Temporal abstraction in reinforcement learning

Temporal-di erence (TD) learning can be used not just to predict rewards, as is commonly done in reinforcement learning, but also to predict states, i.e., to learn a model of the world's dynamics. We present theory and algorithms for intermixing TD models of the world at di erent levels of temporal abstraction within a single structure. Such multi-scale TD models can be used in model-based reinforcement-learning architectures and dynamic programming methods in place of conventional Markov models. This enables planning at higher and varied levels of abstraction, and, as such, may prove useful in formulating methods for hierarchical or multi-level planning and reinforcement learning. In this paper we treat only the prediction problem|that of learning a model and value function for the case of xed agent behavior. Within this context, we establish the theoretical foundations of multi-scale models and derive TD algorithms for learning them. Two small computational experiments are presented to test and illustrate the theory. This work is an extension and generalization of the work of Singh (1992), Dayan (1993), and Sutton & Pinette (1985). 1 Multi-Scale Planning and Modeling Model-based reinforcement learning o ers a potentially elegant solution to the problem of integrating planning into a real-time learning and decisionmaking agent (Sutton, 1990; Barto et al., 1995;  1993, Moore & Atkeson, 1994; Dean et al., in prep). However, most current reinforcementlearning systems assume a single, xed time step: actions take one step to complete, and their immediate consequences become available after one step. This makes it di cult to learn and plan at di erent time scales. For example, commuting to work involves planning at a high level about which route to drive (or whether to take the train) and at a low level about how to steer, when to brake, etc. Planning is necessary at both levels in order to optimize precise low-level movements without becoming lost in a sea of detail when making decisions at a high level. Moreover, these levels cannot be kept totally distinct and separate. They must interrelate at least in the sense that the actions and plans at a high levels must be turned into actual, moment-by-moment decisions at the lowest level. The need for hierarchical and abstract planning is a fundamental problem in AI whether or not one uses the reinforcement-learning framework (e.g., Fikes et al., 1972; Sacerdoti, 1977; Kuipers, 1979; Laird et al., 1986; Korf, 1985; Minton, 1988; Watkins, 1989; Drescher, 1991; Ring, 1991; Wixson, 1991; Schmidhuber, 1991; Tenenberg et al., 1992; Kaelbling, 1993; Lin, 1993; Dayan & Hinton, 1993; Dejong, 1994; Chrisman, 1994; Hansen, 1994; Dean & Lin, in prep). We do not propose to fully solve it in this paper. Rather, we develop an approach to multiple-time-scale modeling of the world that may eventually be useful in such a solution. Our approach is to extend temporal-di erence (TD) methods, which are commonly used in reinforcement learning systems to learn value functions, such that they can be used to learn world models. When TD methods are used, the predictions of the models can naturally extend beyond a single time step. As we will show, they can even make predictions that are not speci c to a single time scale, but intermix many such scales, with no loss of performance when the models are used. This approach is an extension of the ideas of Singh (1992), Dayan (1993), and Sutton & Pinette