SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \mathbb{P}_{p\times q,G}\left(\widehat{\mathrm{FUSE}}_{1}(\boldsymbol {Z})\leq \text{quantile}\widehat{\mathrm{FUSE}}_{1}(\sigma \boldsymbol {Z})\right) \\ \leq \mathbb {P} _ {p \times q} \left(\widehat {\mathrm {F U S E}} _ {1} (\boldsymbol {Z}) \leq \underset {1 - \alpha_ {B}, \mathcal {G}} {\operatorname {q u a n t i l e}} \widehat {\mathrm {F U S E}} _ {1} (\sigma \boldsymbol {Z})\right) + \beta / 2. \\ \end{array}
$$
We also recall that from Theorem 5 when  $\lambda = cn / \kappa$
$$
\widehat{\operatorname{quantile}_{1 - \alpha_{B},\mathcal{G}}\operatorname{FUSE}_{1}} (\sigma \boldsymbol {Z})\leq \frac{C_{1}\kappa(1 + \log\alpha^{-1})}{n},
$$
$$
\mathbb {P} _ {p \times q} \left(\widehat {\mathrm {F U S E}} _ {1} (Z) \leq \underset {1 - \alpha_ {B}, \mathcal {G}} {\text {q u a n t i l e}} \widehat {\mathrm {F U S E}} _ {1} (\sigma \boldsymbol {Z})\right) \leq \mathbb {P} _ {p \times q} \left(\widehat {\mathrm {F U S E}} _ {1} (\boldsymbol {Z}) \leq \frac {C _ {1} \kappa \left(1 + \log \alpha^ {- 1}\right)}{n}\right).
$$
For any  $\rho$ , we define
$$
S _ {\rho} = \mathrm {M M D} _ {K _ {\rho}} ^ {2} (p, q) - \frac {\mathrm {K L} (\rho , \pi)}{\lambda} - \frac {C _ {1} \kappa (1 + \log \alpha^ {- 1})}{n},
$$
which we substitute into
$$
\begin{array}{l} \mathbb {P} _ {p \times q} \left(\widehat {\mathrm {F U S E}} _ {1} (\boldsymbol {Z}) \leq \frac {C _ {1} \kappa (1 + \log \alpha^ {- 1})}{n}\right) \\ = \mathbb {P} \left(\operatorname {M M D} _ {K _ {\rho}} ^ {2} (p, q) - \frac {1}{\lambda} \operatorname {K L} (\rho , \pi) - \widehat {\operatorname {F U S E}} _ {1} (\boldsymbol {Z}) \geq S _ {\rho}\right) \\ = \mathbb {P} \left(\operatorname {M M D} _ {K _ {\rho}} ^ {2} (p, q) - \frac {1}{\lambda} \operatorname {K L} (\rho , \pi) - \sup  _ {\rho^ {\prime}} \left(\widehat {\operatorname {M M D}} _ {K _ {\rho^ {\prime}}} ^ {2} (\boldsymbol {Z}) - \frac {1}{\lambda} \operatorname {K L} (\rho^ {\prime}, \pi)\right) \geq S _ {\rho}\right) \\ \leq \mathbb {P} \left(\operatorname {M M D} _ {K _ {\rho}} ^ {2} (p, q) - \widehat {\operatorname {M M D}} _ {K _ {\rho}} ^ {2} (\boldsymbol {Z}) \geq S _ {\rho}\right) \\ = \frac {1}{S _ {\rho} ^ {2}} \mathbb {V} _ {p \times q} \left[ \widehat {\operatorname {M M D}} _ {K _ {\rho}} ^ {2} (\boldsymbol {Z}) \right] \\ \leq \frac {C _ {2}}{S _ {\rho} ^ {2}} \left(\frac {4 \kappa \mathrm {M M D} ^ {2}}{n} + \frac {\kappa^ {2}}{n ^ {2}}\right). \\ \end{array}
$$
After substituting  $S_{\rho}$ , we used the dual form of  $\widehat{\mathrm{FUSE}}_1$  and the inequalities  $\sup f(\rho') \geq f(\rho)$ , Chebyshev's, and Theorem 14. This term is upper bounded by  $\beta / 2$  if we set
$$
S _ {\rho} ^ {2} > \frac {2 C _ {2}}{\beta} \left(\frac {4 \kappa \mathrm {M M D} ^ {2}}{n} + \frac {\kappa^ {2}}{n ^ {2}}\right).
$$
We also note that for  $a, b, x$  all non-negative, if  $x^2 > a^2 + 2b$ , then  $x^2 > ax + b$ . This works because  $x^2 > ax + b$  is equivalent to  $x^2 > \left(\frac{a}{2} + \sqrt{\frac{a^2}{4} + b}\right)^2$  by taking the positive root, and
$$
\left(\frac {a}{2} + \sqrt {\frac {a ^ {2}}{4} + b}\right) ^ {2} = \frac {a ^ {2}}{2} + b + 2 \frac {a}{2} \sqrt {\frac {a ^ {2}}{4} + b} \leq a ^ {2} + 2 b
$$
using Young's inequality  $2AB \leq A^2 + B^2$ .
Combining the above the Type II error rate is controlled by  $\beta$  provided any of the following statements are true for any  $\rho$  (with each new result implying the former):
$$
\mathrm {M M D} _ {K _ {\rho}} ^ {2} (p, q) > \frac {\mathrm {K L} (\rho , \pi)}{\lambda} + \frac {C _ {1} \kappa (1 + \log \alpha^ {- 1})}{n} + \sqrt {\frac {2 C _ {2}}{\beta} \left(\frac {4 \kappa \mathrm {M M D} ^ {2}}{n} + \frac {\kappa^ {2}}{n ^ {2}}\right)}
$$
$$
\mathrm {M M D} _ {K _ {\rho}} ^ {2} (p, q) > \frac {\mathrm {K L} (\rho , \pi)}{\lambda} + \frac {C _ {1} \kappa (1 + \log \alpha^ {- 1})}{n} + \sqrt {\frac {8 C _ {2} \kappa}{n \beta}} \mathrm {M M D} + \frac {\sqrt {2 C _ {2}} \kappa}{n \sqrt {\beta}}
$$
$$
\mathrm {M M D} _ {K _ {\rho}} ^ {2} (p, q) > \frac {2 \mathrm {K L} (\rho , \pi)}{\lambda} + \frac {2 C _ {1} \kappa (1 + \log \alpha^ {- 1})}{n} + \frac {8 C _ {2} \kappa}{n \beta} + \frac {2 \sqrt {2 C _ {2}} \kappa}{n \sqrt {\beta}}
$$
$$
\mathrm {M M D} _ {K _ {\rho}} ^ {2} (p, q) > \frac {C _ {3} \kappa}{n} \left(\frac {1}{\beta} + \log \frac {1}{\alpha} + \mathrm {K L} (\rho , \pi)\right)
$$
where we used that  $\sqrt{x + y} \leq \sqrt{x} + \sqrt{y}$ , the result above, and  $\lambda = cn / \kappa$ .
The proof of this statement proceeds similarly to the proof of Theorem 2. Using Theorem 5 with  $\lambda = cn$  and Theorem 13 we find that
$$
\begin{array}{l} \mathbb {P} _ {p \times q, G} \left(\widehat {\mathrm {F U S E}} _ {N} (\boldsymbol {Z}) \leq \underset {1 - \alpha , G} {\text {q u a n t i l e}} \widehat {\mathrm {F U S E}} _ {N} (\sigma \boldsymbol {Z})\right) \\ \leq \mathbb {P} _ {p \times q} \left(\widehat {\mathrm {F U S E}} _ {N} (\boldsymbol {Z}) \leq \frac {C _ {1} (1 + \log \alpha^ {- 1})}{n}\right) + \beta / 2. \\ \end{array}
$$
For any  $\rho$  such that  $\mathrm{KL}(\rho, \pi) < \infty$  we define
$$
T _ {\rho} = \frac {C _ {1}}{\kappa} \cdot \mathrm {M M D} _ {K _ {\rho}} ^ {2}
$$
$$
S _ {\rho} = T _ {\rho} - \frac {1}{\lambda} \mathrm {K L} (\rho , \pi) - \frac {C _ {1} \kappa (1 + \log \alpha^ {- 1})}{n}.
$$
We assumed that  $n \leq m \leq cn$  for some  $c \geq 1$ . Based on this, note that  $N_{k} \leq \kappa^{2} / C_{1}^{2}$  for  $C_{1}$  depending only on  $m / n \in [1, c]$ , and (since  $\mathrm{MMD}^2 \geq 0$  is strictly non-negative, unlike  $\widehat{\mathrm{MMD}}^2$  which can be negative),
$$
- \mathbb {E} _ {\rho} \left[ \frac {\operatorname {M M D} _ {k} ^ {2}}{\sqrt {\hat {N} _ {k} (\boldsymbol {Z})}} \right] \leq - \frac {C _ {1}}{\kappa} \mathbb {E} _ {k \sim \rho} \left[ \operatorname {M M D} _ {k} ^ {2} \right] = - T _ {\rho}. \tag {13}
$$
Now we introduce these definitions to bound
$$
\begin{array}{l} \mathbb {P} \left(\widehat {\mathrm {F U S E}} _ {N} (\boldsymbol {Z}) \leq \frac {C _ {1} \left(1 + \log \alpha^ {- 1}\right)}{n}\right) \\ = \mathbb {P} \left(T _ {\rho} - \frac {1}{\lambda} \operatorname {K L} (\rho , \pi) - \widehat {\operatorname {F U S E}} _ {N} (\boldsymbol {Z}) \geq S _ {\rho}\right) \\ = \mathbb {P} \left(T _ {\rho} - \frac {1}{\lambda} \operatorname {K L} (\rho , \pi) - \sup  _ {\rho^ {\prime}} \left(\mathbb {E} _ {k \sim \rho^ {\prime}} \left[ \frac {\widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z})}{\sqrt {\hat {N} _ {k} (\boldsymbol {Z})}} \right] - \frac {1}{\lambda} \operatorname {K L} (\rho^ {\prime}, \pi)\right) \geq S _ {\rho}\right) \\ \leq \mathbb {P} \left(T _ {\rho} - \mathbb {E} _ {k \sim \rho} \left[ \frac {\widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z})}{\sqrt {\hat {N} _ {k} (\boldsymbol {Z})}} \right] \geq S _ {\rho}\right) \quad \sup  f (\rho^ {\prime}) \geq f (\rho) \\ = \mathbb {P} \left(T _ {\rho} - \mathbb {E} _ {k \sim \rho} \left[ \frac {\operatorname {M M D} _ {k} ^ {2}}{\sqrt {\widehat {N} _ {k} (\boldsymbol {Z})}} - \frac {\operatorname {M M D} _ {k} ^ {2}}{\sqrt {\widehat {N} _ {k} (\boldsymbol {Z})}} + \frac {\widehat {\operatorname {M M D} _ {k} ^ {2}} (\boldsymbol {Z})}{\sqrt {\widehat {N} _ {k} (\boldsymbol {Z})}} \right] \geq S _ {\rho}\right) \\ = \mathbb {P} \left(T _ {\rho} - \mathbb {E} _ {k \sim \rho} \left[ \frac {\operatorname {M M D} _ {k} ^ {2}}{\sqrt {\widehat {N} _ {k} (\boldsymbol {Z})}} \right] + \mathbb {E} _ {k \sim \rho} \left[ \frac {\operatorname {M M D} _ {k} ^ {2} - \widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z})}{\sqrt {\widehat {N} _ {k} (\boldsymbol {Z})}} \right] \geq S _ {\rho}\right) \\ = \mathbb {P} \left(T _ {\rho} - \mathbb {E} _ {k \sim \rho} \left[ \frac {\operatorname {M M D} _ {k} ^ {2}}{\sqrt {\widehat {N} _ {k} (\boldsymbol {Z})}} \right] + \mathbb {E} _ {k \sim \rho} \left[ \frac {\operatorname {M M D} _ {k} ^ {2} - \widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z})}{\sqrt {\widehat {N} _ {k} (\boldsymbol {Z})}} \right] \geq S _ {\rho}\right) \\ \end{array}
$$
$$
\begin{array}{l} \leq \mathbb {P} \left(\mathbb {E} _ {k \sim \rho} \left[ \frac {\operatorname {M M D} _ {k} ^ {2} - \widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z})}{\sqrt {\hat {N} _ {k} (\boldsymbol {Z})}} \right] \geq S _ {\rho}\right) \\ \leq \mathbb {P} \left(\left| \mathbb {E} _ {k \sim \rho} \left[ \frac {\operatorname {M M D} _ {k} ^ {2} - \widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z})}{\sqrt {\hat {N} _ {k} (\boldsymbol {Z})}} \right] \right| \geq S _ {\rho}\right) \\ \leq \mathbb {P} \left(\mathbb {E} _ {k \sim \rho} \left[ \left. \frac {\operatorname {M M D} _ {k} ^ {2} - \widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z})}{\sqrt {\hat {N} _ {k} (\boldsymbol {Z})}} \right\rvert   \geq S _ {\rho}\right) \right. \\ = \mathbb {P} \left(\mathbb {E} _ {k \sim \rho} \left[ \frac {\mid \operatorname {M M D} _ {k} ^ {2} - \widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z}) \mid}{\sqrt {\hat {N} _ {k} (\boldsymbol {Z})}} \right] \geq S _ {\rho}\right) \\ \leq \frac {1}{S _ {\rho}} \mathbb {E} _ {\boldsymbol {Z}} \mathbb {E} _ {k \sim \rho} \left[ \frac {| \operatorname {M M D} _ {k} ^ {2} - \widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z}) |}{\sqrt {\widehat {N} _ {k} (\boldsymbol {Z})}} \right] \\ \leq \frac {1}{S _ {\rho}} \sqrt {\mathbb {E} _ {\boldsymbol {Z}} \mathbb {E} _ {k \sim \rho} \left[ | \mathrm {M M D} _ {k} ^ {2} - \widehat {\mathrm {M M D}} _ {k} ^ {2} (\boldsymbol {Z}) | ^ {2} \right] \mathbb {E} _ {\boldsymbol {Z}} \mathbb {E} _ {k \sim \rho} \left[ \frac {1}{\widehat {N} _ {k} (\boldsymbol {Z})} \right]} \\ \end{array}
$$
by Equation (13)
$$
x \leq | x |
$$
$|x|$  convex, Jensen
$\widehat{N}_k$  positive
$\widehat{N}_k$  positive, Markov
Cauchy-Schwarz.
Thus the type II error will be controlled by  $\beta$  if for any  $\rho$
$$
S _ {\rho} > \frac {2}{\beta} \sqrt {\mathbb {E} _ {\boldsymbol {Z}} \mathbb {E} _ {k \sim \rho} \left[ | \operatorname {M M D} _ {k} ^ {2} - \widehat {\operatorname {M M D}} _ {k} ^ {2} (\boldsymbol {Z}) | ^ {2} \right] \mathbb {E} _ {\boldsymbol {Z}} \mathbb {E} _ {k \sim \rho} \left[ \frac {1}{\widehat {N} _ {k} (\boldsymbol {Z})} \right]}.
$$
Provided there is a  $c > 0$  with
$$
\mathbb {E} _ {p \times q} \left[ \frac {1}{\widehat {N} _ {k} (\boldsymbol {Z})} \right] \leq c
$$
for all  $k$ , this reduces to the condition
$$
\mathrm {M M D} _ {K _ {\rho}} ^ {2} (p, q) > C _ {2} \kappa \left(\frac {\log \alpha^ {- 1}}{n} + \frac {1}{\beta} \sqrt {\mathbb {E} _ {\rho} \mathbb {V} _ {p \times q} \left[ \widehat {\mathrm {M M D} _ {k} ^ {2}} (\boldsymbol {Z}) \right]} + \frac {\mathrm {K L} (\rho , \pi)}{n}\right).
$$
Applying Theorem 14, the proof is completed in essentially the same way as in the result for  $\widehat{\mathrm{FUSE}}_1$  (with slightly different  $\beta$  dependence).