  Many inference problems, such as sequential decision problems like A/B
testing, adaptive sampling schemes like bandit selection, are often online in
nature. The fundamental problem for online inference is to provide a sequence
of confidence intervals that are valid uniformly over the growing-into-infinity
sample sizes. To address this question, we provide a near-optimal confidence
sequence for bounded random variables by utilizing Bentkus' concentration
results. We show that it improves on the existing approaches that use the
Cram{\'e}r-Chernoff technique such as the Hoeffding, Bernstein, and Bennett
inequalities. The resulting confidence sequence is confirmed to be favorable in
both synthetic coverage problems and an application to adaptive stopping
algorithms.
