SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \Xi^ {(n, m)} \triangleq \frac {\sqrt {n} \bar {T} ^ {(n , m)}}{\hat {\sigma} ^ {(n , m)}} = \\ = \frac {\sqrt {n} \left(\bar {T} ^ {(n , m)} - \mathbb {E} _ {P ^ {(m)}} [ T _ {1} ^ {(m)} \mid \mathcal {D} _ {t r} ^ {(m)} ]\right)}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} \mathbb {E} _ {P ^ {(m)}} [ T _ {1} ^ {(m)} \mid \mathcal {D} _ {t r} ^ {(m)} ]}{\hat {\sigma} ^ {(n , m)}} \\ = \frac {\sqrt {n} \Big (\bar {T} ^ {(n , m)} - \mathbb {E} _ {P ^ {(m)}} [ T _ {1} ^ {(m)} \mid \mathcal {D} _ {t r} ^ {(m)} ] \Big)}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} \mathbb {E} _ {P ^ {(m)}} [ \ell (\hat {h} ^ {(m)} (Z _ {1}) , Y _ {1}) - \ell (\hat {g} ^ {(m)} (X _ {1} , Z _ {1}) , Y _ {1}) \mid \mathcal {D} _ {t r} ^ {(m)} ]}{\hat {\sigma} ^ {(n , m)}} \\ = \frac {\sqrt {n} \Big (\bar {T} ^ {(n , m)} - \mathbb {E} _ {P ^ {(m)}} [ T _ {1} ^ {(m)} \mid \mathcal {D} _ {t r} ^ {(m)} ] \Big)}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} \mathbb {E} _ {P ^ {(m)}} [ \ell (\int \hat {g} ^ {(m)} (x , Z _ {1}) d \hat {Q} _ {X | Z} ^ {(m)} (x) , Y _ {1}) - \ell (\hat {g} ^ {(m)} (X _ {1} , Z _ {1}) , Y _ {1}) \mid \mathcal {D} _ {t r} ^ {(m)} ]}{\hat {\sigma} ^ {(n , m)}} \\ = \frac {\sqrt {n} \left(\bar {T} ^ {(n , m)} - \mathbb {E} _ {P ^ {(m)}} \left[ T _ {1} ^ {(m)} \mid \mathcal {D} _ {t r} ^ {(m)} \right]\right)}{\hat {\sigma} ^ {(n , m)}} + \\ + \frac {\sqrt {n} \mathbb {E} _ {P ^ {(m)}} [ \ell (\int \hat {g} ^ {(m)} (x , Z _ {1}) d \hat {Q} _ {X | Z} ^ {(m)} (x) , Y _ {1}) - \ell (\int \hat {g} ^ {(m)} (x , Z _ {1}) d Q _ {X | Z} ^ {*} (x) , Y _ {1}) \mid \mathcal {D} _ {t r} ^ {(m)} ]}{\hat {\sigma} ^ {(n , m)}} \\ + \frac {\sqrt {n} \mathbb {E} _ {P ^ {(m)}} \left[ \ell \left(\int \hat {g} ^ {(m)} (x , Z _ {1}) d Q _ {X | Z} ^ {*} (x) , Y _ {1}\right) - \ell \left(\int g _ {P ^ {(m)}} ^ {*} (x , Z _ {1}) d Q _ {X | Z} ^ {*} (x) , Y _ {1}\right) \mid \mathcal {D} _ {t r} ^ {(m)} \right]}{\hat {\sigma} ^ {(n , m)}} \\ + \frac {\sqrt {n} \mathbb {E} _ {P ^ {(m)}} \left[ \ell \left(g _ {P ^ {(m)}} ^ {*} \left(X _ {1} , Z _ {1}\right) , Y _ {1}\right) - \ell \left(\hat {g} ^ {(m)} \left(X _ {1} , Z _ {1}\right) , Y _ {1}\right) \mid \mathcal {D} _ {t r} ^ {(m)} \right]}{\hat {\sigma} ^ {(n , m)}} \\ + \frac {\sqrt {n}}{\hat {\sigma} ^ {(n , m)}} \mathbb {E} _ {P ^ {(m)}} \left[ \left[ \ell \left(\int g _ {P ^ {(m)}} ^ {*} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x), Y\right) - \ell \left(\int g _ {P ^ {(m)}} ^ {*} (x, Z _ {1}) d P _ {X | Z} ^ {(m)} (x), Y\right) \right] + \right. \\ \left. + \left[ \ell \left(\int g _ {P ^ {(m)}} ^ {*} (x, Z _ {1}) d P _ {X | Z} ^ {(m)} (x), Y\right) - \ell \left(g _ {P ^ {(m)}} ^ {*} (X _ {1}, Z _ {1}), Y _ {1}\right) \right] \right] \\ = \frac {\sqrt {n} W _ {1 , P ^ {(m)}} ^ {(m)}}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} W _ {2 , P ^ {(m)}} ^ {(m)}}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} W _ {3 , P ^ {(m)}} ^ {(m)}}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} W _ {4 , P ^ {(m)}} ^ {(m)}}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} \Omega_ {P ^ {(m)}} ^ {\mathrm {R B P T}}}{\hat {\sigma} ^ {(n , m)}} \\ \end{array}
$$
21 Because of the definition of sup.
$$
\begin{array}{l} \mathbb {E} _ {P ^ {(m)}} \left[ \varphi_ {\alpha} ^ {\mathrm {R B P T}} \left(\mathcal {D} _ {t e} ^ {(n)}, \mathcal {D} _ {t r} ^ {(m)}\right) \right] = (B.4) \\ = \mathbb {P} _ {P ^ {(m)}} \left[ p \left(\mathcal {D} _ {t e} ^ {(n)}, \mathcal {D} _ {t r} ^ {(m)}\right) \leq \alpha \right] \\ = \mathbb {P} _ {P ^ {(m)}} \left[ 1 - \Phi \left(\Xi^ {(n, m)}\right) \leq \alpha \right] \\ = \mathbb {P} _ {P ^ {(m)}} \left[ \Xi^ {(n, m)} \geq \tau_ {\alpha} \right] \\ = \mathbb {P} _ {P ^ {(m)}} \left[ \frac {\sqrt {n} W _ {1 , P ^ {(m)}} ^ {(m)}}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} W _ {2 , P ^ {(m)}} ^ {(m)}}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} W _ {3 , P ^ {(m)}} ^ {(m)}}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} W _ {4 , P ^ {(m)}} ^ {(m)}}{\hat {\sigma} ^ {(n , m)}} + \frac {\sqrt {n} \Omega_ {P ^ {(m)}} ^ {\mathrm {R B P T}}}{\hat {\sigma} ^ {(n , m)}} \geq \tau_ {\alpha} \right] \\ = \mathbb {P} _ {P ^ {(m)}} \left[ \frac {\sqrt {n} W _ {1 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} W _ {2 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} W _ {3 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} W _ {4 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} \Omega_ {P ^ {(m)}} ^ {\mathrm {R B P T}}}{\sigma_ {P ^ {(m)}}} + \tau_ {\alpha} - \tau_ {\alpha} \frac {\hat {\sigma} ^ {(n , m)}}{\sigma_ {P ^ {(m)}}} \geq \tau_ {\alpha} \right] \\ = \mathbb {P} _ {P ^ {(m)}} \left[ \frac {\sqrt {n} W _ {1 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}} ^ {(m)}} \frac {\sigma_ {P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} W _ {2 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} W _ {3 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} W _ {4 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \tau_ {\alpha} - \tau_ {\alpha} \frac {\hat {\sigma} ^ {(n , m)}}{\sigma_ {P ^ {(m)}} ^ {(m)}} \frac {\sigma_ {P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} \geq \tau_ {\alpha} - \frac {\sqrt {n} \Omega_ {P ^ {(m)}} ^ {\mathrm {R B P T}}}{\sigma_ {P ^ {(m)}}} \right] \\ = 1 - \Phi \left(\tau_ {\alpha} - \sqrt {\frac {n}{\sigma_ {P ^ {(m)}} ^ {2}}} \Omega_ {P ^ {(m)}} ^ {\mathrm {R B P T}}\right) + o (1) (B.5) \\ \end{array}
$$
$$
\sup  _ {P \in \mathcal {P}} \left| \mathbb {E} _ {P} \left[ \varphi_ {\alpha} ^ {\mathrm {R B P T}} \left(\mathcal {D} _ {t e} ^ {(n)}, \mathcal {D} _ {t r} ^ {(m)}\right)\right] - 1 + \Phi \left(\tau_ {\alpha} - \sqrt {\frac {n}{\sigma_ {P} ^ {2}}} \Omega_ {P} ^ {\mathrm {R B P T}}\right)\right| = o (1) \text {a s} m \rightarrow \infty
$$
Justifying step B.5. First, from a central limit theorem for triangular arrays [6, Corollary 9.5.11], we have that
$$
\begin{array}{l} \frac {\sqrt {n} W _ {1 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}} ^ {(m)}} - \left(\frac {\sqrt {n} W _ {1 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}} ^ {(m)}} \frac {\sigma_ {P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} W _ {2 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} W _ {3 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \frac {\sqrt {n} W _ {4 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} + \tau_ {\alpha} - \tau_ {\alpha} \frac {\hat {\sigma} ^ {(n , m)}}{\sigma_ {P ^ {(m)}} ^ {(m)}} \frac {\sigma_ {P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}}\right) = \\ = \frac {\sqrt {n} W _ {1 , P (m)} ^ {(m)}}{\sigma_ {P (m)} ^ {(m)}} \left(1 - \frac {\sigma_ {P (m)} ^ {(m)}}{\sigma_ {P (m)}}\right) - \frac {\sqrt {n} W _ {2 , P (m)} ^ {(m)}}{\sigma_ {P (m)}} - \frac {\sqrt {n} W _ {3 , P (m)} ^ {(m)}}{\sigma_ {P (m)}} - \frac {\sqrt {n} W _ {4 , P (m)} ^ {(m)}}{\sigma_ {P (m)}} + \\ + \tau_ {\alpha} \left(\frac {\hat {\sigma} ^ {(n , m)}}{\sigma_ {P ^ {(m)}} ^ {(m)}} - 1\right) \left(\frac {\sigma_ {P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} - 1\right) + \tau_ {\alpha} \left(\frac {\sigma_ {P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}}} - 1\right) + \tau_ {\alpha} \left(\frac {\hat {\sigma} ^ {(n , m)}}{\sigma_ {P ^ {(m)}} ^ {(m)}} - 1\right) \\ = o _ {p} (1) \text {a s} m \rightarrow \infty \\ \end{array}
$$
To see why the random quantity above converges to zero in probability, see that because of Assumption 3.3, Lemma $^{22}$  B.1, and continuous mapping theorem, we have that
Additionally, because of Assumptions 4.1, 4.2, and 4.3 and condition  $n = o(m^{\gamma})$ , we have that
22We can apply this STFR's lemma because it still holds when we consider GCM's test statistic.
$$
\begin{array}{l} \left| \frac {\sqrt {n}}{\sigma_ {P ^ {(m)}}} W _ {2, P ^ {(m)}} ^ {(m)} \right| = \left| \frac {\sqrt {n}}{\sigma_ {P ^ {(m)}}} \mathbb {E} _ {P ^ {(m)}} \left[ \ell \left(\int \hat {g} ^ {(m)} (x, Z _ {1}) d \hat {Q} _ {X | Z} ^ {(m)} (x), Y\right) - \ell \left(\int \hat {g} ^ {(m)} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x), Y\right) \mid \mathcal {D} _ {t r} ^ {(m)} \right] \right| \\ \leq \frac {\sqrt {n}}{\sigma_ {P ^ {(m)}}} \mathbb {E} _ {P ^ {(m)}} \left[ \left| \ell \left(\int \hat {g} ^ {(m)} (x, Z _ {1}) d \hat {Q} _ {X | Z} ^ {(m)} (x), Y\right) - \ell \left(\int \hat {g} ^ {(m)} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x), Y\right) \right| \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ \leq \frac {L \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ \left\| \int \hat {g} ^ {(m)} (x, Z _ {1}) d \hat {Q} _ {X | Z} ^ {(m)} (x) - \int \hat {g} ^ {(m)} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x) \right\| _ {2} \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ \leq \frac {L \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ \left\| \int \hat {g} ^ {(m)} (x, Z _ {1}) d \hat {Q} _ {X | Z} ^ {(m)} (x) - \int \hat {g} ^ {(m)} (x, Z _ {1}) d Q _ {X | Z} ^ {*} (x) \right\| _ {1} \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ \leq \frac {L \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ \int \left\| \hat {g} ^ {(m)} (x, Z _ {1}) \right\| _ {1} \cdot | \hat {q} _ {X | Z} ^ {(m)} (x | Z) - q _ {X | Z} ^ {*} (x | Z) | d \mu (x) \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ \leq \frac {L M \sqrt {n}}{\sigma_ {P ^ {(m)}}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ \int | \dot {q} _ {X | Z} ^ {(m)} (x | Z) - q _ {X | Z} ^ {*} (x | Z) | d \mu (x) \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ = \frac {2 L M \sqrt {n}}{\inf  _ {P \in \mathcal {P}} \sigma_ {P}} \cdot \mathbb {E} _ {P ^ {(m)}} \left[ d _ {\mathrm {T V}} \left(\hat {Q} _ {X | Z} ^ {(m)}, Q _ {X | Z} ^ {*}\right) \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ = \frac {2 L M}{\inf  _ {P \in \mathcal {P}} \sigma_ {P}} \cdot \left(o (m ^ {\gamma}) \mathcal {O} _ {p} \left(m ^ {- 2 \gamma}\right)\right) ^ {1 / 2} \\ = o _ {p} (1) \\ \end{array}
$$
$$
\begin{array}{l} \left| \frac {\sqrt {n}}{\sigma_ {P ^ {(m)}}} W _ {4, P ^ {(m)}} ^ {(m)} \right| = \left| \mathbb {E} _ {P ^ {(m)}} \left[ \ell \left(\hat {g} ^ {(m)} \left(X _ {1}, Z _ {1}\right), Y _ {1}\right) - \ell \left(g _ {P ^ {(m)}} ^ {*} \left(X _ {1}, Z _ {1}\right), Y _ {1}\right) \mid \mathcal {D} _ {t r} ^ {(m)} \right] \right| \\ \leq \frac {\sqrt {n}}{\sigma_ {P ^ {(m)}}} \mathbb {E} _ {P ^ {(m)}} \left[ \left| \ell \left(\hat {g} ^ {(m)} \left(X _ {1}, Z _ {1}\right), Y _ {1}\right) - \ell \left(g _ {P ^ {(m)}} ^ {*} \left(X _ {1}, Z _ {1}\right), Y _ {1}\right) \right| \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ \leq \frac {L \sqrt {n}}{\sigma_ {P (m)}} \cdot \mathbb {E} _ {P (m)} \left[ \left\| \hat {g} ^ {(m)} \left(X _ {1}, Z _ {1}\right) - g _ {P (m)} ^ {*} \left(X _ {1}, Z _ {1}\right) \right\| _ {2} \mid \mathcal {D} _ {t r} ^ {(m)} \right] \\ \leq \frac {L \sqrt {n}}{\inf  _ {P \in \mathcal {P}} \sigma_ {P}} \cdot \left(\mathbb {E} _ {P ^ {(m)}} \left[ \left\| \hat {g} ^ {(m)} \left(X _ {1}, Z _ {1}\right) - g _ {P ^ {(m)}} ^ {*} \left(X _ {1}, Z _ {1}\right) \right\| _ {2} ^ {2} \mid \mathcal {D} _ {t r} ^ {(m)} \right]\right) ^ {1 / 2} \\ = \frac {L}{\inf  _ {P \in \mathcal {P}} \sigma_ {P}} \cdot \left(o (m ^ {\gamma}) \mathcal {O} _ {p} (m ^ {- \gamma})\right) ^ {1 / 2} \\ = o _ {p} (1) \\ \end{array}
$$
where  $M$  is an upper bound for the sequence  $\left(\left\| \hat{g}^{(m)}(x,z)\right\| _1\right)_{m\in \mathbb{N}}$  (Assumption 4.3) and  $R$  is an upper bound for the sequence  $\left((\mathbb{E}_{P^{(m)}}\left[\chi^2\left(Q_{X|Z}^*\right||P_{X|Z}\right) + 1]\right)^{1 / 2}\right)_{m\in \mathbb{N}}$  (Assumption 4.2).
$$
\begin{array}{l} \frac {\sqrt {n} W _ {1 , P (m)} ^ {(m)}}{\sigma_ {P (m)} ^ {(m)}} \frac {\sigma_ {P (m)} ^ {(m)}}{\sigma_ {P (m)}} + \frac {\sqrt {n} W _ {2 , P (m)} ^ {(m)}}{\sigma_ {P (m)}} + \frac {\sqrt {n} W _ {3 , P (m)} ^ {(m)}}{\sigma_ {P (m)}} + \frac {\sqrt {n} W _ {4 , P (m)} ^ {(m)}}{\sigma_ {P (m)}} + \tau_ {\alpha} - \tau_ {\alpha} \frac {\hat {\sigma} ^ {(n , m)}}{\sigma_ {P (m)} ^ {(m)}} \frac {\sigma_ {P (m)} ^ {(m)}}{\sigma_ {P (m)}} = \\ = \frac {\sqrt {n} W _ {1 , P (m)} ^ {(m)}}{\sigma_ {P (m)} ^ {(m)}} - \left[ \frac {\sqrt {n} W _ {1 , P (m)} ^ {(m)}}{\sigma_ {P (m)} ^ {(m)}} - \left(\frac {\sqrt {n} W _ {1 , P (m)} ^ {(m)}}{\sigma_ {P (m)} ^ {(m)}} \frac {\sigma_ {P (m)} ^ {(m)}}{\sigma_ {P (m)}} + \frac {\sqrt {n} W _ {2 , P (m)} ^ {(m)}}{\sigma_ {P (m)}} + \frac {\sqrt {n} W _ {3 , P (m)} ^ {(m)}}{\sigma_ {P (m)}} + \frac {\sqrt {n} W _ {4 , P (m)} ^ {(m)}}{\sigma_ {P (m)}} + \tau_ {\alpha} - \tau_ {\alpha} \frac {\tilde {\sigma} ^ {(n , m)}}{\sigma_ {P (m)} ^ {(m)}} \frac {\sigma_ {P (m)} ^ {(m)}}{\sigma_ {P (m)}}\right) \right. \\ = \frac {\sqrt {n} W _ {1 , P ^ {(m)}} ^ {(m)}}{\sigma_ {P ^ {(m)}} ^ {(m)}} + o _ {p} (1) \Rightarrow N (0, 1) \\ \end{array}
$$
by Slutsky's theorem. Because  $N(0,1)$  is a continuous distribution, we have uniform convergence of the distribution function [27][Chapter 8, Exercise 5], and we do not have to worry about the fact that  $\tau_{\alpha} - \frac{\sqrt{n}\Omega_{P(m)}^{\mathrm{RBPT}}}{\sigma_{P(m)}}$  depends on  $m$ .
C.1 Running times
Artificial-data experiments. Regarding running times (average per iteration), RBPT took  $5 \cdot 10^{-4}s$  to run, RBPT2 took  $1.9s$ , STFR took  $7.5^{-4}s$ , RESIT took  $1.3 \cdot 10^{-1}s$ , GCM took  $6 \cdot 10^{-4}s$ , CRT took  $1.7 \cdot 10^{-2}s$ , and CPT took  $6.2 \cdot 10^{-1}s$ , all in a MacBook Air 2020 M1.
Real-data experiments. Regarding running times (average per iteration), RBPT took  $1.1 \cdot 10^{-1}s$  to run, RBPT2 took  $2.4 \cdot 10^{-1}s$ , STFR took  $10^{-3}s$ , GCM took  $10^{-3}s$ , CRT took  $2.5 \cdot 10^{-2}s$ , and CPT took  $7.2 \cdot 10^{-1}s$ , all in a MacBook Air 2020 M1.
C.2 Extra results
We include extra experiments in which  $Y \mid X, Z$  has skewed normal distributions with location  $\mu = cX + a^{\top}Z + \gamma (b^{\top}Z)^{2}$ , scale  $\sigma = 1$ , and shape  $s = 3$  (shape  $s = 0$  lead to the normal distribution). From the following plots, we can learn that the skewness often impacts negatively in Type-I error control.
Figure 7: Type-I error rates  $(c = 0)$ . In the first two plots, we set  $\theta = 0$  for RBPT, permitting Type-I error control across different  $d_{Z}$  values (Theorem 4.4), while RBPT2 allows Type-I error control for moderate  $d_{Z}$ . All the baselines fail to control Type-I errors regardless of  $d_{Z}$ . The last two plots illustrate that CRT emerges as the least robust test in this context, succeeded by RBPT and CPT.
Figure 8: (i) Making RBPT2 more robust using unlabeled data. With  $d_Z = 40$ , we gradually increase the unlabeled sample size from 0 to 1000 when fitting  $\hat{h}$ . The results show that a larger unlabeled sample size leads to effective Type-I error control. Even though we present this result for RBPT2, the same pattern is expected for RBPT in the presence of unlabeled data. (ii) Power curves for different methods. We compare our methods with CPT (when  $\theta = 0$ ), which seems to have practical robustness against misspecified inductive biases. RBPT2 and CPT have similar power while RBPT is slightly more conservative.