Okay, we've seen how Policy Iteration finds the optimal policy $\pi^*$ by iterating between evaluating the current policy and improving it. Now, let's look at another dynamic programming algorithm called Value Iteration (VI). Value Iteration provides a different, often more computationally efficient, way to find the optimal value function $V^*$ directly, bypassing the need for explicit policy evaluation steps within the main loop.

The Core Idea: Direct Bellman Optimality Updates

Recall the Bellman Optimality Equation for $V^*(s)$ :

V^*(s) = \max_{a} \sum_{s', r} p(s', r | s, a) [r + \gamma V^*(s')]

This equation states that the optimal value of a state $s$ is the expected return achieved by taking the best possible action $a$ from that state, and then following the optimal policy thereafter.

Value Iteration leverages this equation directly as an update rule. It starts with an initial guess for the value function, $V_0(s)$ , for all states (often initialized to zero). Then, it iteratively refines this estimate by applying the Bellman optimality update across all states until the value function converges.

The Value Iteration Algorithm

Here's the process step-by-step:

Initialization: Choose an arbitrary initial value function $V_0(s)$ for all $s \in S$ . A small tolerance threshold $\theta > 0$ is also set to check for convergence.
Iteration Loop: Repeat for $k = 0, 1, 2, ...$ $k = 0, 1, 2, ...$ :
- Initialize a variable to track the maximum change in this iteration: $\Delta \leftarrow 0$ .
- For each state $s \in S$ $s \in S$ :
  - Store the current value: $v \leftarrow V_k(s)$ .
  - Compute the new potential value by finding the best action's expected return using the current value function $V_k$ for successor states: $V_{k+1}(s) \leftarrow \max_{a} \sum_{s', r} p(s', r | s, a) [r + \gamma V_k(s')]$
  - Update the maximum change seen so far: $\Delta \leftarrow \max(\Delta, |v - V_{k+1}(s)|)$ .
- Convergence Check: If $\Delta < \theta$ , stop the loop. The value function $V_{k+1}$ is considered a close approximation of the optimal value function $V^*$ .

Essentially, each iteration involves a sweep through the state space, updating each state's value based on the current values of its potential successor states, using the maximization over actions from the Bellman Optimality Equation. This update is sometimes called a "Bellman backup" or "optimality backup".

This diagram illustrates how the value $V_{k+1}(s)$ is calculated based on the maximum expected future rewards obtainable from state $s$ , using the values $V_k(s')$ from the previous iteration for successor states $s'$ .

Convergence and Finding the Optimal Policy

Why does this work? The Bellman optimality operator used in the update step is a contraction mapping. This mathematical property guarantees that applying the update repeatedly will cause the value function $V_k$ to converge to a unique fixed point, which is precisely the optimal value function $V^*$ . We won't go into the formal proof here, but the intuition is that each iteration brings the estimated values closer to the true optimal values.

Once Value Iteration converges (i.e., the changes in the value function $\Delta$ become very small), we have a good approximation of $V^*$ . How do we get the optimal policy $\pi^*$ from $V^*$ ? We can extract it directly by choosing the action that maximizes the expected return from each state, looking one step ahead using the converged $V^*$ :

For each state $s$ :

\pi^*(s) = \arg\max_{a} \sum_{s', r} p(s', r | s, a) [r + \gamma V^*(s')]

This step involves computing the expected value for each action in each state using the final value function $V^*$ and selecting the action yielding the highest value. Note that this policy extraction step is performed only once after the value function has converged.

Comparison with Policy Iteration

Policy Iteration: Alternates between full Policy Evaluation (computing $V^{\pi}$ for the current $\pi$ ) and Policy Improvement (making $\pi$ greedy with respect to $V^{\pi}$ ). Requires multiple sweeps for evaluation in each iteration.
Value Iteration: Directly iterates towards $V^*$ using the Bellman Optimality Equation. Each iteration involves one sweep applying the optimality update. It implicitly combines evaluation and improvement.

In practice, Value Iteration often converges faster than Policy Iteration, especially when the number of iterations needed for policy evaluation in PI is large. However, each VI iteration requires calculating the maximum over actions, which can be computationally intensive if the action space is large.

Like Policy Iteration, Value Iteration falls under the umbrella of Dynamic Programming methods. It finds the optimal solution but relies heavily on having a complete and accurate model of the environment's dynamics ( $p(s', r | s, a)$ ). This requirement is a significant limitation, which we'll address when we move to model-free learning methods in the upcoming chapters.