We introduce a new programming paradigm called oracle-guided decision programming in which a program specifies a Markov Decision Process (MDP) and the language provides a universal policy. We prototype a new programming language, Dodona, that manifests this paradigm using a primitive 'choose' representing nondeterministic choice. The Dodona interpreter returns either a value or a choicepoint that includes a lossless encoding of all information necessary in principle to make an optimal decision. Meta-interpreters query Dodona's (neural) oracle on these choicepoints to get policy and value estimates, which they can use to perform heuristic search on the underlying MDP. We demonstrate Dodona's potential for zero-shot heuristic guidance by meta-learning over hundreds of synthetic tasks that simulate basic operations over lists, trees, Church datastructures, polynomials, first-order terms and higher-order terms.
School of Electrical and Computer Engineering, Purdue University（普渡大学电气与计算机工程学院）
We select policies for large Markov Decision Processes (MDPs) with compact first-order representations. We find policies that generalize well as the number of objects in the domain grows, potentially without bound. Existing dynamic-programming approaches based on flat, propositional, or first-order representations either are impractical here or do not naturally scale as the number of objects grows without bound. We implement and evaluate an alternative approach that induces first-order policies using training data constructed by solving small problem instances using PGraphplan (Blum & Langford…
The use of deep neural networks as function approximators has led to striking progress for reinforcement learning algorithms and applications. Yet the knowledge we have on decision boundary geometry and the loss landscape of neural policies is still quite limited. In this paper we propose a framework to investigate the decision boundary and loss landscape similarities across states and across MDPs. We conduct experiments in various games from Arcade Learning Environment, and discover that high sensitivity directions for neural policies are correlated across MDPs. We argue that these high sensi…
Policies of Markov Decision Processes (MDPs) determine the next action to execute from the current state and, possibly, the history (the past states). When the number of states is large, succinct representations are often used to compactly represent both the MDPs and the policies in a reduced amount of space. In this paper, some problems related to the size of succinctly represented policies are analyzed. Namely, it is shown that some MDPs have policies that can only be represented in space super-polynomial in the size of the MDP, unless the polynomial hierarchy collapses. This fact motivates …
Current work in explainable reinforcement learning generally produces policies in the form of a decision tree over the state space. Such policies can be used for formal safety verification, agent behavior prediction, and manual inspection of important features. However, existing approaches fit a decision tree after training or use a custom learning procedure which is not compatible with new learning techniques, such as those which use neural networks. To address this limitation, we propose a novel Markov Decision Process (MDP) type for learning decision tree policies: Iterative Bounding MDPs (…
In this paper we propose a method that learns to play Pac-Man. We define a set of high-level observation and action modules. Actions are temporally extended, and multiple action modules may be in effect concurrently. A decision of the agent is represented as a rule-based policy. For learning, we apply the cross-entropy method, a recent global optimization algorithm. The learned policies reached better score than the hand-crafted policy, and neared the score of average human players. We argue that learning is successful mainly because (i) the policy space includes the combination of individual …
KAIST（韩国科学技术院）；Department of Computer Science（计算机科学系）
Markov decision processes (MDPs) are widely used in modeling decision making problems in stochastic environments. However, precise specification of the reward functions in MDPs is often very difficult. Recent approaches have focused on computing an optimal policy based on the minimax regret criterion for obtaining a robust policy under uncertainty in the reward function. One of the core tasks in computing the minimax regret policy is to obtain the set of all policies that can be optimal for some candidate reward function. In this paper, we propose an efficient algorithm that exploits the geome…
In these notes we will tackle the problem of finding optimal policies for Markov decision processes (MDPs) which are not fully known to us. Our intention is to slowly transition from an offline setting to an online (learning) setting. Namely, we are moving towards reinforcement learning.
Singapore Management University（新加坡管理大学）；University of Toronto（多伦多大学）
Recent advances in deep learning have enabled optimization of deep reactive policies (DRPs) for continuous MDP planning by encoding a parametric policy as a deep neural network and exploiting automatic differentiation in an end-to-end model-based gradient descent framework. This approach has proven effective for optimizing DRPs in nonlinear continuous MDPs, but it requires a large number of sampled trajectories to learn effectively and can suffer from high variance in solution quality. In this work, we revisit the overall model-based DRP objective and instead take a minorization-maximization p…
Technion（以色列理工学院）；National University of Singapore（新加坡国立大学）
Robust Markov decision processes (MDPs) provide a general framework to model decision problems where the system dynamics are changing or only partially known. Efficient methods for some \texttt{sa}-rectangular robust MDPs exist, using its equivalence with reward regularized MDPs, generalizable to online settings. In comparison to \texttt{sa}-rectangular robust MDPs, \texttt{s}-rectangular robust MDPs are less restrictive but much more difficult to deal with. Interestingly, recent works have established the equivalence between \texttt{s}-rectangular robust MDPs and policy regularized MDPs. But …
University of New Hampshire（新罕布什尔大学）；City University of Hong Kong（香港城市大学）
We address the problem of computing reliable policies in reinforcement learning problems with limited data. In particular, we compute policies that achieve good returns with high confidence when deployed. This objective, known as the \emph{percentile criterion}, can be optimized using Robust MDPs~(RMDPs). RMDPs generalize MDPs to allow for uncertain transition probabilities chosen adversarially from given ambiguity sets. We show that the RMDP solution's sub-optimality depends on the spans of the ambiguity sets along the value function. We then propose new algorithms that minimize the span of a…
Masaryk University（马萨里克大学）；Institute of Science and Technology Austria（奥地利科学技术研究所）
Markov decision processes (MDPs) are the defacto frame-work for sequential decision making in the presence ofstochastic uncertainty. A classical optimization criterion forMDPs is to maximize the expected discounted-sum pay-off, which ignores low probability catastrophic events withhighly negative impact on the system. On the other hand,risk-averse policies require the probability of undesirableevents to be below a given threshold, but they do not accountfor optimization of the expected payoff. We consider MDPswith discounted-sum payoff with failure states which repre-sent catastrophic outcomes…
Tsinghua University（清华大学）；Carnegie Mellon University（卡内基梅隆大学）；University of Toronto（多伦多大学）
Deep Reinforcement Learning (RL) is successful in solving many complex Markov Decision Processes (MDPs) problems. However, agents often face unanticipated environmental changes after deployment in the real world. These changes are often spurious and unrelated to the underlying problem, such as background shifts for visual input agents. Unfortunately, deep RL policies are usually sensitive to these changes and fail to act robustly against them. This resembles the problem of domain generalization in supervised learning. In this work, we study this problem for goal-conditioned RL agents. We propo…
Markov decision processes (MDPs) are a popular model for performance analysis and optimization of stochastic systems. The parameters of stochastic behavior of MDPs are estimates from empirical observations of a system; their values are not known precisely. Different types of MDPs with uncertain, imprecise or bounded transition rates or probabilities and rewards exist in the literature. Commonly, analysis of models with uncertainties amounts to searching for the most robust policy which means that the goal is to generate a policy with the greatest lower bound on performance (or, symmetrically, …
A Relational Markov Decision Process (RMDP) is a first-order representation to express all instances of a single probabilistic planning domain with possibly unbounded number of objects. Early work in RMDPs outputs generalized (instance-independent) first-order policies or value functions as a means to solve all instances of a domain at once. Unfortunately, this line of work met with limited success due to inherent limitations of the representation space used in such policies or value functions. Can neural models provide the missing link by easily representing more complex generalized policies,…
Optimal policies in Markov decision processes (MDPs) are very sensitive to model misspecification. This raises serious concerns about deploying them in high-stake domains. Robust MDPs (RMDP) provide a promising framework to mitigate vulnerabilities by computing policies with worst-case guarantees in reinforcement learning. The solution quality of an RMDP depends on the ambiguity set, which is a quantification of model uncertainties. In this paper, we propose a new approach for optimizing the shape of the ambiguity sets for RMDPs. Our method departs from the conventional idea of constructing a …
Missingness-MDPs: Bridging the Theory of Missing Data and POMDPs
Ruhr University Bochum（博德姆鲁尔大学）；Brno University of Technology（布拉格技术大学）；Radboud University Nijmegen（拉德博德大学奈杰姆）；Harvard University（哈佛大学）；Eindhoven University of Technology（埃因霍温理工大学）
Google Research（谷歌研究院）；Inria（法国国家信息与自动化研究所）；CRIStAL, CNRS, Université de Lille（里尔大学法国国家科学研究中心CRIStAL实验室）
Traditionally, Reinforcement Learning (RL) aims at deciding how to act optimally for an artificial agent. We argue that deciding when to act is equally important. As humans, we drift from default, instinctive or memorized behaviors to focused, thought-out behaviors when required by the situation. To enhance RL agents with this aptitude, we propose to augment the standard Markov Decision Process and make a new mode of action available: being lazy, which defers decision-making to a default policy. In addition, we penalize non-lazy actions in order to encourage minimal effort and have agents focu…
California Institute of Technology（加州理工学院）；NASA Jet Propulsion Laboratory（NASA喷气推进实验室）
A large class of decision making under uncertainty problems can be described via Markov decision processes (MDPs) or partially observable MDPs (POMDPs), with application to artificial intelligence and operations research, among others. Traditionally, policy synthesis techniques are proposed such that a total expected cost or reward is minimized or maximized. However, optimality in the total expected cost sense is only reasonable if system behavior in the large number of runs is of interest, which has limited the use of such policies in practical mission-critical scenarios, wherein large deviat…
We present a method for solving implicit (factored) Markov decision processes (MDPs) with very large state spaces. We introduce a property of state space partitions which we call epsilon-homogeneity. Intuitively, an epsilon-homogeneous partition groups together states that behave approximately the same under all or some subset of policies. Borrowing from recent work on model minimization in computer-aided software verification, we present an algorithm that takes a factored representation of an MDP and an 0<=epsilon<=1 and computes a factored epsilon-homogeneous partition of the state space. Th…
Data61/CSIRO（澳大利亚联邦科学与工业研究组织数据61）；Research School of Computer Science, ANU（澳大利亚国立大学计算机科学研究学院）
Markov decision processes (MDPs) are the standard formalism for modelling sequential decision making in stochastic environments. Policy synthesis addresses the problem of how to control or limit the decisions an agent makes so that a given specification is met. In this paper we consider PCTL*, the probabilistic counterpart of CTL*, as the specification language. Because in general the policy synthesis problem for PCTL* is undecidable, we restrict to policies whose execution history memory is finitely bounded a priori. Surprisingly, no algorithm for policy synthesis for this natural and express…
University of Central Florida（中佛罗里达大学）；Air Force Research Laboratory（空军研究实验室）
The planning domain has experienced increased interest in the formal synthesis of decision-making policies. This formal synthesis typically entails finding a policy which satisfies formal specifications in the form of some well-defined logic. While many such logics have been proposed with varying degrees of expressiveness and complexity in their capacity to capture desirable agent behavior, their value is limited when deriving decision-making policies which satisfy certain types of asymptotic behavior in general system models. In particular, we are interested in specifying constraints on the s…
BBN Technologies（BBN技术公司）；University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）
To deal with the prohibitive complexity of calculating policies in Decentralized MDPs, researchers have proposed models that exploit structured agent interactions. Settings where most agent actions are independent except for few actions that affect the transitions and/or rewards of other agents can be modeled using Event-Driven Interactions with Complex Rewards (EDI-CR). Finding the optimal joint policy can be formulated as an optimization problem. However, existing formulations are too verbose and/or lack optimality guarantees. We propose a compact Mixed Integer Linear Program formulation of …
School of Electrical Engineering and Computer Science, Oregon State University（俄勒冈州立大学电气工程与计算机科学学院）；School of Electrical and Computer Engineering, Purdue University（普渡大学电气与计算机工程学院）
We study an approach to policy selection for large relational Markov Decision Processes (MDPs). We consider a variant of approximate policy iteration (API) that replaces the usual value-function learning step with a learning step in policy space. This is advantageous in domains where good policies are easier to represent and learn than the corresponding value functions, which is often the case for the relational MDPs we are interested in. In order to apply API to such problems, we introduce a relational policy language and corresponding learner. In addition, we introduce a new bootstrapping ro…
We consider the problem of learning the optimal threshold policy for control problems. Threshold policies make control decisions by evaluating whether an element of the system state exceeds a certain threshold, whose value is determined by other elements of the system state. By leveraging the monotone property of threshold policies, we prove that their policy gradients have a surprisingly simple expression. We use this simple expression to build an off-policy actor-critic algorithm for learning the optimal threshold policy. Simulation results show that our policy significantly outperforms othe…
Arizona State University（亚利桑那州立大学）；United Technologies Research Center（联合技术研究中心）；University of Massachusetts（马萨诸塞大学）
Integrated task and motion planning has emerged as a challenging problem in sequential decision making, where a robot needs to compute high-level strategy and low-level motion plans for solving complex tasks. While high-level strategies require decision making over longer time-horizons and scales, their feasibility depends on low-level constraints based upon the geometries and continuous dynamics of the environment. The hybrid nature of this problem makes it difficult to scale; most existing approaches focus on deterministic, fully observable scenarios. We present a new approach where the high…
In hierarchical planning for Markov decision processes (MDPs), temporal abstraction allows planning with macro-actions that take place at different time scale in form of sequential composition. In this paper, we propose a novel approach to compositional reasoning and hierarchical planning for MDPs under temporal logic constraints. In addition to sequential composition, we introduce a composition of policies based on generalized logic composition: Given sub-policies for sub-tasks and a new task expressed as logic compositions of subtasks, a semi-optimal policy, which is optimal in planning with…
We introduce a new approximate solution technique for first-order Markov decision processes (FOMDPs). Representing the value function linearly w.r.t. a set of first-order basis functions, we compute suitable weights by casting the corresponding optimization as a first-order linear program and show how off-the-shelf theorem prover and LP software can be effectively used. This technique allows one to solve FOMDPs independent of a specific domain instantiation; furthermore, it allows one to determine bounds on approximation error that apply equally to all domain instantiations. We apply this solu…
We consider a Reinforcement Learning setup where an agent interacts with an environment in observation-reward-action cycles without any (esp.\ MDP) assumptions on the environment. State aggregation and more generally feature reinforcement learning is concerned with mapping histories/raw-states to reduced/aggregated states. The idea behind both is that the resulting reduced process (approximately) forms a small stationary finite-state MDP, which can then be efficiently solved or learnt. We considerably generalize existing aggregation results by showing that even if the reduced process is not an…
Solving general Markov decision processes (MDPs) is a computationally hard problem. Solving finite-horizon MDPs, on the other hand, is highly tractable with well known polynomial-time algorithms. What drives this extreme disparity, and do problems exist that lie between these diametrically opposed complexities? In this paper we identify and analyse a sub-class of stochastic shortest path problems (SSPs) for general state-action spaces whose dynamics satisfy a particular drift condition. This construction generalises the traditional, temporal notion of a horizon via decreasing reachability: a p…
In many real-life reinforcement learning (RL) problems, deploying new policies is costly. In those scenarios, algorithms must solve exploration (which requires adaptivity) while switching the deployed policy sparsely (which limits adaptivity). In this paper, we go beyond the existing state-of-the-art on this problem that focused on linear Markov Decision Processes (MDPs) by considering linear Bellman-complete MDPs with low inherent Bellman error. We propose the ELEANOR-LowSwitching algorithm that achieves the near-optimal regret with a switching cost logarithmic in the number of episodes and l…
Relational Markov Decision Processes are a useful abstraction for complex reinforcement learning problems and stochastic planning problems. Recent work developed representation schemes and algorithms for planning in such problems using the value iteration algorithm. However, exact versions of more complex algorithms, including policy iteration, have not been developed or analyzed. The paper investigates this potential and makes several contributions. First we observe two anomalies for relational representations showing that the value of some policies is not well defined or cannot be calculated…
RWTH Aachen University（亚琛工业大学）；University of California, Berkeley（加州大学伯克利分校）
The verification problem in MDPs asks whether, for any policy resolving the nondeterminism, the probability that something bad happens is bounded by some given threshold. This verification problem is often overly pessimistic, as the policies it considers may depend on the complete system state. This paper considers the verification problem for partially observable MDPs, in which the policies make their decisions based on (the history of) the observations emitted by the system. We present an abstraction-refinement framework extending previous instantiations of the Lovejoy-approach. Our experime…
We propose a new approach to the problem of searching a space of policies for a Markov decision process (MDP) or a partially observable Markov decision process (POMDP), given a model. Our approach is based on the following observation: Any (PO)MDP can be transformed into an "equivalent" POMDP in which all state transitions (given the current state and action) are deterministic. This reduces the general problem of policy search to one in which we need only consider POMDPs with deterministic transitions. We give a natural way of estimating the value of all policies in these transformed POMDPs. P…
In many sequential decision-making problems we may want to manage risk by minimizing some measure of variability in costs in addition to minimizing a standard criterion. Conditional value-at-risk (CVaR) is a relatively new risk measure that addresses some of the shortcomings of the well-known variance-related risk measures, and because of its computational efficiencies has gained popularity in finance and operations research. In this paper, we consider the mean-CVaR optimization problem in MDPs. We first derive a formula for computing the gradient of this risk-sensitive objective function. We …
N-discount optimality was introduced as a hierarchical form of policy- and value-function optimality, with Blackwell optimality lying at the top level of the hierarchy Veinott (1969); Blackwell (1962). We formalize notions of myopic discount factors, value functions and policies in terms of Blackwell optimality in MDPs, and we provide a novel concept of regret, called Blackwell regret, which measures the regret compared to a Blackwell optimal policy. Our main analysis focuses on long horizon MDPs with sparse rewards. We show that selecting the discount factor under which zero Blackwell regret …
Deep active inference has been proposed as a scalable approach to perception and action that deals with large policy and state spaces. However, current models are limited to fully observable domains. In this paper, we describe a deep active inference model that can learn successful policies directly from high-dimensional sensory inputs. The deep learning architecture optimizes a variant of the expected free energy and encodes the continuous state representation by means of a variational autoencoder. We show, in the OpenAI benchmark, that our approach has comparable or better performance than d…
Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Computer Science Department, Princeton University（普林斯顿大学计算机科学系）；Department of Computer Science and Statistics, Stanford University（斯坦福大学计算机科学与统计系）
We compare the model-free reinforcement learning with the model-based approaches through the lens of the expressive power of neural networks for policies, $Q$-functions, and dynamics. We show, theoretically and empirically, that even for one-dimensional continuous state space, there are many MDPs whose optimal $Q$-functions and policies are much more complex than the dynamics. We hypothesize many real-world MDPs also have a similar property. For these MDPs, model-based planning is a favorable algorithm, because the resulting policies can approximate the optimal policy significantly better than…
Planning in Markov decision processes (MDPs) typically optimises the expected cost. However, optimising the expectation does not consider the risk that for any given run of the MDP, the total cost received may be unacceptably high. An alternative approach is to find a policy which optimises a risk-averse objective such as conditional value at risk (CVaR). However, optimising the CVaR alone may result in poor performance in expectation. In this work, we begin by showing that there can be multiple policies which obtain the optimal CVaR. This motivates us to propose a lexicographic approach which…
DeepMind（深度思维）；University College London（伦敦大学学院）；University of Alberta（阿尔伯塔大学）；Omron Sinic X（欧姆龙Sinic X）
We consider approximate dynamic programming in $γ$-discounted Markov decision processes and apply it to approximate planning with linear value-function approximation. Our first contribution is a new variant of Approximate Policy Iteration (API), called Confident Approximate Policy Iteration (CAPI), which computes a deterministic stationary policy with an optimal error bound scaling linearly with the product of the effective horizon $H$ and the worst-case approximation error $ε$ of the action-value functions of stationary policies. This improvement over API (whose error scales with $H^2$) comes…
Many large MDPs can be represented compactly using a dynamic Bayesian network. Although the structure of the value function does not retain the structure of the process, recent work has shown that value functions in factored MDPs can often be approximated well using a decomposed value function: a linear combination of <I>restricted</I> basis functions, each of which refers only to a small subset of variables. An approximate value function for a particular policy can be computed using approximate dynamic programming, but this approach (and others) can only produce an approximation relative to a…
University of Virginia（弗吉尼亚大学）；Shanghai Jiao Tong University（上海交通大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Algorithmic decisions made by machine learning models in high-stakes domains may have lasting impacts over time. However, naive applications of standard fairness criterion in static settings over temporal domains may lead to delayed and adverse effects. To understand the dynamics of performance disparity, we study a fairness problem in Markov decision processes (MDPs). Specifically, we propose return parity, a fairness notion that requires MDPs from different demographic groups that share the same state and action spaces to achieve approximately the same expected time-discounted rewards. We fi…
