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A classic problem in statistics is the estimation of the expectation ofrandom variables from samples. This gives rise to the tightly connectedproblems of deriving concentration inequalities and confidence sequences, thatis confidence intervals that hold uniformly over time. Previous work has shownhow to easily convert the regret guarantee of an online betting algorithm intoa time-uniform concentration inequality. In this paper, we show that we can goeven further: We show that the regret of universal portfolio algorithms giverise to new implicit time-uniform concentrations and state-of-the-artempirically calculated confidence sequences. In particular, our numericallyobtained confidence sequences can never be vacuous, even with a single sample,and satisfy the law of iterated logarithm.

Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio