So far we have been imprecise regarding the objective of learning. We
have said that the agent's goal is to maximize the reward it receives in the long
run. How might this be formally defined? If the sequence of rewards received after
is denoted , then what precise aspect of this
sequence do we wish to maximize? In general, we seek to maximize the expected
return, where the return, , is defined as some specific function of the reward
sequence. In the simplest case the return is the sum of the rewards:
On the other hand, in many cases the agent-environment interaction does not break naturally into identifiable episodes, but goes on continually without limit. For example, this would be the natural way to formulate a continual process-control task, or an application to a robot with a long life span. We call these continuing tasks. The return formulation (3.1) is problematic for continuing tasks because the final time step would be , and the return, which is what we are trying to maximize, could itself easily be infinite. (For example, suppose the agent receives a reward of at each time step.) Thus, in this book we usually use a definition of return that is slightly more complex conceptually but much simpler mathematically.
The additional concept that we need is that of discounting.
According to this approach, the agent tries to select actions so that the
sum of the discounted rewards it receives over the future is maximized. In
particular, it chooses to maximize the expected discounted
The discount rate determines the present value of future rewards: a reward received time steps in the future is worth only times what it would be worth if it were received immediately. If , the infinite sum has a finite value as long as the reward sequence is bounded. If , the agent is "myopic" in being concerned only with maximizing immediate rewards: its objective in this case is to learn how to choose so as to maximize only . If each of the agent's actions happened to influence only the immediate reward, not future rewards as well, then a myopic agent could maximize (3.2) by separately maximizing each immediate reward. But in general, acting to maximize immediate reward can reduce access to future rewards so that the return may actually be reduced. As approaches 1, the objective takes future rewards into account more strongly: the agent becomes more farsighted.
Example 3.4: Pole-Balancing Figure 3.2 shows a task that served as an early illustration of reinforcement learning. The objective here is to apply forces to a cart moving along a track so as to keep a pole hinged to the cart from falling over. A failure is said to occur if the pole falls past a given angle from vertical or if the cart runs off the track. The pole is reset to vertical after each failure. This task could be treated as episodic, where the natural episodes are the repeated attempts to balance the pole. The reward in this case could be for every time step on which failure did not occur, so that the return at each time would be the number of steps until failure. Alternatively, we could treat pole-balancing as a continuing task, using discounting. In this case the reward would be on each failure and zero at all other times. The return at each time would then be related to , where is the number of time steps before failure. In either case, the return is maximized by keeping the pole balanced for as long as possible.
Exercise 3.4 Suppose you treated pole-balancing as an episodic task but also used discounting, with all rewards zero except for upon failure. What then would the return be at each time? How does this return differ from that in the discounted, continuing formulation of this task?
Exercise 3.5 Imagine that you are designing a robot to run a maze. You decide to give it a reward of for escaping from the maze and a reward of zero at all other times. The task seems to break down naturally into episodes--the successive runs through the maze--so you decide to treat it as an episodic task, where the goal is to maximize expected total reward (3.1). After running the learning agent for a while, you find that it is showing no improvement in escaping from the maze. What is going wrong? Have you effectively communicated to the agent what you want it to achieve?