  This paper considers the problem of constructing a confidence sequence, which
is a sequence of confidence intervals that hold uniformly over time, for
estimating the mean of bounded real-valued random processes. This paper
revisits the gambling-based approach established in the recent literature from
a natural \emph{two-horse race} perspective, and demonstrates new properties of
the resulting algorithm induced by Cover (1991)'s universal portfolio. The main
result of this paper is a new algorithm based on a mixture of lower bounds,
which closely approximates the performance of Cover's universal portfolio with
constant per-round time complexity. A higher-order generalization of a lower
bound on a logarithmic function in (Fan et al., 2015), which is developed as a
key technique for the proposed algorithm, may be of independent interest.
