To develop a Python program to find the optimal policy for the given MDP using the value iteration algorithm.

The FrozenLake environment in OpenAI Gym is a gridworld problem that challenges reinforcement learning agents to navigate a slippery terrain to reach a goal state while avoiding hazards. Note that the environment is closed with a fence, so the agent cannot leave the gridworld.

• Value iteration is a method of computing an optimal MDP policy and its value.
• It begins with an initial guess for the value function, and iteratively updates it towards the optimal value function, according to the Bellman optimality equation.
• The algorithm is guaranteed to converge to the optimal value function, and in the process of doing so, also converges to the optimal policy.

The algorithm is as follows:

1.Initialize the value function V(s) arbitrarily for all states s.

2.Repeat until convergence:

• Initialize aaction-value function Q(s, a) arbitrarily for all states s and actions a.
• For all the states s and all the action a of every state:
  • Update the action-value function Q(s, a) using the Bellman equation.
  • Take the value function V(s) to be the maximum of Q(s, a) over all actions a.
  • Check if the maximum difference between Old V and new V is less than theta.
  • Where theta is a small positive number that determines the accuracy of estimation. 3.If the maximum difference between Old V and new V is greater than theta, then
• Update the value function V with the maximum action-value from Q.
• Go to step 2. 4.The optimal policy can be constructed by taking the argmax of the action-value function Q(s, a) over all actions a. 5.Return the optimal policy and the optimal value function.

def value_iteration(P, gamma=1.0, theta=1e-10):
    V = np.zeros(len(P), dtype=np.float64)
    while True:
      Q=np.zeros((len(P),len(P[0])),dtype=np.float64)
      for s in range(len(P)):
        for a in range(len(P[s])):
          for prob,next_state,reward,done in P[s][a]:
            Q[s][a]+=prob*(reward+gamma*V[next_state]*(not done))
      if np.max(np.abs(V-np.max(Q,axis=1)))<theta:
        break
      V=np.max(Q,axis=1)
    pi=lambda s:{s:a for s,a in enumerate(np.argmax(Q,axis=1))}[s]
    return V, pi

Thus, a Python program is developed to find the optimal policy for the given MDP using the value iteration algorithm.