6.1 TD Prediction

Both TD and Monte Carlo methods use experience to solve the prediction problem. Given some experience following a policy , both methods update their estimate of . If a nonterminal state is visited at time , then both methods update their estimate based on what happens after that visit. Roughly speaking, Monte Carlo methods wait until the return following the visit is known, then use that return as a target for . A simple every-visit Monte Carlo method suitable for nonstationary environments is

(6.1)

where is the actual return following time and is a constant step-size parameter (c.f., Equation (2.5)). Let us call this method constant-$\alpha$ MC. Whereas Monte Carlo methods must wait until the end of the episode to determine the increment to (only then is known), TD methods need wait only until the next time step. At time they immediately form a target and make a useful update using the observed reward and the estimate . The simplest TD method, known as TD(0), is

(6.2)

In effect, the target for the Monte Carlo update is , whereas the target for the TD update is .

Because the TD method bases its update in part on an existing estimate, we say that it is a bootstrapping method, like DP. We know from Chapter 3 that

			(6.3)
			(6.4)

Roughly speaking, Monte Carlo methods use an estimate of (6.3) as a target, whereas DP methods use an estimate of (6.4) as a target. The Monte Carlo target is an estimate because the expected value in (6.3) is not known; a sample return is used in place of the real expected return. The DP target is an estimate not because of the expected values, which are assumed to be completely provided by a model of the environment, but because is not known and the current estimate, , is used instead. The TD target is an estimate for both reasons: it samples the expected values in (6.4) and it uses the current estimate instead of the true . Thus, TD methods combine the sampling of Monte Carlo with the bootstrapping of DP. As we shall see, with care and imagination this can take us a long way toward obtaining the advantages of both Monte Carlo and DP methods.

Figure  6.1 specifies TD(0) completely in procedural form, and Figure  6.2 shows its backup diagram. The value estimate for the state node at the top of the backup diagram is updated on the basis of the one sample transition from it to the immediately following state. We refer to TD and Monte Carlo updates as sample backups because they involve looking ahead to a sample successor state (or state-action pair), using the value of the successor and the reward along the way to compute a backed-up value, and then changing the value of the original state (or state-action pair) accordingly. Sample backups differ from the full backups of DP methods in that they are based on a single sample successor rather than on a complete distribution of all possible successors.

Figure 6.1: Tabular TD(0) for estimating .

 

Figure 6.2: The backup diagram for TD(0).

 

Example 6.1: Driving Home   Each day as you drive home from work, you try to predict how long it will take to get home. When you leave your office, you note the time, the day of week, and anything else that might be relevant. Say on this Friday you are leaving at exactly 6 o'clock, and you estimate that it will take 30 minutes to get home. As you reach your car it is 6:05, and you notice it is starting to rain. Traffic is often slower in the rain, so you reestimate that it will take 35 minutes from then, or a total of 40 minutes. Fifteen minutes later you have completed the highway portion of your journey in good time. As you exit onto a secondary road you cut your estimate of total travel time to 35 minutes. Unfortunately, at this point you get stuck behind a slow truck, and the road is too narrow to pass. You end up having to follow the truck until you turn onto the side street where you live at 6:40. Three minutes later you are home. The sequence of states, times, and predictions is thus as follows:

	Elapsed Time	Predicted	Predicted
State	(minutes)	Time to Go	Total Time
leaving office, friday at 6	0	30	30
reach car, raining	5	35	40
exiting highway	20	15	35
2ndary road, behind truck	30	10	40
entering home street	40	3	43
arrive home	43	0	43

The rewards in this example are the elapsed times on each leg of the journey.^6.1 We are not discounting (), and thus the return for each state is the actual time to go from that state. The value of each state is the expected time to go. The second column of numbers gives the current estimated value for each state encountered.

Figure 6.3: Changes recommended by Monte Carlo methods in the driving home example.

 

A simple way to view the operation of Monte Carlo methods is to plot the predicted total time (the last column) over the sequence, as in Figure  6.3. The arrows show the changes in predictions recommended by the constant-$\alpha$ MC method (6.1), for . These are exactly the errors between the estimated value (predicted time to go) in each state and the actual return (actual time to go). For example, when you exited the highway you thought it would take only 15 minutes more to get home, but in fact it took 23 minutes. Equation (6.1) applies at this point and determines an increment in the estimate of time to go after exiting the highway. The error, , at this time is eight minutes. Suppose the step-size parameter, , is . Then the predicted time to go after exiting the highway would be revised upward by four minutes as a result of this experience. This is probably too large a change in this case; the truck was probably just an unlucky break. In any event, the change can only be made off-line, that is, after you have reached home. Only at this point do you know any of the actual returns.

Is it necessary to wait until the final outcome is known before learning can begin? Suppose on another day you again estimate when leaving your office that it will take 30 minutes to drive home, but then you become stuck in a massive traffic jam. Twenty-five minutes after leaving the office you are still bumper-to-bumper on the highway. You now estimate that it will take another 25 minutes to get home, for a total of 50 minutes. As you wait in traffic, you already know that your initial estimate of 30 minutes was too optimistic. Must you wait until you get home before increasing your estimate for the initial state? According to the Monte Carlo approach you must, because you don't yet know the true return.

According to a TD approach, on the other hand, you would learn immediately, shifting your initial estimate from 30 minutes toward 50. In fact, each estimate would be shifted toward the estimate that immediately follows it. Returning to our first day of driving, Figure  6.4 shows the same predictions as Figure  6.3, except with the changes recommended by the TD rule (6.2) (these are the changes made by the rule if ). Each error is proportional to the change over time of the prediction, that is, to the temporal differences in predictions.

Figure 6.4: Changes recommended by TD methods in the driving home example.

 

Besides giving you something to do while waiting in traffic, there are several computational reasons why it is advantageous to learn based on your current predictions rather than waiting until termination when you know the actual return. We briefly discuss some of these next.