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 time step t 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 just the sum of the rewards:
where T is a final time step. This approach makes sense in applications in which there is a natural notion of final time step, that is, when the agent-environment interaction breaks naturally into subsequences, which we call episodes, such as plays of a game, trips through a maze, or any sort of repeated interactions. Each episode ends in a special state called the terminal state, followed by a reset to a standard starting state or to a standard distribution of starting states. Tasks with episodes of this kind are called episodic tasks. In episodic tasks we sometimes need to distinguish the set of all non-terminal states, denoted , and the set of all states plus the terminal state, denoted .
On the other hand, in many cases that the agent-environment interaction does not break naturally into identifiable episodes, but simply goes on and on 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 lifespan. We call these continual tasks. The return formulation (3.1) is problematic for continual 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 +1 at each time step.) Thus, in this book we 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 return:
where is a parameter, , called the discount rate.
The discount rate determines the present value of future rewards: a reward received k 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 only concerned with maximizing immediate rewards: its objective in this case is to learn how to choose so as maximize only . If each of the agent's actions happened only to influence 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 one, the objective takes future rewards into account more strongly: the agent becomes more far-sighted.
Figure 3.2: The pole-balancing task
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 exceeds the limits of 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 +1 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 continual task, using discounting. In this case the reward would be -1 on each failure and zero at all other times. The return at each time would then be related to , where k 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.
Suppose you treated pole-balancing as an episodic task but also used discounting, with all rewards zero except for a -1 upon failure. What then would the return be at each time? How does this return differ from that in the discounted, continual formulation of this task?
Imagine that you are designing a robot to run a maze. You decide to give it a reward of +1 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?