On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies

TL;DR

This work advances time-uniform confidence sequences for the mean of bounded processes by reframing the problem as a gambling game over a two-horse race. It formalizes a principled mixture approach that closely emulates Cover's universal portfolio with constant per-round complexity, and introduces a higher-order lower-bound technique (LBUP) to achieve near-UP performance with scalable computation. The paper provides exact mixture-wealth formulations, proves interval-type confidence sets via Ville’s inequality, and demonstrates through experiments that the proposed methods yield tight, practically useful confidence sequences, including a hybrid strategy that blends UP and LBUP for the best of both worlds. The results have practical impact for sequential mean estimation under boundedness, offering efficient, robust tools for time-adaptive inference and safe stopping rules in online settings.

Abstract

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.

Figures (3)

• Figure 1: The evolution of the wealth processes with respect to single realizations of i.i.d. i.i.d. Bern(0.25), Beta(1,3), and Beta(10,30) processes. Note that the true mean parameter $\mu$ is 0.25 for all three cases and is indicated by the vertical lines, while the variances are decreasing in the displayed order. HR and UP correspond to the two-horse-race-based algorithm and the UP-based algorithm, respectively. The $x$-axis corresponds to the parameter $m$, and the $y$-axis indicates the logarithmic cumulative wealth of each strategy for the game with a corresponding parameter $m$. The horizontal lines indicate an example threshold $\ln\frac{1}{0.05}\approx2.996$ for $\delta=0.05$.
• Figure 2: Examples of confidence sequences with respect to i.i.d. i.i.d. $\mathrm{Bern}(0.5)$, $\mathrm{Beta}(1,1)$, and $\mathrm{Beta}(10,10)$ processes at level $0.95$. For each distribution, shown here are the averages of five independent runs. The first column plots the confidence sequences. The second column plots the gap of the size of the confidence sequences from that of UP in log scale, where we took maximums of the differences and $10^{-7}$ for visualization. (Note that for the binary process example, HR is equivalent to UP.) The last column shows the elapsed time for each algorithm. For the continuous process examples in (b) and (c), the results are shown in two rows to avoid clutters. The results from UP, PRECiSE-CO96, LBUP's are shown in both rows for comparison, and the second row contains results from constant-per-step-complexity algorithms PRECISE-A-CO96 and HybridUP's.
• Figure 3: Examples of confidence sequences with respect to i.i.d. i.i.d. $\mathrm{Bern}(0.25)$, $\mathrm{Beta}(1,3)$, and $\mathrm{Beta}(10,30)$ processes at level $0.95$. See the caption of Figure \ref{['fig:exps_0.5']} for the details.

Theorems & Definitions (49)

• Theorem 1: Ville's inequality
• Proposition 2
• proof
• Corollary 3
• proof
• Remark 4: Different gambling parameterization
• Theorem 5
• Corollary 6
• proof
• Theorem 7