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## Correctness of Value Iteration Algorithm

In the following theorem we show that the algorithm finds an -optimal policy in a finite number of steps. Note that the selected policy might in fact be the optimal policy, but we have no way of knowing that in advance.

Theorem 6.1   For the series and the policy computed by VI the following holds:
1.
2.
3.
The policy is -optimal
4.
If then

Proof:Parts (1) and (2) follow directly from the properties of the series vn+1 = Lvn, that were shown in Lecture 5.
For part (3) we assume that , as is the case when the algorithm stops, and show that , which would make the policy -optimal.

 (6.1)

We now bound each part of the sum individually:

Since is maximal over the actions using vn+1, it implies that and we conclude that:

From the inequality it follows:

For the second part of the sum we derive similarly that:

From which part (4) of the theorem also follows.
Returning to inequality , it follows:

Therefore the selected policy is -optimal.

Next: Convergence of Value Iteration Up: Finding the Optimal Policy: Previous: The Value Iteration Algorithm
Yishay Mansour
1999-12-18
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