  A classic problem in statistics is the estimation of the expectation of
random variables from samples. This gives rise to the tightly connected
problems of deriving concentration inequalities and confidence sequences, that
is confidence intervals that hold uniformly over time. Previous work has shown
how to easily convert the regret guarantee of an online betting algorithm into
a time-uniform concentration inequality. In this paper, we show that we can go
even further: We show that the regret of universal portfolio algorithms give
rise to new implicit time-uniform concentrations and state-of-the-art
empirically calculated confidence sequences. In particular, our numerically
obtained confidence sequences can never be vacuous, even with a single sample,
and satisfy the law of iterated logarithm.
