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.

• 5 Terminal States:

  G - (Goal): The state the agent aims to reach.

  H - (Hole): A hazardous state that the agent must avoid at all costs.

• Non-Terminal States:

  S - (Starting state): The initial position of the agent.

  Intermediate states: Grid cells forming a layout that the agent must traverse.

The agent can take 4 actions in each state:

• LEFT
• RIGHT
• UP
• DOWN

The environment is stochastic, meaning that the outcome of an action is not always certain.

• 33.33% chance of moving in the intended direction.
• 66.66% chance of moving in a orthogonal directions. This uncertainty adds complexity to the agent's navigation.

• +1 for reaching the goal state(G).
• 0 reward for all other states, including the starting state (S) and intermediate states.

The episode terminates when the agent reaches the goal state (G) or falls into a hole (H).

• 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.