Fitted-Q Iteration

Main Idea

Q-function. The Q-function-based approach aims to direct learn the state-action value function (referred to as the Q-function)

(102)\[\begin{eqnarray} Q^\pi(a,s)&= \mathbb{E}^{\pi} (\sum_{t=0}^{+\infty} \gamma^t R_{t}|A_{0}=a,S_{0}=s) \end{eqnarray}\]

of either the policy \(\pi\) that we aim to evaluate or the optimal policy \(\pi = \pi^*\).

Bellman optimality equations. The Q-learning-type policy learning is commonly based on the Bellman optimality equation, which characterizes the optimal policy \(\pi^*\) and is commonly used in policy optimization. Specifically, \(Q^*\) is the unique solution of

(103)\[\begin{equation} Q(a, s) = \mathbb{E} \Big(R_t + \gamma \arg \max_{a'} Q(a, S_{t+1}) | A_t = a, S_t = s \Big). \;\;\;\;\; \text{(2)} \end{equation}\]

FQI. Similar to FQE, the fitted-Q iteration (FQI) [EGW05] algorithm is also popular due to its simple form and good numerical performance. It is mainly motivated by the fact that, the optimal value function \(Q^*\) is the unique solution to the Bellman optimality equation (2). Besides, the right-hand side of (2) is a contraction mapping. Therefore, we can consider a fixed-point method: with an initial estimate \(\widehat{Q}^{0}\), FQI iteratively solves the following optimization problem,

(104)\[\begin{eqnarray} \widehat{Q}^{{\ell}}=\arg \min_{Q} \sum_{\substack{i \le n}}\sum_{t<T} \Big\{ \gamma \max_{a'} \widehat{Q}^{\ell-1}(a',S_{i, t+1}) +R_{i,t}- Q(A_{i, t}, S_{i, t}) \Big\}^2, \end{eqnarray}\]

for \(\ell=1,2,\cdots\), until convergence. The final estimate is denoted as \(\widehat{Q}_{FQI}\).

References

[EGW05]

Damien Ernst, Pierre Geurts, and Louis Wehenkel. Tree-based batch mode reinforcement learning. Journal of Machine Learning Research, 2005.