SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathbb {E} \left[ C ^ {- 1} \sum_ {j = 1} ^ {t - 1} \mathbb {I} \left\{v _ {j} \leq \gamma \right\} v _ {j} \right] \leq \frac {\left(1 - B / T _ {\max }\right)}{0 . 9 8 (1 - \epsilon) ^ {2}} \left(k - (k - 1) \left(\frac {1 - \epsilon}{B / T _ {\max } - \epsilon}\right) ^ {\frac {1}{k - 1}}\right)
$$
holds with probability at least  $1 - \exp \left(-\frac{\epsilon^2b^2(t - 1)}{(k - 2)^2(u - b)^2}\right) - \exp \left(-\frac{2(t - 1)(1 - B / T_{\max})^2}{(1 - \epsilon)^2}\right)$ . Taking a union bound gives the desired result.