Greedy Policy from Action Values

1. Background Knowledge

In Reinforcement Learning, an agent interacts with an environment by selecting actions based on a policy $\pi$. A policy maps states to actions. When we have estimates of the expected return for taking each action in a given state, we can improve the policy by making it greedy: at every state, choose the action with the highest estimated value. This process is known as policy improvement and is a core component of algorithms like Policy Iteration and Q-Learning.

The input to this problem is a matrix $Q$, often called the action-value function or Q-function. Each row $Q[s]$ corresponds to a state $s$ and contains the estimated value of taking each possible action in that state. The goal is to derive a deterministic policy from these values. A deterministic policy selects exactly one action per state, as opposed to a stochastic policy which might distribute probability mass across multiple actions.

A critical detail in this problem is the tie-breaking rule: if multiple actions share the same maximum value in a given state, the policy must select the action with the lowest index. This ensures the resulting policy is well-defined and reproducible, which is essential for consistent evaluation and further learning steps.

2. Algorithm Approach

The problem reduces to a straightforward argmax operation applied independently to each row of the matrix $Q$. For each state $s$, we need to find the index $a$ that maximizes $Q[s][a]$. If there are ties, we select the smallest such index $a$.

This is a classic pattern in optimization and machine learning: given a vector of scores, return the index of the best score. The key insight is that this operation is state-local; the decision at state $s$ depends only on the values in $Q[s]$ and not on values in other rows. Therefore, we can process each row independently in a single pass.

3. Step-by-Step Strategy