the amenability of the system, recovering an asymmetric LLL condition in the noiseless case.Comment: 18 page

Expectation Optimization with Probabilistic Guarantees in POMDPs with Discounted-sum Objectives

Partially-observable Markov decision processes (POMDPs) with discounted-sum payoff are a standard framework to model a wide range of problems related to decision making under uncertainty. Traditionally, the goal has been to obtain policies that optimize the expectation of the discounted-sum payoff. A key drawback of the expectation measure is that even low probability events with extreme payoff can significantly affect the expectation, and thus the obtained policies are not necessarily risk-averse. An alternate approach is to optimize the probability that the payoff is above a certain threshold, which allows obtaining risk-averse policies, but ignores optimization of the expectation. We consider the expectation optimization with probabilistic guarantee (EOPG) problem, where the goal is to optimize the expectation ensuring that the payoff is above a given threshold with at least a specified probability. We present several results on the EOPG problem, including the first algorithm to solve it.Comment: Full version of a paper published at IJCAI/ECAI 201

LNCS

In the analysis of reactive systems a quantitative objective assigns a real value to every trace of the system. The value decision problem for a quantitative objective requires a trace whose value is at least a given threshold, and the exact value decision problem requires a trace whose value is exactly the threshold. We compare the computational complexity of the value and exact value decision problems for classical quantitative objectives, such as sum, discounted sum, energy, and mean-payoff for two standard models of reactive systems, namely, graphs and graph games