SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Applying this to the mixture yields
$$
\begin{array}{l} M _ {t} ^ {\mathrm {E B}} \geqslant \frac {C (\tau) e ^ {\tau + V _ {t} + S _ {t}}}{(\tau + V _ {t} + S _ {t}) ^ {\tau + V _ {t}}} \Gamma (\tau + V _ {t}) - \frac {C (\tau)}{S _ {t}} \\ \geqslant \frac {C (\tau) e ^ {\tau + V _ {t} + S _ {t}}}{(\tau + V _ {t} + S _ {t}) ^ {\tau + V _ {t}}} \Gamma (\tau + V _ {t}) - 1, \tag {†} \\ \end{array}
$$
where  $(\dagger)$  follows from the self-imposed constraint  $S_{t} \geqslant C(\tau)$ . This yields crossing boundary
$$
\begin{array}{l} \alpha^ {- 1} = \frac {C (\tau) e ^ {\tau + V _ {t} + S _ {t}}}{(\tau + V _ {t} + S _ {t}) ^ {\tau + V _ {t}}} \Gamma (\tau + V _ {t}) - 1, \\ \frac {e ^ {\tau + V _ {t} + S _ {t}}}{\left(1 + \frac {S _ {t}}{\tau + V _ {t}}\right) ^ {\tau + V _ {t}}} = \left(\frac {(\tau + V _ {t}) ^ {\tau + V _ {t}}}{\Gamma (\tau + V _ {t})}\right) \left(\frac {1 + \alpha^ {- 1}}{C (\tau)}\right) \doteq \left(\frac {(\tau + V _ {t}) ^ {\tau + V _ {t}}}{\Gamma (\tau + V _ {t})}\right) \phi_ {t} (\tau , \alpha), \\ \frac {e ^ {1 + \frac {S _ {t}}{\tau + V _ {t}}}}{\left(1 + \frac {S _ {t}}{\tau + V _ {t}}\right)} = \left(\frac {\left(\tau + V _ {t}\right) ^ {\tau + V _ {t}}}{\Gamma \left(\tau + V _ {t}\right)}\right) ^ {\frac {1}{\tau + V _ {t}}} \phi_ {t} (\tau , \alpha) ^ {\frac {1}{\tau + V _ {t}}} \doteq z _ {t}, \\ S _ {t} = \left(\tau + V _ {t}\right) \left(- 1 - W _ {- 1} \left(- z _ {t} ^ {- 1}\right)\right). \\ \end{array}
$$
Chatzigeorgiou [2013, Theorem 1] states
$$
\begin{array}{l} W _ {- 1} (- e ^ {- u - 1}) \in - 1 - \sqrt {2 u} + \left[ - u, - \frac {2}{3} u \right] \\ \Longrightarrow - 1 - W _ {- 1} \left(- e ^ {- u - 1}\right) \in \sqrt {2 u} + \left[ \frac {2}{3} u, u \right]. \\ \end{array}
$$
Substituting yields
$$
\left(\tau + V _ {t}\right) \left(- 1 - W _ {- 1} \left(- z _ {t} ^ {- 1}\right)\right) \leqslant \left(\tau + V _ {t}\right) \left(\sqrt {2 \log \left(\frac {z _ {t}}{e ^ {1}}\right)} + \log \left(\frac {z _ {t}}{e ^ {1}}\right)\right). \tag {19}
$$
From Feller [1958, Equation (9.8)] we have
$$
\begin{array}{l} \Gamma (1 + n) \in \sqrt {2 \pi n} \left(\frac {n}{e ^ {1}}\right) ^ {n} \left[ e ^ {\frac {1}{1 2 n + 1}}, e ^ {\frac {1}{1 2 n}} \right] \\ \Longrightarrow \left(\frac {(\tau + V _ {t}) ^ {\tau + V _ {t}}}{\Gamma (\tau + V _ {t})}\right) ^ {\frac {1}{\tau + V _ {t}}} \in \left(\frac {\tau + V _ {t}}{2 \pi}\right) ^ {\frac {1}{2 (\tau + V _ {t})}} e ^ {1} \left[ e ^ {- \frac {1}{1 2 (\tau + V _ {t}) ^ {2}}}, e ^ {- \frac {1}{1 2 (\tau + V _ {t}) ^ {2} + (\tau + V _ {t})}} \right]. \\ \end{array}
$$
$$
\begin{array}{l} \left(\tau + V _ {t}\right) \sqrt {2 \log \left(\frac {z _ {t}}{e ^ {1}}\right)} \leqslant \left(\tau + V _ {t}\right) \sqrt {2 \log \left(\left(\frac {\tau + V _ {t}}{2 \pi}\right) ^ {\frac {1}{2 (\tau + V _ {t})}} e ^ {- \frac {1}{1 2 (\tau + V _ {t}) ^ {2} + (\tau + V _ {t})}} \phi_ {t} (\tau , \alpha) ^ {\frac {1}{\tau + V _ {t}}}\right)} \\ = \sqrt {2 (\tau + V _ {t}) \log \left(\sqrt {\frac {\tau + V _ {t}}{2 \pi}} e ^ {- \frac {1}{1 2 (\tau + V _ {t}) + 1}} \phi_ {t} (\tau , \alpha)\right)}, \tag {20} \\ \end{array}
$$
$$
\begin{array}{l} (\tau + V _ {t}) \log \left(\frac {z _ {t}}{e ^ {1}}\right) \leqslant (\tau + V _ {t}) \log \left(\left(\frac {\tau + V _ {t}}{2 \pi}\right) ^ {\frac {1}{2 (\tau + V _ {t})}} e ^ {- \frac {1}{1 2 (\tau + V _ {t}) ^ {2} + (\tau + V _ {t})}} \phi_ {t} (\tau , \alpha) ^ {\frac {1}{\tau + V _ {t}}}\right) \\ = \log \left(\sqrt {\frac {\tau + V _ {t}}{2 \pi}} e ^ {- \frac {1}{1 2 (\tau + V _ {t}) + 1}} \phi_ {t} (\tau , \alpha)\right). \tag {21} \\ \end{array}
$$
Combining Equations (19) to (21) yields the crossing boundary
$$
\begin{array}{l} \frac {S _ {t}}{t} = \sqrt {2 \left(\frac {\left(\tau + V _ {t}\right) / t}{t}\right) \log \left(\sqrt {\frac {\tau + V _ {t}}{2 \pi}} e ^ {- \frac {1}{1 2 \left(\tau + V _ {t}\right) + 1}} \left(\frac {1 + \alpha^ {- 1}}{C (\tau)}\right)\right)} \\ + \frac {1}{t} \log \left(\sqrt {\frac {\tau + V _ {t}}{2 \pi}} e ^ {- \frac {1}{1 2 (\tau + V _ {t}) + 1}} \left(\frac {1 + \alpha^ {- 1}}{C (\tau)}\right)\right). \\ \end{array}
$$
F Simulations
F.1 i.i.d. setting
For non-importance-weighted simulations, we use the Beta-Binomial boundary of Howard et al. [2021] for  $\Lambda_t$  and  $\Xi_t$ . The curved boundary is induced by the test NSM
$$
\begin{array}{l} W _ {t} (b; \hat {q} _ {t}, q _ {t}) = \frac {\int_ {q _ {t}} ^ {1} d \mathrm {B e t a} (p ; b q _ {t} , b (1 - q _ {t})) \left(\frac {p}{q _ {t}}\right) ^ {t \hat {q} _ {t}} \left(\frac {1 - p}{1 - q _ {t}}\right) ^ {t (1 - \hat {q} _ {t})}}{\int_ {q _ {t}} ^ {1} d \mathrm {B e t a} (p ; b q _ {t} , b (1 - q _ {t}))} \\ = \frac {1}{\left(1 - q _ {t}\right) ^ {t (1 - \hat {q} _ {t})} q _ {t} ^ {t \hat {q} _ {t}}} \left(\frac {\operatorname {B e t a} \left(q _ {t} , 1 , b q _ {t} + t \hat {q} _ {t} , b \left(1 - q _ {t}\right) + t \left(1 - \hat {q} _ {t}\right)\right)}{\operatorname {B e t a} \left(q _ {t} , 1 , b q _ {t} , b \left(1 - q _ {t}\right)\right)}\right) \\ \end{array}
$$
with prior parameter  $b = 1$ . Further documentation and details are in the reference implementation `csnsquantile.ipynb`.
The importance-weighted simulations use the constructions from Appendix E: the reference implementation is in csnsopquantile.ipynb for the DDRM variant and csnsopquantile-ebern.ipynb for the empirical Bernstein variant.
Figure 8: Comparison to naive time-uniform DKW (which is only valid in the i.i.d. setting) for Beta distributions of varying smoothness. Decreasing smoothness degrades our bound.
Figure 9: CDF bounds approaching the true CDF when sampling i.i.d. from a lognormal(0,1) distribution. Recall these bounds are simultaneously valid for all times and values.
Figure 10: CDF bounds approaching the true CDF when sampling i.i.d. from a Gaussian(0, 1) distribution. Recall these bounds are simultaneously valid for all times and values.
Figure 11: Demonstration of the variant described in Section 3.3 and Appendix D.1 for distributions with arbitrary support, based on i.i.d. sampling from a variety of Gaussian distributions. Logarithmic range dependence is evident.
Figure 12: Maximum bound width, scaled by  $\sqrt{t / \log(t)}$  to remove the primary trend, as a function of  $t$ , for nonstationary Polya simulations with different  $\gamma_{t}$  schedules. See Section 4.2
Figure 13: CDF bounds approaching the true counterfactual CDF when sampling i.i.d. from a Beta(6,3) with finite-variance importance weights, using DDRM for the oracle confidence sequence.
Figure 14: CDF bounds approaching the true counterfactual CDF when sampling i.i.d. from a  $\mathrm{Beta}(6,3)$  with finite-variance importance weights, using Empirical Bernstein for the oracle confidence sequence.
Figure 15: CDF bounds approaching the true counterfactual CDF when sampling i.i.d. from a  $\mathrm{Beta}(6,3)$  with infinite-variance importance weights, using Empirical Bernstein for the oracle confidence sequence. Despite apparent convergence, eventually this simulation would reset the Empirical Bernstein oracle confidence sequence to trivial bounds.