Bellman goes Relational ( extended abstract ) 1

We introduce ReBel, a relational Bellman update operator that can be used for Markov Decision Processes in – possibly infinite – relational domains. Using ReBel we develop a relational value iteration algorithm. 1 Relational Markov Decision Processes Many reinforcement learning (RL) and dynamic programming techniques have been developed for solving Markov Decision Processes (MDP). Until recently, the representation of MDPs was limited to states and actions that were discrete symbols or factored into propositions or attribute-value pairs. Recent approaches have shown that this representation can be upgraded towards relational (or first-order) representations – thereby enabling the use of first-order logical languages to represent states and actions in terms of objects and relations such as inRoom(robot, kitchen) and driveTo(library). Although some approaches have appeared for solving relational MDPs (RMDP) (e.g. relational RL [2]), not much work has considered exact, model-based solution techniques. The only example is the work by Boutilier et al. (2001) which employs situation calculus for modeling an RMDP which is then solved by a value iteration algorithm. However, because of the complexity of the language, the algorithm was not fully automated yet, e.g. the simplification of expressions obtained is done manually. Here we show that by using a restricted language, the simplification is computationally feasible and we develop a fully automatic value iteration algorithm. Another aim of the paper is to give some insights into the framework of relational RL. 2 Relational Bellman Operator The first step is the introduction of a logical formalism to specify MDPs over relational domains. A constraint logic programming language is used to define four The full paper appeared in the Proceedings of the 21st International Conference on Machine Learning (ICML’04 ) [July 4–8, 2004, Banff, Canada]. ingredients: (1) abstract states are conjunctions of logical atoms aggregating sets of concrete states. (2) abstract actions are similar to first-order, probabilistic STRIPS operators. (3) abstract rewards form a reward model describing state-based rewards. (4) integrity constraints describe domain constraints. Based on this formalism a relational upgrade of the Bellman backup operator can be defined. Starting from an initial state value function, each value backup step consists of the following operations: (1) For each action rule we employ regression to compute weakest preconditions (wp) of abstract states and abstract action rules. (2) For each of these wps derived by some action rule, we compute a Qvalue, defining a Q-rule. (3) Each Q-rule defines a partial Q-value, because it is based on one outcome of the action. All the Q-rules for different outcomes of the same action are combined into a complete Q-rule. (4) Because V (s) = maxa Q(s, a) we maximize over the set of Q-rules to get an equivalent state-value function. Overall, these four steps refine the current value partition V t into V t+1 Relational value iteration can now be implemented by repeated application of the described operations. Starting from the state-value function defining goal state rewards (V ) we compute the series V 1 . V n until some stopping criterium.