Speaker: Aaditya Ramdas

Abstract: This paper develops a class of exponential bounds for the probability that a martingale sequence crosses a time-dependent linear threshold. Our key insight is that it is both natural and fruitful to formulate exponential concentration inequalities in this way. We illustrate this point by recovering and improving many tail bounds for martingales, including classical inequalities (1960-80) by Bernstein, Bennett, Hoeffding, and Freedman; contemporary inequalities (1980-2000) by Shorack and Wellner, Pinelis, Blackwell, van de Geer, and de la Pena; and several modern inequalities (post-2000) by Khan, Tropp, Bercu and Touati, Delyon, and others. In each of these cases, we give the strongest and most general statements to date, quantifying the time-uniform concentration of scalar, matrix, and Banach-space-valued martingales, under a variety of nonparametric assumptions in discrete and continuous time. By choosing the optimal linear bound for a given time, we bridge the gap between existing line-crossing inequalities, the sequential probability ratio test, the Cramer-Chernoff method, self-normalized processes, and other parts of the literature.