#### Authors

Eddie Rafols, Anna Koop, Richard S. Sutton

#### Abstract

We present a generalization of temporal-difference networks to include temporally abstract options on the links of the question network. Temporal-difference (TD) networks have been proposed as a way of representing and learning a wide variety of predictions about the interaction between an agent and its environment. These predictions are compositional in that their targets are defined in terms of other predictions, and subjunctive in that that they are about what would happen if an action or sequence of actions were taken. In conventional TD networks, the inter-related predictions are at successive time steps and contingent on a single action; here we generalize them to accommodate extended time intervals and contingency on whole ways of behaving. Our generalization is based on the options framework for temporal abstraction. The primary contribution of this paper is to introduce a new algorithm for intra-option learning in TD networks with function approximation and eligibility traces. We present empirical examples of our algorithm's effectiveness and of the greater representational expressiveness of temporallyabstract TD networks. The primary distinguishing feature of temporal-difference (TD) networks (Sutton & Tanner, 2005) is that they permit a general compositional specification of the goals of learning. The goals of learning are thought of as predictive questions being asked by the agent in the learning problem, such as "What will I see if I step forward and look right?" or "If I open the fridge, will I see a bottle of beer?" Seeing a bottle of beer is of course a complicated perceptual act. It might be thought of as obtaining a set of predictions about what would happen if certain reaching and grasping actions were taken, about what would happen if the bottle were opened and turned upside down, and of what the bottle would look like if viewed from various angles. To predict seeing a bottle of beer is thus to make a prediction about a set of other predictions. The target for the overall prediction is a composition in the mathematical sense of the first prediction with each of the other predictions. TD networks are the first framework for representing the goals of predictive learning in a compositional, machine-accessible form. Each node of a TD network represents an individual question--something to be predicted--and has associated with it a value representing an answer to the question--a prediction of that something. The questions are represented by a set of directed links between nodes. If node 1 is linked to node 2, then node 1 rep-

1

The options framework

In this section we present the essential elements of the options framework (Sutton, Precup & Singh, 1999) that we will need for our extension of TD networks. In this framework, an agent and an environment interact at discrete time steps t = 1, 2, 3.... In each state st S , the agent selects an action at A, determining the next state st+1 .1 An action is a way of behaving for one time step; the options framework lets us talk about temporally extended ways of behaving. An individual option consists of three parts. The first is the initiation set, I S , the subset of states in which the option can be started. The second component of an option is its policy, : S A [0, 1], specifying how the agent behaves when 1 Although the options framework includes rewards, we omit them here because we are concerned only with prediction, not control.

following the option. Finally, a termination function, : S A [0, 1], specifies how the option ends: (s) denotes the probability of terminating when in state s. The option is thus completely and formally defined by the 3-tuple (I , , ).

2

Conventional TD networks

In this section we briefly present the details of the structure and the learning algorithm comprising TD networks as introduced by Sutton and Tanner (2005). TD networks address a prediction problem in which the agent may not have direct access to the state of the environment. Instead, at each time step the agent receives an observation ot O dependent on the state. The experience stream thus consists of a sequence of alternating actions and observations, o1 , a1 , o2 , a2 , o3 . The TD network consists of a set of nodes, each representing a single scalar prediction, interlinked by the question and answer networks as suggested previously. For a network 1 n of n nodes, the vector of all predictions at time step t is denoted yt = (yt , . . . , yt )T . The predictions are estimates of the expected value of some scalar quantity, typically of a bit, in which case they can be interpreted as estimates of probabilities. The predictions are updated at each time step according to a vector-valued function u with modifiable parameter W, which is often taken to be of a linear form: yt = u(yt-1 , at-1 , ot , Wt ) = (Wt xt ), (1) where xt m is an m-vector of features created from (yt-1 , at-1 , ot ), Wt is an n m matrix (whose elements are sometimes referred to as weights), and is the n-vector form of either the identity function or the S-shaped logistic function (s) = 1+1 -s . The e feature vector is an arbitrary vector-valued function of yt-1 , at-1 , and ot . For example, in the simplest case the feature vector is a unit basis vector with the location of the one communicating the current state. In a partially observable environment, the feature vector may be a combination of the agent's action, observations, and predictions from the previous time step. The overall update u defines the answer network. The question network consists of a set of target functions, z i : O n , and condition i y functions, ci : A n [0, 1]n . We define zt = z i (ot+1 , ~t+1 ) as the target for prediction i2 i i yt . Similarly, we define ct = c (at , yt ) as the condition at time t. The learning algorithm i for each component wtj of Wt can then be written i zi c yt ij i i wt+1 = wtj + t - yt i , (2) t i wt j where is a positive step-size parameter. Note that the targets here are functions of the observation and predictions exactly one time step later, and that the conditions are functions of a single primitive action. This is what makes this algorithm suitable only for learning about one-step TD relationships. By chaining together multiple nodes, Sutton and Tanner (2005) used it to predict k steps ahead, for various particular values of k , and to predict the outcome of specific action sequences (as in PSRs, e.g., Littman et al., 2002; Singh et al., 2004). Now we consider the extension to temporally abstract actions.

3

Option-extended TD networks

In this section we present our intra-option learning algorithm for TD networks with options and eligibility traces. As suggested earlier, each node's outgoing link in the question The quantity ~ is almost the same as y, and we encourage the reader to think of them as identical y here. The difference is that ~ is calculated by weights that are one step out of date as compared to y, y i.e., ~t = u(yt-1 , at-1 , ot , Wt-1 ) (cf. equation 1). y 2

network will now correspond to an option applying over possibly many steps. The policy of the ith node's option corresponds to the condition function ci , which we think of as a recognizer for the option. It inspects each action taken to assess whether the option is being followed: ci = 1 if the agent is acting consistently with the option policy and ci = 0 othert t wise (intermediate values are also possible). When an agent ceases to act consistently with the option policy, we say that the option has diverged. The possibility of recognizing more than one action as consistent with the option is a significant generalization of the original idea of options. If no actions are recognized as acceptable in a state, then the option cannot be followed and thus cannot be initiated. Here we take the set of states with at least one recognized action to be the initiation set of the option. The option-termination function generalizes naturally to TD networks. Each node i is i given a corresponding termination function, i : O n [0, 1], where t = i (ot+1 , yt ) i is the probability of terminating at time t.3 t = 1 indicates that the option has terminated i at time t; t = 0 indicates that it has not, and intermediate values of correspond to soft i or stochastic termination conditions. If an option terminates, then zt acts as the target, but if the option is ongoing without termination, then the node's own next value, yt+1 , should ~i be the target. The termination function specifies which of the two targets (or mixture of the two targets) is used to produce a form of TD error for each node i: i ii ii i t = t zt + (1 - t )yt+1 - yt . ~

(3)

Our option-extended algorithm incorporates eligibility traces (see Sutton & Barto, 1998) as short-term memory variables organized in an n m matrix E, paralleling the weight matrix. The traces are a record of the effect that each weight could have had on each node's prediction during the time the agent has been acting consistently with the node's option. The components eij of the eligibility matrix are updated by , i yt ij ij i i (4) et = ct et-1 (1 - t ) + i wt j where 0 1 is the trace-decay parameter familiar from the TD() learning algorithm. Because of the ci factor, all of a node's traces will be immediately reset to zero whenever t the agent deviates from the node's option's policy. If the agent follows the policy and the option does not terminate, then the trace decays by and increments by the gradient in the way typical of eligibility traces. If the policy is followed and the option does terminate, then the trace will be reset to zero on the immediately following time step, and a new trace will start building. Finally, our algorithm updates the weights on each time step by ij i i wt+1 = wtj + t eij . t