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This paper considers the problem of constructing a confidence sequence forbounded random processes. Building upon the gambling approach pioneered byHendriks (2018) and Jun and Orabona (2019) and following the recent work ofWaudby-Smith and Ramdas (2020) and Orabona and Jun (2021), this paper revisitsthe idea of Cover (1991)'s universal portfolio in constructing confidencesequences and demonstrates new properties, based on a natural \emph{two-horserace} perspective on the gambling approach. The main result of this paper is anew algorithm based on a mixture of lower bounds, which closely approximatesthe performance of Cover's universal portfolio with only constant per-roundtime complexity. A higher-order generalization of a lower bound in (Fan et al,2015), which is invoked in the proposed algorithm, may be of independentinterest.

On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies