SlowGuess/ACL_2020_OCR huggingface dataset:
$$
\mathbb {E} _ {\mathbf {y} \sim q} \left[ \sum_ {i} \phi^ {i} (\mathbf {y} ^ {i}) \right] \leq \mathbf {0}
$$
$$
\sum_ {i} \mathbb {E} _ {\mathbf {y} ^ {i} \sim q (\cdot , i)} \left[ \phi^ {i} \left(\mathbf {y} ^ {i}\right) \right] \leq \mathbf {0}
$$
Thus, it is equivalent as  $\forall v^{*}$
$$
\sum_ {i} \mathbb {E} _ {\mathbf {y} ^ {i} \sim q (\cdot , i)} \left[ \phi_ {v ^ {*}, -} ^ {i} (\mathbf {y} ^ {i}) \right] \leq \mathbf {0},
$$
$$
\sum_ {i} \mathbb {E} _ {\mathbf {y} ^ {i} \sim q (\cdot , i)} \left[ \phi_ {v ^ {*}, +} ^ {i} (\mathbf {y} ^ {i}) \right] \leq \mathbf {0}.
$$
The inequality about  $\phi_{v^{*}, - }^{i}$  can be derived as
$$
\sum_ {i} \mathbb {E} _ {\mathbf {y} ^ {i} \sim q (\cdot , i)} \left[ \phi_ {v ^ {*}, -} ^ {i} \left(\mathbf {y} ^ {i}\right) \right] \leq \mathbf {0}
$$
$$
\sum_ {i} \sum_ {\mathbf {y} ^ {i}} q (\mathbf {y} ^ {i}, i) \phi_ {v ^ {*}, -} ^ {i} (\mathbf {y} ^ {i}) \leq \mathbf {0}
$$
$$
\begin{array}{l} \sum_ {i} \sum_ {\mathbf {y} ^ {i}: \mathbf {y} _ {v} ^ {i} = v ^ {*}, \mathbf {y} _ {r} ^ {i} \in M} (1 - b ^ {*} - \gamma) q (\mathbf {y} ^ {i}, i) - \\ \sum_ {i} \sum_ {\mathbf {y} ^ {i}: \mathbf {y} _ {v} ^ {i} = v ^ {*}, \mathbf {y} _ {r} ^ {i} \in W} (b ^ {*} + \gamma) q (\mathbf {y} ^ {i}, i) \leq \mathbf {0} \\ \frac {\sum_ {i} \sum_ {\mathbf {y} ^ {i} : \mathbf {y} _ {v} ^ {i} = v ^ {*} , \mathbf {y} _ {r} ^ {i} \in M} q (\mathbf {y} ^ {i} , i)}{\sum_ {i} \sum_ {\mathbf {y} ^ {i} : \mathbf {y} _ {v} ^ {i} = v ^ {*} , \mathbf {y} _ {r} ^ {i} \in M \cup W} q (\mathbf {y} ^ {i} , i)} \leq b ^ {*} + \gamma \\ B (q, v ^ {*}, \cdot) \leq b ^ {*} + \gamma \\ \end{array}
$$
The inequality about  $\phi_{v^*, - }^i$  can be derived similarly.