SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Take any  $\mu \in \mathcal{P}_M(\mathcal{H}_b)$  and consider any weighted ensemble  $h^\alpha$  over  $\mathcal{H}_b$  of the form  $h^\alpha(x) = \mathbb{1}\{\sum_{m\in [M]}p_mh_m > \alpha\}$  with  $p_m = \mu(h_m)$ . Define the function  $g^\alpha : \{0,1\}^M \to \{0,1\}$  as  $g^\alpha(z_1\ldots z_M) = \mathbb{1}\left\{\sum_{m=1}^{M}p_mz_m > \alpha\right\}$ . Then,  $h^\alpha$  can be written as  $g^\alpha(h_1(x)\ldots h_M(x))$ . Because the  $p_m$  are positive, the function  $g^\alpha$  is a monotone boolean function. As any monotone boolean function,  $g^\alpha$  can be written as a disjunctive normal form (DNF) without negations [9]. Thus, the set  $A_{\mathrm{ENS}} = \{x \in \mathcal{X} : h^\alpha(x) = 1\}$  is a union of intersections over the sets  $A_1\ldots A_M$  where  $A_k = h_m^{-1}(1)$ . Because  $\mathcal{A}$  is closed under union and intersection,  $A_{\mathrm{ENS}} \in \mathcal{A}$ , which means that  $h^\alpha \in \mathcal{H}_b$ . Thus,  $\overline{\mathcal{H}_b}(\mathcal{P}_M(\mathcal{H}_b)) \subseteq \mathcal{H}_b$ .
As all the hypothesis of Theorem 5.1 are met, we can conclude that  $\inf_{h\in \mathcal{H}_b}\mathcal{R}_{\epsilon}(h) = \inf_{\mu \in \mathcal{P}_M(\mathcal{H}_b)}\mathcal{R}_{\epsilon}(\mathbf{h}_{\mu})$