Arvojatkaisulle

Arvojatkaisulle, often translated as "value iteration," is a fundamental algorithm in reinforcement learning. It is used to find an optimal policy for a Markov Decision Process (MDP). The core idea behind value iteration is to iteratively update the value function of each state until it converges. The value function, denoted as V(s), represents the expected cumulative future reward starting from state s and following a particular policy.

The algorithm begins by initializing the value function for all states, often to zero. In each iteration, the value function for every state is updated using the Bellman optimality equation. This equation relates the value of a state to the values of its successor states, considering the possible actions and their associated rewards and transition probabilities. Specifically, for each state s, the new value V_{k+1}(s) is calculated as the maximum expected future reward achievable from s, which is the maximum over all possible actions a of the immediate reward plus the discounted expected value of the next state s'. This process is repeated until the difference between successive value function estimates is below a predefined small threshold, indicating convergence.

Once the value function has converged to the optimal value function V*(s), an optimal policy can be easily derived. The optimal policy, denoted as π*(s), specifies the action that maximizes the expected future reward from each state s. Value iteration guarantees convergence to the optimal value function and thus to an optimal policy, provided that the MDP is finite and the discount factor is less than one. It is a foundational algorithm that underlies many more advanced reinforcement learning techniques.