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
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:
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,3.5 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 sample from 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 nonterminal
states, denoted 
, from the set of all states plus the terminal state,
denoted 
.
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
return:
is a parameter, 
, called the discount rate.
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?
