Path planning using reinforcement learning in a n x n grid environment:

Starting position at (0, 0) (top left), and goal position at (n-1, n-1) (bottom right).

Current techniques

1. Monte-Carlo control without exploring starts
2. SARSA with an $\epsilon$-greedy behavior policy
3. Q-learning with an $\epsilon$-greedy behavior policy

Update rule:

$$Q\left(S_t,A_t\right)← Q\left(S_t,A_t\right)+\alpha\left[R_{t+1}+\gamma Q\left(S_{t+1}, A_{t+1}\right)-Q\left(S_t,A_t\right)\right]$$

Update rule:

$$Q\left(S_t,A_t\right)← Q\left(S_t,A_t\right)+\alpha\left[R_{t+1}+\gamma \text{max}_ {a'}Q\left(S_{t+1}, a'\right)-Q\left(S_t,A_t\right)\right]$$

$$\pi\left(a|s\right) \begin{cases} 1-\epsilon+\frac{\epsilon}{\left|A\left(s\right)\right|}, & \text{if}\ a=A^{*}≜\text{argmax}_{a}Q\left(s,a\right) \\ \frac{\epsilon}{\left|A\left(s\right)\right|}, & \text{if}\ a\neq A^{*} \end{cases}$$