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.

翻译：本文研究了构建置信序列的问题，该序列是一组随时间均匀成立的置信区间，用于估计有界实值随机过程的均值。本文从自然的“双马赛马”视角重新审视了近期文献中基于赌博的方法，并展示了由Cover（1991）通用投资组合所衍生算法的新性质。本文的主要成果是基于下界混合的新算法，该算法在常数级每轮时间复杂度下紧密逼近Cover通用投资组合的性能。作为所提算法关键技术的对数函数下界的高阶推广（Fan等，2015），可能具有独立的研究价值。