# Why does Q-learning converge under 100% exploration rate?

I am working on this assignment where I made the agent learn state-action values (Q-values) with Q-learning and 100% exploration rate. The environment is the classic gridworld as shown in the following picture.

Here are the values of my parameters.

• Learning rate = 0.1
• Discount factor = 0.95
• Default reward = 0

Reaching the trophy is the final reward, no negative reward is given for bumping into walls or for taking a step.

After 500 episodes, the arrows have converged. As shown in the figure, some states have longer arrows than others (i.e., larger Q-values). Why is this so? I don't understand how the agent learns and finds the optimal actions and states when the exploration rate is 100% (each action: N-S-E-W has 25% chance to be selected)

The reason that different state-action pairs have longer arrows, i.e. higher Q-values, is simply because the value of being in that state-action pair is higher. An example would be the arrow pointing down right above the trophy -- obviously this has the highest Q-value as the return is 1. For all other states it will be $$\gamma^k$$ for some $$k$$ -- to see this remember that a Q-value is defined as
$$Q(s, a) = \mathbb{E}_\pi \left[\sum_{j=0}^\infty \gamma^j R_{t+j+1} |S_t = s, A_t = a \right]\;;$$ so for any state-action pair that is not the block above the trophy with the down arrow $$\sum_{j=0}^\infty \gamma^j R_{t+j+1}$$ will be a sum of $$0$$'s plus $$\gamma^T$$ where $$T$$ is the time that you finally reach the trophy (assuming you give a reward of 1 for reaching the trophy).