Recall that $Q^{\ast }(s,a)$, where $a$ is an action and $s$ is a state, is the expected value (cumulative discounted reward) of doing $a$ in state $s$ and then following the optimal policy.

Q-learning uses temporal differences to estimate the value of $Q^{\ast }(s,a)$. In Q-learning, the agent maintains a table of $Q[S,A]$, where $S$ is the set of states and $A$ is the set of actions. $Q[s,a]$ represents its current estimate of $Q^{\ast }(s,a)$.

An experience provides one data point for the value of $Q(s,a)$. The data point is that the agent received the future value of $r+\gamma V(s^{\prime })$, where $V(s^{\prime })={\max }_{a^{\prime }}Q(s^{\prime },a^{\prime })$; this is the actual current reward plus the discounted estimated future value. This new data point is called a return. The agent can use the temporal difference equation (13.1) to update its estimate for $Q(s,a)$:

or, equivalently:

Figure 13.3 shows a Q-learning controller, where the agent is acting and learning at the same time. The do command, $do(a)$, on line 17 specifies that the action $a$ is the command the controller sends to the body. The reward and the resulting state are the percepts the controller receives from the body.

The algorithm of Figure 13.3 also maintains an array $N[s,a]$, which counts the number of times action $a$ was performed in state $s$. The function $alpha\mathrm{\_}fun$ computes $\alpha$ from the count. $alpha\mathrm{\_}fun(c)=10/(9+c)$ often works well; see Exercise 13.6. When $\alpha$ is fixed, the $N$ array does not need to be maintained (but it is also used for some exploration strategies; see below).

The Q-learner learns (an approximation of) the optimal $Q$-function as long as the agent explores enough, and there is no bound on the number of times it tries an action in any state (i.e., it does not always do the same subset of actions in a state).