We introduce a model-free algorithm for learning in Markov decision processes with parameterized actions-discrete actions with continuous parameters. At each step the agent must select both which action to use and which parameters to use with that action. We introduce the Q-PAMDP algorithm for learning in these domains, show that it converges to a local optimum, and compare it to direct policy search in the goal-scoring and Platform domains.
A hierarchical MDP is an MDP where each action has subtasks. A subtask is itself an MDP with its own states and actions which may have their own subtasks. Hierarchical MDPs are well-suited for representing parameterized actions as we could consider selecting the parameters for a discrete action as a subtask. MAXQ is a method for value function decomposition of hierarchical MDPs @cite_10 . One possiblity is to use MAXQ for learning the action-values in a parameterized action problem.
@cite_10
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This paper presents a new approach to hierarchical reinforcement learning based on decomposing the target Markov decision process (MDP) into a hierarchy of smaller MDPs and decomposing the value function of the target MDP into an additive combination of the value functions of the smaller MDPs. The decomposition, known as the MAXQ decomposition, has both a procedural semantics--as a subroutine hierarchy--and a declarative semantics--as a representation of the value function of a hierarchical policy. MAXQ unifies and extends previous work on hierarchical reinforcement learning by Singh, Kaelbling, and Dayan and Hinton. It is based on the assumption that the programmer can identify useful subgoals and define subtasks that achieve these subgoals. By defining such subgoals, the programmer constrains the set of policies that need to be considered during reinforcement learning. The MAXQ value function decomposition can represent the value function of any policy that is consistent with the given hierarchy. The decomposition also creates opportunities to exploit state abstractions, so that individual MDPs within the hierarchy can ignore large parts of the state space. This is important for the practical application of the method. This paper defines the MAXQ hierarchy, proves formal results on its representational power, and establishes five conditions for the safe use of state abstractions. The paper presents an online model-free learning algorithm, MAXQ-Q, and proves that it converges with probability 1 to a kind of locally-optimal policy known as a recursively optimal policy, even in the presence of the five kinds of state abstraction. The paper evaluates the MAXQ representation and MAXQ-Q through a series of experiments in three domains and shows experimentally that MAXQ-Q (with state abstractions) converges to a recursively optimal policy much faster than flat Q learning. The fact that MAXQ learns a representation of the value function has an important benefit: it makes it possible to compute and execute an improved, non-hierarchical policy via a procedure similar to the policy improvement step of policy iteration. The paper demonstrates the effectiveness of this nonhierarchical execution experimentally. Finally, the paper concludes with a comparison to related work and a discussion of the design tradeoffs in hierarchical reinforcement learning. This paper presents a new approach to hierarchical reinforcement learning based on the MAXQ decomposition of the value function. The MAXQ decomposition has both a procedural semantics—as a subroutine hierarchy—and a declarative semantics—as a representation of the value function of a hierarchical policy. MAXQ unifies and extends previous work on hierarchical reinforcement learning by Singh, Kaelbling, and Dayan and Hinton. Conditions under which the MAXQ decomposition can represent the optimal value function are derived. The paper defines a hierarchical Q learning algorithm, proves its convergence, and shows experimentally that it can learn much faster than ordinary “flat” Q learning. Finally, the paper discusses some interesting issues that arise in hierarchical reinforcement learning including the hierarchical credit assignment problem and non-hierarchical execution of the MAXQ hierarchy. Recently, structured methods for solving factored Markov decisions processes (MDPs) with large state spaces have been proposed recently to allow dynamic programming to be applied without the need for complete state enumeration. We propose and examine a new value iteration algorithm for MDPs that uses algebraic decision diagrams (ADDs) to represent value functions and policies, assuming an ADD input representation of the MDP. Dynamic programming is implemented via ADD manipulation. We demonstrate our method on a class of large MDPs (up to 63 million states) and show that significant gains can be had when compared to tree-structured representations (with up to a thirty-fold reduction in the number of nodes required to represent optimal value functions). This paper addresses the question how robot planning (e.g. for navigation) can be done with hierarchical maps. We present an algorithm for hierarchical path planning for stochastic tasks, based on Markov decision processes (MDPs) and dynamic programming. It is more efficient than standard dynamic programming for "flat" MDPs, because it reduces the state space for all levels in its hierarchy and it allows reuse of previously computed partial policies. This computational advantage comes at the cost of some extra memory and overhead to represent and coordinate the hierarchical system, and in some cases somewhat longer paths to target locations. We demonstrate the method on artificially generated MDP data, and on real robot data from our vision-controlled robot navigating in an office environment.