SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathbb {P} _ {0} ^ {n} \left(\int_ {\Theta} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \tilde {\Pi} (\theta) \leq e ^ {- (1 + C _ {3}) n \epsilon^ {2}}\right) = \mathbb {P} _ {0} ^ {n} \left(\left[ \int_ {\Theta} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \tilde {\Pi} (\theta) \right] ^ {- \lambda} \geq e ^ {(1 + C _ {3}) \lambda n \epsilon^ {2}}\right).
$$
Then, using the Markov's inequality in the last equality above, we have
$$
\begin{array}{l} \mathbb {P} _ {0} ^ {n} \left(\int_ {\Theta} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \tilde {\Pi} (\theta) \leq e ^ {- (1 + C _ {3}) n \epsilon^ {2}}\right) \leq e ^ {- (1 + C _ {3}) \lambda n \epsilon^ {2}} \mathbb {E} _ {P _ {0} ^ {n}} \left(\left[ \int_ {\Theta} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \tilde {\Pi} (\theta) \right] ^ {- \lambda}\right) \\ \leq e ^ {- (1 + C _ {3}) \lambda n \epsilon^ {2}} \left[ \int_ {\Theta} \mathbb {E} _ {P _ {0} ^ {n}} \left(\left[ \mathcal {L R} _ {n} (\theta , \theta_ {0}) \right] ^ {- \lambda}\right) d \tilde {\Pi} (\theta) \right] \\ = e ^ {- (1 + C _ {3}) \lambda n \epsilon^ {2}} \left[ \int_ {\Theta} \exp \left(\lambda D _ {\lambda + 1} \left(P _ {0} ^ {n} \| P _ {\theta} ^ {n}\right)\right) d \tilde {\Pi} (\theta) \right] \\ \leq e ^ {- (1 + C _ {3}) \lambda n \epsilon^ {2}} e ^ {\lambda C _ {3} n \epsilon_ {n} ^ {2}} \leq \epsilon^ {- \lambda n \epsilon^ {2}}, \tag {25} \\ \end{array}
$$
where the second inequality follows from first applying Jensen's inequality (on the term inside  $[\cdot ]$ ) and then using Fubini's theorem, and the penultimate inequality follows from Assumption 2.3 and the definition of  $\tilde{\Pi} (\theta)$ .
Next, define the set  $K_{n} \coloneqq \{\theta \in \Theta : L_{n}(\theta, \theta_{0}) > C_{1} n \epsilon^{2}\}$ . Notice that set  $K_{n}$  is the set of alternate hypothesis as defined in Assumption 2.1. We bound the calibrated posterior probability of this set  $K_{n}$  to get a bound on the first term in the RHS of equation (19). Recall the sequence of test function  $\{\phi_{n,\epsilon}\}$  from Assumption 2.1. Observe that
$$
\begin{array}{l} \mathbb {E} _ {P _ {0} ^ {n}} \left[ \frac {\int_ {K _ {n}} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L} \mathcal {R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L} \mathcal {R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] \\ = \mathbb {E} _ {P _ {0} ^ {n}} \left[ (\phi_ {n, \epsilon} + 1 - \phi_ {n, \epsilon}) \frac {\int_ {K _ {n}} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] \\ \leq \mathbb {E} _ {P _ {0} ^ {n}} [ \phi_ {n, \epsilon} ] + \mathbb {E} _ {P _ {0} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \mathbb {1} _ {B _ {n} ^ {C}} \right] \\ + \mathbb {E} _ {P _ {0} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \mathbb {1} _ {B _ {n}} \frac {\int_ {K _ {n}} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L} \mathcal {R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L} \mathcal {R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] \\ \leq \mathbb {E} _ {P _ {0} ^ {n}} \phi_ {n, \epsilon} + \mathbb {E} _ {P _ {0} ^ {n}} \left[ \mathbb {1} _ {B _ {n} ^ {C}} \right] + \mathbb {E} _ {P _ {0} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \mathbb {1} _ {B _ {n}} \frac {\int_ {K _ {n}} e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right], \tag {26} \\ \end{array}
$$
where in the second inequality, we first divide the second term over set  $B_{n}$  and its complement and then use the fact that  $\frac{\int_{K_n}e^{\gamma R(a',\theta)}\mathcal{L}\mathcal{R}_n(\theta,\theta_0)d\Pi(\theta)}{\int_{\Theta}e^{\gamma R(a',\theta)}\mathcal{L}\mathcal{R}_n(\theta,\theta_0)d\Pi(\theta)} \leq 1$ . The third inequality is due the fact that  $\phi_{n,\epsilon} \in [0,1]$ . Next, using Assumption 2.3 and 2.5 observe that on set  $B_{n}$
$$
\begin{array}{l} \int_ {\Theta} e ^ {\gamma R \left(a ^ {\prime}, \theta\right)} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta) \geq W ^ {\gamma} \int_ {\Theta} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta) \\ \geq W ^ {\gamma} e ^ {- (1 + C _ {2} + C _ {3}) n \epsilon_ {n} ^ {2}} \geq W ^ {\gamma} e ^ {- (1 + C _ {2} + C _ {3}) n \epsilon^ {2}}. \\ \end{array}
$$
Substituting the equation above in the third term of equation (26), we obtain
$$
\begin{array}{l} \mathbb {E} _ {P _ {0} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \mathbb {1} _ {B _ {n}} \frac {\int_ {K _ {n}} e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right] \\ \leq W ^ {- \gamma} e ^ {(1 + C _ {2} + C _ {3}) n \epsilon^ {2}} \mathbb {E} _ {P _ {0} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \mathbb {1} _ {B _ {n}} \int_ {K _ {n}} e ^ {\gamma R (a ^ {\prime}, \theta)} \mathcal {L} \mathcal {R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta) \right] \\ \leq W ^ {- \gamma} e ^ {(1 + C _ {2} + C _ {3}) n \epsilon^ {2}} \mathbb {E} _ {P _ {0} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \int_ {K _ {n}} e ^ {\gamma R \left(a ^ {\prime}, \theta\right)} \mathcal {L} \mathcal {R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta) \right]. \quad (\star) \\ \end{array}
$$
Now using Fubini's theorem observe that,
$$
\begin{array}{l} (\star) = W ^ {- \gamma} e ^ {(1 + C _ {2} + C _ {3}) n \epsilon^ {2}} \int_ {K _ {n}} e ^ {\gamma R (a ^ {\prime}, \theta)} \mathbb {E} _ {P _ {\theta} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \right] d \Pi (\theta) \\ \leq W ^ {- \gamma} e ^ {(1 + C _ {2} + C _ {3} + C _ {4} (\gamma)) n \epsilon^ {2}} \left[ \int_ {K _ {n} \cap \{e ^ {\gamma R (a ^ {\prime}, \theta)} \leq e ^ {C _ {4} (\gamma) n \epsilon^ {2}} \}} \mathbb {E} _ {P _ {\theta} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \right] d \Pi (\theta) \right. \\ + e ^ {- C _ {4} (\gamma) n \epsilon^ {2}} \int_ {K _ {n} \cap \{e ^ {\gamma R (a ^ {\prime}, \theta)} > e ^ {C _ {4} (\gamma) n \epsilon^ {2}} \}} e ^ {\gamma R (a ^ {\prime}, \theta)} d \Pi (\theta) \Bigg ], \\ \end{array}
$$
where in the last inequality, we first divide the integral over set  $\{\theta \in \Theta : e^{\gamma R(a',\theta)} \leq e^{C_4(\gamma)n\epsilon^2}\}$  and its complement and then use the upper bound on  $e^{\gamma R(a',\theta)}$  in the first integral. Now, it follows that
$$
\begin{array}{l} (\star) \leq W ^ {- \gamma} e ^ {\left(1 + C _ {2} + C _ {3} + C _ {4} (\gamma)\right) n \epsilon^ {2}} \left[ \int_ {K _ {n}} \mathbb {E} _ {P _ {\theta} ^ {n}} \left[ \left(1 - \phi_ {n, \epsilon}\right) \right] d \Pi (\theta) \right. \\ \left. + e ^ {- C _ {4} (\gamma) n \epsilon^ {2}} \int_ {\{e ^ {\gamma R (a ^ {\prime}, \theta)} > e ^ {C _ {4} (\gamma) n \epsilon^ {2}} \}} e ^ {\gamma R (a ^ {\prime}, \theta)} d \Pi (\theta) \right] \\ = W ^ {- \gamma} e ^ {(1 + C _ {2} + C _ {3} + C _ {4} (\gamma)) n \epsilon^ {2}} \left[ \int_ {K _ {n} \cap \Theta_ {n} (\epsilon)} \mathbb {E} _ {P _ {\theta} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \right] d \Pi (\theta) \right. \\ + \int_ {K _ {n} \cap \Theta_ {n} (\epsilon) ^ {c}} \mathbb {E} _ {P _ {\theta} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \right] d \Pi (\theta) + e ^ {- C _ {4} (\gamma) n \epsilon^ {2}} \int_ {\left\{e ^ {\gamma R \left(a ^ {\prime}, \theta\right)} > e ^ {C _ {4} (\gamma) n \epsilon^ {2}} \right\}} e ^ {\gamma R \left(a ^ {\prime}, \theta\right)} d \Pi (\theta) \Bigg ] \\ \leq W ^ {- \gamma} e ^ {(1 + C _ {2} + C _ {3} + C _ {4} (\gamma)) n \epsilon^ {2}} \left[ \int_ {K _ {n} \cap \Theta_ {n} (\epsilon)} \mathbb {E} _ {P _ {\theta} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \right] d \Pi (\theta) + \Pi \left(\Theta_ {n} (\epsilon) ^ {c}\right) \right. \\ + e ^ {- C _ {4} (\gamma) n \epsilon^ {2}} \int_ {\{e ^ {\gamma R (a ^ {\prime}, \theta)} > e ^ {C _ {4} (\gamma) n \epsilon^ {2}} \}} e ^ {\gamma R (a ^ {\prime}, \theta)} d \Pi (\theta) \Bigg ], \\ \end{array}
$$
where the second equality is obtained by dividing the first integral on set  $\Theta_{n}(\epsilon)$  and its complement, and the second inequality is due the fact that  $\phi_{n,\epsilon}\in [0,1]$ . Now, using the equation above and Assumption 2.1, 2.2, and 2.4 observe that
$$
\begin{array}{l} \mathbb {E} _ {P _ {0} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \mathbb {1} _ {B _ {n}} \frac {\int_ {K _ {n}} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] \\ \leq W ^ {- \gamma} e ^ {(1 + C _ {2} + C _ {3} + C _ {4} (\gamma)) n \epsilon^ {2}} \left[ 2 e ^ {- C n \epsilon^ {2}} + e ^ {- (C _ {5} (\gamma) + C _ {4} (\gamma)) n \epsilon^ {2}} \right]. \\ \end{array}
$$
Hence, choosing  $C, C_2, C_3, C_4(\gamma)$  and  $C_5(\gamma)$  such that  $-1 > 1 + C_2 + C_3 + C_4(\gamma) - \min(C, (C_4(\gamma) + C_5(\gamma)))$  implies
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ (1 - \phi_ {n, \epsilon}) \mathbb {I} _ {B _ {n}} \frac {\int_ {K _ {n}} e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right] \leq 3 W ^ {- \gamma} e ^ {- n \epsilon^ {2}}. \tag {27}
$$
$$
\mathbb {E} _ {P _ {0} ^ {n}} \phi_ {n, \epsilon} \leq C _ {0} e ^ {- C n \epsilon^ {2}}. \tag {28}
$$
Therefore, substituting equation (25), equation (27), and (28) into (26), we obtain
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \frac {\int_ {K _ {n}} e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right] \leq (1 + C _ {0} + 3 W ^ {- \gamma}) e ^ {- C _ {1 0} C _ {1} n \epsilon^ {2}}, \tag {29}
$$
where  $C_{10} = \min \{\lambda, C, 1\} / C_1$ . Using Fubini's theorem, observe that the LHS in the equation (29) can be expressed as  $\mu(K_n)$ , where
$$
d \mu (\theta) = \mathbb {E} _ {P _ {0} ^ {n}} \left[ \frac {\mathcal {L R} _ {n} (\theta , \theta_ {0})}{\int_ {\Theta} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] \Pi (\theta) e ^ {\gamma R (a ^ {\prime}, \theta)} d \theta .
$$
Next, recall that the set  $K_{n} = \{\theta \in \Theta : L_{n}(\theta ,\theta_{0}) > C_{1}n\epsilon^{2}\}$ . Applying Lemma C.2 above with  $X = L_{n}(\theta ,\theta_{0})$ ,  $c_{1} = (1 + C_{0} + 3W^{-\gamma})$ ,  $c_{2} = C_{10}$ ,  $t_0 = C_1n\epsilon_n^2$ , and for  $0 < \zeta \leq C_{10} / 2$ , we obtain
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} e ^ {\zeta L _ {n} (\theta , \theta_ {0})} \frac {e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} d \theta \right] \leq e ^ {\zeta C _ {1} n \epsilon_ {n} ^ {2}} + (1 + C _ {0} + 3 W ^ {- \gamma}). \tag {30}
$$
Further, we have another technical lemma, that will be crucial in proving the subsequent lemma that upper bounds the last term in the equation (19).
Lemma C.4. Suppose a positive random variable  $X$  satisfies
$$
\mathbb {P} (X \geq e ^ {t}) \leq c _ {1} \exp (- (c _ {2} + 1) t),
$$
for all  $t \geq t_0 > 0$ ,  $c_1 > 0$ , and  $c_2 > 0$ . Then,
$$
\mathbb {E} [ X ] \leq \exp (t _ {0}) + \frac {c _ {1}}{c _ {2}}.
$$
Proof. For any  $Z_0 > 1$
$$
\begin{array}{l} \mathbb {E} [ X ] \leq Z _ {0} + \int_ {Z _ {0}} ^ {\infty} \mathbb {P} (X \geq x) d x \\ = Z _ {0} + \int_ {\ln Z _ {0}} ^ {\infty} \mathbb {P} (X \geq e ^ {y}) e ^ {y} d y \leq Z _ {0} + c _ {1} \int_ {\ln Z _ {0}} ^ {\infty} \exp (- c _ {2} y) d y. \\ \end{array}
$$
Therefore, choosing  $Z_0 = \exp(t_0)$ ,
$$
\mathbb {E} [ X ] \leq \exp (t _ {0}) + \frac {c _ {1}}{c _ {2}} \exp (- c _ {2} t _ {0}) \leq \exp (t _ {0}) + \frac {c _ {1}}{c _ {2}}.
$$
Next, we establish the following bound on the last term in equation (19).
Lemma C.5. Under Assumptions 2.1, 2.2, 2.3, 2.4, 2.5, and for  $C_5(\gamma) > C_2 + C_3 + 2$
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} \frac {e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right] \leq e ^ {C _ {4} (\gamma) n \epsilon_ {n} ^ {2}} + 2 C _ {4} (\gamma). \tag {31}
$$
for any  $\lambda \geq 1 + C_4(\gamma)$
Proof. Define the set
$$
M _ {n} := \left\{\theta \in \Theta : e ^ {\gamma R \left(a ^ {\prime}, \theta\right)} > e ^ {C _ {4} (\gamma) n \epsilon^ {2}} \right\}. \tag {32}
$$
Using the set  $B_{n}$  in equation (23), observe that the measure of the set  $M_{n}$ , under the posterior distribution satisfies,
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \frac {\int_ {M _ {n}} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right] \leq \mathbb {E} _ {P _ {0} ^ {n}} \left[ \mathbb {1} _ {B _ {n} ^ {c}} \right] + \mathbb {E} _ {P _ {0} ^ {n}} \left[ \mathbb {1} _ {B _ {n}} \frac {\int_ {M _ {n}} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right]. \tag {33}
$$
Now, the second term of equation (33) can be bounded as follows: recall Assumption 2.3 and the definition of set  $B_{n}$ , both together imply that,
$$
\begin{array}{l} \mathbb {E} _ {P _ {0} ^ {n}} \left[ \mathbb {1} _ {B _ {n}} \frac {\int_ {M _ {n}} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] \leq e ^ {(1 + C _ {2} + C _ {3}) n \epsilon^ {2}} \mathbb {E} _ {P _ {0} ^ {n}} \left[ \mathbb {1} _ {B _ {n}} \int_ {M _ {n}} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta) \right] \\ \leq e ^ {(1 + C _ {2} + C _ {3}) n \epsilon^ {2}} \mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {M _ {n}} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta) \right]. \quad (\star \star) \\ \end{array}
$$
Then, using Fubini's Theorem  $(\star \star) = e^{(1 + C_2 + C_3)n\epsilon^2}\Pi (M_n)$ . Next, using the definition of set  $M_{n}$  and then Assumption 2.4, we obtain
$$
\begin{array}{l} \mathbb {E} _ {P _ {0} ^ {n}} \left[ \mathbb {1} _ {B _ {n}} \frac {\int_ {M _ {n}} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] \leq e ^ {(1 + C _ {2} + C _ {3}) n \epsilon^ {2}} e ^ {- C _ {4} (\gamma) n \epsilon^ {2}} \int_ {M _ {n}} e ^ {\gamma R (a ^ {\prime}, \theta)} d \Pi (\theta) \\ \leq e ^ {(1 + C _ {2} + C _ {3}) n \epsilon^ {2}} e ^ {- C _ {4} (\gamma) n \epsilon^ {2}} e ^ {- C _ {5} (\gamma) n \epsilon^ {2}}, \\ \end{array}
$$
Hence, choosing the constants  $C_2, C_3, C_4(\gamma)$  and  $C_5(\gamma)$  such that  $-1 > 1 + C_2 + C_3 - C_5(\gamma)$  implies
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \mathbb {1} _ {B _ {n}} \frac {\int_ {M _ {n}} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right] \leq e ^ {- (1 + C _ {4} (\gamma)) n \epsilon^ {2}} \tag {34}
$$
Therefore, substituting (25) and (34) into (33)
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \frac {\int_ {M _ {n}} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right] \leq 2 e ^ {- C _ {4} (\gamma) \left(C _ {1 1} (\gamma) + 1\right) n \epsilon^ {2}}, \tag {35}
$$
where  $C_{11}(\gamma) = \min \{\lambda, 1 + C_4(\gamma)\} / C_4(\gamma) - 1$ . Using Fubini's theorem, observe that the RHS in (35) can be expressed as  $\nu(M_n)$ , where the measure
$$
d \nu (\theta) = \mathbb {E} _ {P _ {0} ^ {n}} \left[ \frac {\mathcal {L R} _ {n} (\theta , \theta_ {0})}{\int_ {\Theta} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] d \Pi (\theta).
$$
Applying Lemma C.4 for  $X = e^{\gamma R(a',\theta)},c_{1} = 2,c_{2} = C_{11}(\gamma),t_{0} = C_{4}(\gamma)n\epsilon_{n}^{2}$  and  $\lambda \geq 1 + C_4(\gamma)$  we obtain
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} \frac {e ^ {\gamma R \left(a ^ {\prime} , \theta\right)} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L} \mathcal {R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right] \leq e ^ {C _ {4} (\gamma) n \epsilon_ {n} ^ {2}} + \frac {2}{C _ {1 1} (\gamma)} \leq e ^ {C _ {4} (\gamma) n \epsilon_ {n} ^ {2}} + 2 C _ {4} (\gamma). \tag {36}
$$
Proof. Proof of Theorem 3.1: Finally, recall (19),
$$
\begin{array}{l} \zeta \mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} L _ {n} (\theta , \theta_ {0}) d Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n}) \right] \\ \leq \log \mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} e ^ {\zeta L _ {n} (\theta , \theta_ {0})} \frac {e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)}{\int_ {\Theta} e ^ {\gamma R (a ^ {\prime} , \theta)} \mathcal {L R} _ {n} (\theta , \theta_ {0}) d \Pi (\theta)} \right] + \inf  _ {Q \in \mathcal {Q}} \mathbb {E} _ {P _ {0} ^ {n}} \left[ \mathrm {K L} (Q \| \Pi_ {n}) \right. \\ \left. - \gamma \inf  _ {a \in \mathcal {A}} \mathbb {E} _ {Q} [ R (a, \theta) ] \right] + \log \mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} e ^ {\gamma R \left(a ^ {\prime}, \theta\right)} \frac {\mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)}{\int_ {\Theta} \mathcal {L R} _ {n} \left(\theta , \theta_ {0}\right) d \Pi (\theta)} \right]. \tag {37} \\ \end{array}
$$
Substituting (31) and (22) into the equation above and then using the definition of  $\eta_n^R (\gamma)$ , we get
$$
\begin{array}{l} \mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} L _ {n} (\theta , \theta_ {0}) d Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n}) \right] \\ \leq \frac {1}{\zeta} \left\{\log \left(e ^ {\zeta C _ {1} n \epsilon_ {n} ^ {2}} + (1 + C _ {0} + 3 W ^ {- \gamma})\right) + \log \left(e ^ {C _ {4} (\gamma) n \epsilon_ {n} ^ {2}} + 2 C _ {4} (\gamma)\right) + n \eta_ {n} ^ {R} (\gamma) \right\} \\ \leq \left(C _ {1} + \frac {1}{\zeta} C _ {4} (\gamma)\right) n \epsilon_ {n} ^ {2} + \frac {1}{\zeta} n \eta_ {n} ^ {R} (\gamma) + \frac {(1 + C _ {0} + 3 W ^ {- \gamma}) e ^ {(- \zeta C _ {1} n \epsilon_ {n} ^ {2})} + 2 C _ {4} (\gamma) e ^ {- C _ {4} (\gamma) n \epsilon_ {n} ^ {2}}}{\zeta}, \\ \end{array}
$$
where the last inequality uses the fact that  $\log x \leq x - 1$ . Choosing  $\zeta = C_{10} / 2 = \frac{\min(C, \lambda, 1)}{2C_1}$ ,
$$
\begin{array}{l} \mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} L _ {n} (\theta , \theta_ {0}) d Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n}) \right] \\ \leq M (\gamma) n \left(\epsilon_ {n} ^ {2}\right) + M ^ {\prime} n \eta_ {n} ^ {R} (\gamma) + \frac {2 \left(1 + C _ {0} + 3 W ^ {- \gamma}\right) e ^ {\left(- \frac {C _ {1 0}}{2} n \epsilon_ {n} ^ {2}\right)} + 4 C _ {4} (\gamma) e ^ {- C _ {4} (\gamma) n \epsilon_ {n} ^ {2}}}{C _ {1 0}} \tag {38} \\ \end{array}
$$
where  $M(\gamma) = C_1 + \frac{1}{\zeta} C_4(\gamma)$  and  $M' = \frac{1}{\zeta}$  depend on  $C, C_1, C_4(\gamma), W$  and  $\lambda$ . Since the last two terms in (38) decrease and the first term increases as  $n$  increases, we can choose  $M'$  large enough, such that for all  $n \geq 1$
$$
M ^ {\prime} n \eta_ {n} ^ {R} (\gamma) > \frac {2 (1 + C _ {0} + 3 W ^ {- \gamma})}{C _ {1 0}} + \frac {4 C _ {4} (\gamma)}{C _ {1 0}},
$$
and therefore for  $M = 2M^{\prime}$
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \int_ {\Theta} L _ {n} \left(\theta , \theta_ {0}\right) d Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n}) \right] \leq M (\gamma) n \left(\epsilon_ {n} ^ {2}\right) + M n \eta_ {n} ^ {R} (\gamma). \tag {39}
$$
Also, observe that the LHS in the above equation is always positive, therefore  $M(\gamma)\epsilon_{n}^{2} + M\eta_{n}^{R}(\gamma) \geq 0 \forall n \geq 1$  and  $\gamma > 0$ .
C.3 Proof of Theorem 3.2
Lemma C.6. Given  $a' \in \mathcal{A}$  and for a constant  $M$ , as defined in Theorem 3.1
$$
\mathbb {E} _ {P _ {0} ^ {n}} \left[ \sup  _ {a \in \mathcal {A}} \left| \mathbb {E} _ {Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n})} [ R (a, \theta) ] - R (a, \theta_ {0}) \right| \right] \leq \left[ M (\gamma) \epsilon_ {n} ^ {2} + M \eta_ {n} ^ {R} (\gamma) \right] ^ {\frac {1}{2}}. \tag {40}
$$
$$
\begin{array}{l} \left(\sup  _ {a \in \mathcal {A}} \left| \mathbb {E} _ {Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n})} [ R (a, \theta) ] - R (a, \theta) \right|\right) ^ {2} \leq \left(\mathbb {E} _ {Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n})} \left[ \sup  _ {a \in \mathcal {A}} | R (a, \theta) - R (a, \theta_ {0}) | \right]\right) ^ {2} \\ \leq \mathbb {E} _ {Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n})} \left[ \left(\sup  _ {a \in \mathcal {A}} | R (a, \theta) - R (a, \theta_ {0}) |\right) ^ {2} \right], \\ \end{array}
$$
where the last inequality follows from Jensen's inequality. Now, using the Jensen's inequality again
$$
\begin{array}{l} \left. \left(\mathbb {E} _ {P _ {0} ^ {n}} \left[ \sup  _ {a \in \mathcal {A}} \left| \mathbb {E} _ {Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n})} [ R (a, \theta) ] - R (a, \theta_ {0}) \right| \right]\right)\right) ^ {2} \\ \leq \mathbb {E} _ {P _ {0} ^ {n}} \left[ \left(\sup  _ {a \in \mathcal {A}} \left| \mathbb {E} _ {Q _ {a ^ {\prime}, \gamma} ^ {*} (\theta | \tilde {X} _ {n})} [ R (a, \theta) ] - R (a, \theta_ {0}) \right|\right) ^ {2} \right]. \\ \end{array}
$$
Now, using Theorem 3.1 the result follows immediately.
Proof of Theorem 3.2. Observe that
$$
\begin{array}{l} R \left(\mathbf {a} _ {\mathrm {R S}} ^ {*}, \theta_ {0}\right) - \inf  _ {z \in \mathcal {A}} R (z, \theta_ {0}) \\ = \left| R \left(\mathsf {a} _ {\mathbb {R S}} ^ {*}, \theta_ {0}\right) - \inf  _ {z \in \mathcal {A}} R (z, \theta_ {0}) \right| \\ = R \left(\mathbf {a} _ {\mathrm {R S}} ^ {*}, \theta_ {0}\right) - \mathbb {E} _ {Q _ {\mathbf {a} _ {\mathrm {R S}} ^ {*}, \gamma} ^ {*} (\theta | \tilde {X} _ {n})} [ R \left(\mathbf {a} _ {\mathrm {R S}} ^ {*}, \theta\right) ] + \mathbb {E} _ {Q _ {\mathbf {a} _ {\mathrm {R S}} ^ {*}, \gamma} ^ {*} (\theta | \tilde {X} _ {n})} [ R \left(\mathbf {a} _ {\mathrm {R S}} ^ {*}, \theta\right) ] - \inf  _ {z \in \mathcal {A}} R (z, \theta_ {0}) \\ \leq \left| R \left(\mathbf {a} _ {\mathrm {R S}} ^ {*}, \theta_ {0}\right) - \mathbb {E} _ {Q _ {\mathbf {a} _ {\mathrm {R S}} ^ {*}, \gamma} ^ {*}, \gamma (\theta | \tilde {X} _ {n})} [ R \left(\mathbf {a} _ {\mathrm {R S}} ^ {*}, \theta\right) ] \right| + \left| \mathbb {E} _ {Q _ {\mathbf {a} _ {\mathrm {R S}} ^ {*}, \gamma} ^ {*}, \gamma (\theta | \tilde {X} _ {n})} [ R \left(\mathbf {a} _ {\mathrm {R S}} ^ {*}, \theta\right) ] - \inf  _ {a \in \mathcal {A}} R (a, \theta_ {0}) \right| \\ \leq 2 \sup  _ {a \in \mathcal {A}} \left| \int R (a, \theta) d Q _ {a _ {\mathrm {R S}} ^ {*}, \gamma} ^ {*} (\theta | \tilde {X} _ {n}) - R (a, \theta_ {0}) \right|. \tag {41} \\ \end{array}
$$
Given  $\mathsf{a}_{\mathrm{RS}}^{*} \in \mathcal{A}$  and for a constant M (defined in Theorem 3.1), we have from Lemma C.6 for  $a' = \mathsf{a}_{\mathrm{RS}}^{*}$