Q-learning is an algorithm, that contains many of the basic structures required for reinforcement learning and acts as the basis for many more sophisticated algorithms. The Q-learning algorithm can be seen as an (asynchronous) implementation of the Robbins-Munro procedure for finding fixed points. For this reason we will require results from Robbins-Munro when proving convergence.

A key ingredient is the notion of a -factor as described in Section [IDP]. Recall that optimal -factor, , is the value of starting in state taking action and thereafter following the optimal policy. In Infinite Time Horizon, Prop 2 we showed that this solved the recursion:

**Def.** [Q-learning] Given a state , an action , its reward and the next state , -learning performs the update

where positive (learning rate) parameter. Recall means reset with such that .

To implement this as an algorithm, we assume that we have a sequence of state-action-reward-next_state quadruplets and we apply the above update to each of the terms in this sequence.

**Thrm 1.** For a sequence of state-action-reward triples Consider the Q-learning update for

if the sequence of state-action-reward triples visits each state and action infinitely often, and if the learning rate is an adapted sequence satisfying the Robbins-Munro condition

then, with probability ,

where is the optimal value function.

**Proof.** We essentially show that the result is a consequence of Theorem 3 in Robbins-Munro. We note that the optimal -function, satisfies a fixed point equation

with

for each and . We know from Prop 2 in the post Infinite Time Horizon, that for discounted programming is a contraction. I.e.

Now notice that the -learning algorithm performs the update

where

for . The update above is a Robbin’s Munro update. Further Notice remains the same for all other values of , the update is asynchronous. It is not hard to see that when we condition on the set of previous actions and states that

and, a quick calculation shows,^{1} that

From this we see that we are working in the setting of Theorem 3 in Robbins-Munro and that the conditions of that theorem are satisfied. Thus it must be that

where satisfies . In otherwords, as required, it satisfies the Bellman equation for the optimal -function and thus is optimal.

- Note ↩