We consider a reinforcement learning (RL) setting in which the agent interacts with a sequence of episodic MDPs. At the start of each episode the agent has access to some side-information or context that determines the dynamics of the MDP for that episode. Our setting is motivated by applications in healthcare where baseline measurements of a patient at the start of a treatment episode form the context that may provide information about how the patient might respond to treatment decisions. We propose algorithms for learning in such Contextual Markov Decision Processes (CMDPs) under an assumption that the unobserved MDP parameters vary smoothly with the observed context. We also give lower and upper PAC bounds under the smoothness assumption. Because our lower bound has an exponential dependence on the dimension, we consider a tractable linear setting where the context is used to create linear combinations of a finite set of MDPs. For the linear setting, we give a PAC learning algorithm based on KWIK learning techniques.
For smooth CMDPs (), we pool observations across similar contexts and reduce the problem to learning policies for finitely many MDPs. An alternative approach is to consider an infinite MDP whose state representation is augmented by the context, and apply PAC-MDP methods for metric state spaces (e.g., C-PACE proposed by @cite_4 ). However, doing so might increase the sample and computational complexity unnecessarily, because we no longer leverage the structure that a particular component of the (augmented) state, namely the context, remains the same in an episode. Concretely, the augmenting approach needs to perform planning in the augmented MDP over states and contexts, which makes its computational storage requirement worse than our solution: we only perform planning in MDPs defined on @math , whose computational characteristics have no dependence on the context space. In addition, we allow the context sequence to be chosen in an adversarial manner. This corresponds to adversarially chosen initial states in MDPs, which is usually not handled by PAC-MDP methods.
@cite_4
2109910161 2784847637 2950462606 1512919909
Learning, planning, and representing knowledge at multiple levels of temporal ab- straction are key, longstanding challenges for AI. In this paper we consider how these challenges can be addressed within the mathematical framework of reinforce- ment learning and Markov decision processes (MDPs). We extend the usual notion of action in this framework to include options—closed-loop policies for taking ac- tion over a period of time. Examples of options include picking up an object, going to lunch, and traveling to a distant city, as well as primitive actions such as mus- cle twitches and joint torques. Overall, we show that options enable temporally abstract knowledge and action to be included in the reinforcement learning frame- work in a natural and general way. In particular, we show that options may be used interchangeably with primitive actions in planning methods such as dynamic pro- gramming and in learning methods such as Q-learning. Formally, a set of options defined over an MDP constitutes a semi-Markov decision process (SMDP), and the theory of SMDPs provides the foundation for the theory of options. However, the most interesting issues concern the interplay between the underlying MDP and the SMDP and are thus beyond SMDP theory. We present results for three such cases: 1) we show that the results of planning with options can be used during execution to interrupt options and thereby perform even better than planned, 2) we introduce new intra-option methods that are able to learn about an option from fragments of its execution, and 3) we propose a notion of subgoal that can be used to improve the options themselves. All of these results have precursors in the existing literature; the contribution of this paper is to establish them in a simpler and more general setting with fewer changes to the existing reinforcement learning framework. In particular, we show that these results can be obtained without committing to (or ruling out) any particular approach to state abstraction, hierarchy, function approximation, or the macro-utility problem. This thesis addresses the question of planning under uncertainty within a time-dependent changing environment. Original motivation for this work came from the problem of building an autonomous agent able to coordinate with its uncertain environment; this environment being composed of other agents communicating their intentions or non-controllable processes for which some discrete-event model is available. We investigate several approaches for modeling continuous time-dependency in the framework of Markov Decision Processes (MDPs), leading us to a definition of Temporal Markov Decision Problems. Then our approach focuses on two separate paradigms. First, we investigate time-dependent problems as processes and describe them through the formalism of Time-dependent MDPs (TMDPs). We extend the existing results concerning optimality equations and present a new Value Iteration algorithm based on piecewise polynomial function representations in order to solve a more general class of TMDPs. This paves the way to a more general discussion on parametric actions in hybrid state and action spaces MDPs with continuous time. In a second time, we investigate the option of separately modeling the concurrent contributions of exogenous events. This approach of modeling leads to the use of Generalized Semi-Markov Decision Processes (GSMDP). We establish a link between the general framework of Discrete Events Systems Specification (DEVS) and the formalism of GSMDP, allowing us to build sound discrete-event compatible simulators. Then we introduce a simulation-based Policy Iteration approach for explicit-event Temporal Markov Decision Problems. This algorithmic contribution brings together results from simulation theory, forward search in MDPs, and statistical learning theory. The implicit-event approach was tested on a specific version of the Mars rover planning problem and on a drone patrol mission planning problem while the explicit-event approach was evaluated on a subway network control problem. 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. Policy search is then simply performed by searching for a policy with high estimated value. We also establish conditions under which our value estimates will be good, recovering theoretical results similar to those of Kearns, Mansour and Ng (1999), but with "sample complexity" bounds that have only a polynomial rather than exponential dependence on the horizon time. Our method applies to arbitrary POMDPs, including ones with infinite state and action spaces. We also present empirical results for our approach on a small discrete problem, and on a complex continuous state continuous action problem involving learning to ride a bicycle. A critical issue for the application of Markov decision processes (MDPs) to realistic problems is how the complexity of planning scales with the size of the MDP. In stochastic environments with very large or infinite state spaces, traditional planning and reinforcement learning algorithms may be inapplicable, since their running time typically grows linearly with the state space size in the worst case. In this paper we present a new algorithm that, given only a generative model (a natural and common type of simulator) for an arbitrary MDP, performs on-line, near-optimal planning with a per-state running time that has no dependence on the number of states. The running time is exponential in the horizon time (which depends only on the discount factor γ and the desired degree of approximation to the optimal policy). Our algorithm thus provides a different complexity trade-off than classical algorithms such as value iteration—rather than scaling linearly in both horizon time and state space size, our running time trades an exponential dependence on the former in exchange for no dependence on the latter. Our algorithm is based on the idea of sparse sampling. We prove that a randomly sampled look-ahead tree that covers only a vanishing fraction of the full look-ahead tree nevertheless suffices to compute near-optimal actions from any state of an MDP. Practical implementations of the algorithm are discussed, and we draw ties to our related recent results on finding a near-best strategy from a given class of strategies in very large partially observable MDPs (Kearns, Mansour, & Ng. Neural information processing systems 13, to appear).