SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathcal {M} (X) = \left(\mathcal {M} _ {1} \left(X _ {1}\right), \mathcal {M} _ {2} \left(X _ {2}, \mathcal {M} _ {1} \left(X _ {1}\right)\right), \dots , \mathcal {M} _ {k} \left(X _ {k}, \mathcal {M} _ {1} \left(X _ {1}\right), \dots , \mathcal {M} _ {k - 1} \left(X _ {k - 1}\right)\right)\right).
$$
$$
\max _ {X \sim Y} D _ {f} \big (\mathcal {M} (X) | | \mathcal {M} (Y) \big) \leq \max _ {i \in [ k ]} \max _ {X \sim Y} D _ {f} \big (\mathcal {M} _ {i} (X) | | \mathcal {M} _ {i} (Y) \big).
$$
Proof. Let  $X$  and  $Y$  be divided into  $k$  equal-sized disjoint shards and suppose  $Y$  is a neighbouring dataset such that  $X$  and  $Y$  differ in  $j^{th}$  shard, i.e.,  $X_{j} \sim Y_{j}$  and  $Y = (Y_{1}, Y_{2}, \ldots, Y_{k}) = (X_{1}, \ldots, X_{j-1}, Y_{j}, X_{j+1}, \ldots, X_{k})$ .
$$
\begin{array}{l} \begin{array}{l} \mathbb {P} \big (\mathcal {M} (X) = (a _ {1}, \ldots , a _ {k}) \big) \\ \hline \mathbb {P} \big (\mathcal {M} (Y) = (a _ {1}, \ldots , a _ {k}) \big) \end{array} \\ = \frac {\mathbb {P} \left(\mathcal {M} _ {1} \left(X _ {1}\right) = a _ {1}\right) \cdot \mathbb {P} \left(\mathcal {M} _ {1} \left(X _ {2} , a _ {1}\right) = a _ {2}\right) \cdots \mathbb {P} \left(\mathcal {M} _ {k} \left(X _ {k} , a _ {1} , \dots , a _ {k - 1}\right) = a _ {k}\right)}{\mathbb {P} \left(\mathcal {M} _ {1} \left(Y _ {1}\right) = a _ {1}\right) \cdot \mathbb {P} \left(\mathcal {M} _ {1} \left(Y _ {2} , a _ {1}\right) = a _ {2}\right) \cdots \mathbb {P} \left(\mathcal {M} _ {k} \left(Y _ {k} , a _ {1} , \dots , a _ {k - 1}\right) = a _ {k}\right)} \\ = \frac {\mathbb {P} \big (\mathcal {M} _ {j} (X _ {j} , a _ {1} , \ldots , a _ {j - 1}) = a _ {j} \big)}{\mathbb {P} \big (\mathcal {M} _ {j} (Y _ {j} , a _ {1} , \ldots , a _ {j - 1}) = a _ {j} \big)}. \\ \end{array}
$$
and furthermore, denoting  $a = (a_{1},\ldots ,a_{k})$
$$
\begin{array}{l} D _ {f} \left(\mathcal {M} (X) | | \mathcal {M} (Y))\right) \\ = \int f \left(\frac {\mathbb {P} (\mathcal {M} (X) = a)}{\mathbb {P} (\mathcal {M} (Y) = a)}\right) \mathbb {P} (\mathcal {M} (Y) = a) d a \\ = \int f \left(\frac {\mathbb {P} \left(\mathcal {M} _ {j} \left(X _ {j} , a _ {1} , \dots , a _ {j - 1}\right) = a _ {j}\right)}{\mathbb {P} \left(\mathcal {M} _ {j} \left(Y _ {j} , a _ {1} , \dots , a _ {j - 1}\right) = a _ {j}\right)}\right) \mathbb {P} \left(\mathcal {M} (Y) = a\right) d a \\ = \int f \left(\frac {\mathbb {P} \left(\mathcal {M} _ {j} \left(X _ {j} , a _ {1} , \dots , a _ {j - 1}\right) = a _ {j}\right)}{\mathbb {P} \left(\mathcal {M} _ {j} \left(Y _ {j} , a _ {1} , \dots , a _ {j - 1}\right) = a _ {j}\right)}\right) \mathbb {P} \left(\mathcal {M} _ {1} (Y _ {1}) = a _ {1}\right) \cdot \mathbb {P} \left(\mathcal {M} _ {1} \left(Y _ {2}, a _ {1}\right) = a _ {2}\right) \dots \\ \mathbb {P} \left(\mathcal {M} _ {j} \left(Y _ {j}, a _ {1}, \dots , a _ {j - 1}\right) = a _ {j}\right) \mathrm {d} a _ {1} \dots \mathrm {d} a _ {j} \\ = D _ {f} \left(\mathcal {M} _ {j} \left(X _ {j}\right) \mid \mid \mathcal {M} _ {j} \left(Y _ {j}\right)\right)) \cdot \int \mathbb {P} \left(\mathcal {M} _ {1} \left(Y _ {1}\right) = a _ {1}\right) \cdot \mathbb {P} \left(\mathcal {M} _ {1} \left(Y _ {2}, a _ {1}\right) = a _ {2}\right) \dots \\ \mathbb {P} \left(\mathcal {M} _ {j - 1} \left(Y _ {j - 1}, a _ {1}, \dots , a _ {j - 2}\right) = a _ {j - 1}\right) \mathrm {d} a _ {1} \dots \mathrm {d} a _ {j - 1} \\ = D _ {f} \left(\mathcal {M} _ {j} \left(X _ {j}\right) \mid \mid \mathcal {M} _ {j} \left(Y _ {j}\right)\right)\left. \right). \\ \end{array}
$$
$$
D _ {f} (\mathcal {M} (X) | | \mathcal {M} (Y)) = D _ {f} (\mathcal {M} _ {j} (X _ {j}) | | \mathcal {M} _ {j} (Y _ {j})) \leq \max  _ {X \sim Y} D _ {f} (\mathcal {M} _ {j} (X) | | \mathcal {M} _ {j} (Y))
$$
and also, we have that
$$
\max _ {X \sim Y} D _ {f} (\mathcal {M} (X) | | \mathcal {M} (Y)) = \max _ {i \in [ k ]} \max _ {X \sim Y} D _ {f} (\mathcal {M} _ {i} (X) | | \mathcal {M} _ {i} (Y)).
$$
Corollary 13. Suppose a dataset  $X \in \mathcal{X}^N$  is divided into  $k$  disjoint shards  $X_{i}$ ,  $i \in [k]$ , and mechanisms  $\mathcal{M}_i$ ,  $i \in [k]$ , are applied to the shards, respectively. Consider the mechanism
$$
\mathcal {M} (X) = \left(\mathcal {M} _ {1} \left(X _ {1}\right), \dots , \mathcal {M} _ {k} \left(X _ {k}\right)\right).
$$
Suppose each  $\mathcal{M}_i$  is  $(\alpha, \varepsilon_i(\alpha))$ -RDP, respectively. Then,  $\mathcal{M}$  is  $(\alpha, \max_{i \in [k]} \varepsilon_i(\alpha))$ -RDP.
Proof. This follows from Lemma 12 since
$$
\exp \left((\alpha - 1) D _ {\alpha} (\mathcal {M} (X) | | \mathcal {M} (Y))\right) = \int \left(\frac {\mathbb {P} \big (\mathcal {M} (X) = a \big)}{\mathbb {P} \big (\mathcal {M} (Y) = a \big)}\right) ^ {\alpha} \mathbb {P} \big (\mathcal {M} (Y) = a \big) d a
$$
is an  $f$ -divergence for  $f(x) = x^{\alpha}$ . Thus, by Lemma 12 we have that
$$
\max  _ {X \sim Y} \exp \left((\alpha - 1) D _ {\alpha} (\mathcal {M} (X) | | \mathcal {M} (Y))\right) \leq \max  _ {i \in [ k ]} \max  _ {X \sim Y} \exp \left((\alpha - 1) D _ {\alpha} (\mathcal {M} _ {i} (X) | | \mathcal {M} _ {i} (Y))\right)
$$
from which it follows that
$$
\max  _ {X \sim Y} D _ {\alpha} (\mathcal {M} (X) | | \mathcal {M} (Y)) \leq \max  _ {i \in [ k ]} \max  _ {X \sim Y} D _ {\alpha} (\mathcal {M} _ {i} (X) | | \mathcal {M} _ {i} (Y)) = \max  _ {i \in [ k ]} \varepsilon_ {i} (\alpha).
$$