Update qlearning.md: gradient descent coefficients

chapter_reinforcement-learning/qlearning.md

@@ -22,19 +22,19 @@ As we discussed, implementing this algorithm requires knowing the MDP, specifica
 
 Let us imagine that the robot uses a policy $\pi_e(a \mid s)$ to take actions. Just like the previous chapter, it collects a dataset of $n$ trajectories of $T$ timesteps each $\{ (s_t^i, a_t^i)_{t=0,\ldots,T-1}\}_{i=1,\ldots, n}$. Recall that value iteration is really a set of constraints that ties together the action-value $Q^*(s, a)$ of different states and actions to each other. We can implement an approximate version of value iteration using the data that the robot has collected using $\pi_e$ as
 
-$$\hat{Q} = \min_Q \underbrace{\frac{1}{nT} \sum_{i=1}^n \sum_{t=0}^{T-1} (Q(s_t^i, a_t^i) - r(s_t^i, a_t^i) - \gamma \max_{a'} Q(s_{t+1}^i, a'))^2}_{\stackrel{\textrm{def}}{=} \ell(Q)}.$$
+$$\hat{Q} = \min_Q \underbrace{\frac{1}{nT} \sum_{i=1}^n \sum_{t=0}^{T-1} \frac{1}{2} (Q(s_t^i, a_t^i) - r(s_t^i, a_t^i) - \gamma \max_{a'} Q(s_{t+1}^i, a'))^2}_{\stackrel{\textrm{def}}{=} \ell(Q)}.$$
 :eqlabel:`q_learning_optimization_problem`
 
 Let us first observe the similarities and differences between this expression and value iteration above. If the robot's policy $\pi_e$ were equal to the optimal policy $\pi^*$, and if it collected an infinite amount of data, then this optimization problem would be identical to the optimization problem underlying value iteration. But while value iteration requires us to know $P(s' \mid s, a)$, the optimization objective does not have this term. We have not cheated: as the robot uses the policy $\pi_e$ to take an action $a_t^i$ at state $s_t^i$, the next state $s_{t+1}^i$ is a sample drawn from the transition function. So the optimization objective also has access to the transition function, but implicitly in terms of the data collected by the robot.
 
 The variables of our optimization problem are $Q(s, a)$ for all $s \in \mathcal{S}$ and $a \in \mathcal{A}$. We can minimize the objective using gradient descent. For every pair $(s_t^i, a_t^i)$ in our dataset, we can write
 
-$$\begin{aligned}Q(s_t^i, a_t^i) &\leftarrow Q(s_t^i, a_t^i) - \alpha \nabla_{Q(s_t^i,a_t^i)} \ell(Q) \\&=(1 - \alpha) Q(s_t^i,a_t^i) - \alpha \Big( r(s_t^i, a_t^i) + \gamma \max_{a'} Q(s_{t+1}^i, a') \Big),\end{aligned}$$
+$$\begin{aligned}Q(s_t^i, a_t^i) &\leftarrow Q(s_t^i, a_t^i) - \alpha \nabla_{Q(s_t^i,a_t^i)} \ell(Q) \\&=(1 - \alpha) Q(s_t^i,a_t^i) + \alpha \Big( r(s_t^i, a_t^i) + \gamma \max_{a'} Q(s_{t+1}^i, a') \Big),\end{aligned}$$
 :eqlabel:`q_learning`
 
 where $\alpha$ is the learning rate. Typically in real problems, when the robot reaches the goal location, the trajectories end. The value of such a terminal state is zero because the robot does not take any further actions beyond this state. We should modify our update to handle such states as
 
-$$Q(s_t^i, a_t^i) =(1 - \alpha) Q(s_t^i,a_t^i) - \alpha \Big( r(s_t^i, a_t^i) + \gamma (1 - \mathbb{1}_{s_{t+1}^i \textrm{ is terminal}} )\max_{a'} Q(s_{t+1}^i, a') \Big).$$
+$$Q(s_t^i, a_t^i) =(1 - \alpha) Q(s_t^i,a_t^i) + \alpha \Big( r(s_t^i, a_t^i) + \gamma (1 - \mathbb{1}_{s_{t+1}^i \textrm{ is terminal}} )\max_{a'} Q(s_{t+1}^i, a') \Big).$$
 
 where $\mathbb{1}_{s_{t+1}^i \textrm{ is terminal}}$ is an indicator variable that is one if $s_{t+1}^i$ is a terminal state and zero otherwise. The value of state-action tuples $(s, a)$ that are not a part of the dataset is set to $-\infty$. This algorithm is known as Q-Learning.