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Policy iteration is another dynamic programming algorithm used in reinforcement learning to find the optimal policy in a Markov decision process (MDP). Like value iteration, policy iteration is an iterative algorithm that alternates between two steps: policy evaluation and policy improvement. The algorithm continues these steps until convergence to find the optimal policy.

The step-by-step policy iteration is:

1. Initialization: Start with an initial policy π, which is a mapping from states to actions. This policy could be arbitrary.

2. Policy Evaluation: Evaluate the value function for the current policy. This involves determining the expected cumulative reward for each state under the current policy. The goal is to estimate the value of each state, denoted as V^π(s).

   The value function is calculated using the Bellman expectation equation:

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This equation expresses the expected cumulative reward for being in state s and following policy π.

3. Policy Improvement: Improve the current policy by selecting actions that maximize the expected cumulative reward in each state. This involves updating the policy based on the current value function:

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The new policy π' is greedy with respect to the current value function.

4. Convergence Check: Check whether the policy has converged. If the policy remains unchanged or the change is negligible, the algorithm stops. Otherwise, go back to step 2.

In some cases, hybrid approaches that combine elements of both value iteration and policy iteration (see page3678) can be effective. For example, one might use a few iterations of value iteration to quickly obtain a reasonable approximation and then switch to policy iteration for fine-tunin

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