First we consider how to compute the state-value function for an
arbitrary policy . This is called policy evaluation in the DP
literature. We also refer to it as the prediction problem. Recall from
Chapter 3 that, for all
If the environment's dynamics are
completely known, then (4.4) is a system of simultaneous
linear equations in
unknowns (the , ). In principle, its solution is a
straightforward, if tedious, computation. For our purposes, iterative solution
methods are most suitable. Consider a sequence of approximate value functions
, each mapping to . The initial
, is chosen arbitrarily (except that the terminal state, if any, must be
given value 0), and each successive approximation is obtained by using the
Bellman equation for (3.10) as an update rule:
To produce each successive approximation, from , iterative policy evaluation applies the same operation to each state : it replaces the old value of with a new value obtained from the old values of the successor states of , and the expected immediate rewards, along all the one-step transitions possible under the policy being evaluated. We call this kind of operation a full backup. Each iteration of iterative policy evaluation backs up the value of every state once to produce the new approximate value function . There are several different kinds of full backups, depending on whether a state (as here) or a state-action pair is being backed up, and depending on the precise way the estimated values of the successor states are combined. All the backups done in DP algorithms are called full backups because they are based on all possible next states rather than on a sample next state. The nature of a backup can be expressed in an equation, as above, or in a backup diagram like those introduced in Chapter 3. For example, Figure 3.4a is the backup diagram corresponding to the full backup used in iterative policy evaluation.
To write a sequential computer program to implement iterative policy evaluation, as given by (4.5), you would have to use two arrays, one for the old values, , and one for the new values, . This way, the new values can be computed one by one from the old values without the old values being changed. Of course it is easier to use one array and update the values "in place," that is, with each new backed-up value immediately overwriting the old one. Then, depending on the order in which the states are backed up, sometimes new values are used instead of old ones on the right-hand side of (4.5). This slightly different algorithm also converges to ; in fact, it usually converges faster than the two-array version, as you might expect, since it uses new data as soon as they are available. We think of the backups as being done in a sweep through the state space. For the in-place algorithm, the order in which states are backed up during the sweep has a significant influence on the rate of convergence. We usually have the in-place version in mind when we think of DP algorithms.
Another implementation point concerns the termination of the algorithm. Formally, iterative policy evaluation converges only in the limit, but in practice it must be halted short of this. A typical stopping condition for iterative policy evaluation is to test the quantity after each sweep and stop when it is sufficiently small. Figure 4.1 gives a complete algorithm for iterative policy evaluation with this stopping criterion.
Example 4.1 Consider the gridworld shown below.
Exercise 4.1 If is the equiprobable random policy, what is ? What is ?
Exercise 4.2 Suppose a new state 15 is added to the gridworld just below state 13, and its actions, left, up, right, and down, take the agent to states 12, 13, 14, and 15, respectively. Assume that the transitions from the original states are unchanged. What, then, is for the equiprobable random policy? Now suppose the dynamics of state 13 are also changed, such that action down from state 13 takes the agent to the new state 15. What is for the equiprobable random policy in this case?
Exercise 4.3 What are the equations analogous to (4.3), (4.4), and (4.5) for the action-value function and its successive approximation by a sequence of functions ?
Exercise 4.3.5 In some undiscounted episodic tasks there may be policies for which eventual termination is not guaranteed. For example, in the grid problem above it is possible to go back and forth between two states forever. In a task that is otherwise perfectly sensible, may be negative infinity for some policies and states, in which case the algorithm for iterative policy evaluation given in Figure 4.1 will not terminate. As a purely practical matter, how might we amend this algorithm to assure termination even in this case? Assume that eventual termination is guaranteed under the optimal policy.