  We take the first step in studying general sequential decision-making under
two adaptivity constraints: rare policy switch and batch learning. First, we
provide a general class called the Eluder Condition class, which includes a
wide range of reinforcement learning classes. Then, for the rare policy switch
constraint, we provide a generic algorithm to achieve a
$\widetilde{\mathcal{O}}(\log K) $ switching cost with a
$\widetilde{\mathcal{O}}(\sqrt{K})$ regret on the EC class. For the batch
learning constraint, we provide an algorithm that provides a
$\widetilde{\mathcal{O}}(\sqrt{K}+K/B)$ regret with the number of batches $B.$
This paper is the first work considering rare policy switch and batch learning
under general function classes, which covers nearly all the models studied in
the previous works such as tabular MDP (Bai et al. 2019; Zhang et al. 2020),
linear MDP (Wang et al. 2021; Gao et al. 2021), low eluder dimension MDP (Kong
et al. 2021; Gao et al. 2021), generalized linear function approximation (Qiao
et al. 2023), and also some new classes such as the low $D_\Delta$-type Bellman
eluder dimension problem, linear mixture MDP, kernelized nonlinear regulator
and undercomplete partially observed Markov decision process (POMDP).
