\BOOKMARK [1][-]{section.0.1}{Introduction}{}% 1 \BOOKMARK [1][-]{section.0.2}{Exponential Families and Bregman Divergence}{}% 2 \BOOKMARK [1][-]{section.0.3}{Time-uniform Bregman Concentration}{}% 3 \BOOKMARK [2][-]{subsection.0.3.1}{A generic deviation inequality}{section.0.3}% 4 \BOOKMARK [2][-]{subsection.0.3.2}{Specification to classical families}{section.0.3}% 5 \BOOKMARK [2][-]{subsection.0.3.3}{Proof Sketch: Theorem 3}{section.0.3}% 6 \BOOKMARK [1][-]{section.0.4}{Numerical Experiments}{}% 7 \BOOKMARK [1][-]{section.0.5}{Application: Generalized Likelihood Ratio Test in Exponential Families}{}% 8 \BOOKMARK [1][-]{section.0.6}{Conclusion}{}% 9 \BOOKMARK [1][-]{section.0.A}{Proof of the Main Result About Bregman Deviations}{}% 10 \BOOKMARK [2][-]{subsection.0.A.1}{Proof of Bregman concentration using mixture parameter c}{section.0.A}% 11 \BOOKMARK [2][-]{subsection.0.A.2}{Bregman concentration using Legendre function L0}{section.0.A}% 12 \BOOKMARK [1][-]{section.0.B}{Properties of Bregman Divergences}{}% 13 \BOOKMARK [1][-]{section.0.C}{Specification to Illustrative Exponential Families}{}% 14 \BOOKMARK [2][-]{subsection.0.C.1}{Gaussian with unknown mean, known variance}{section.0.C}% 15 \BOOKMARK [2][-]{subsection.0.C.2}{Gaussian with known mean, unknown variance}{section.0.C}% 16 \BOOKMARK [2][-]{subsection.0.C.3}{Gaussian with unknown mean and variance}{section.0.C}% 17 \BOOKMARK [2][-]{subsection.0.C.4}{Bernoulli}{section.0.C}% 18 \BOOKMARK [2][-]{subsection.0.C.5}{Exponential}{section.0.C}% 19 \BOOKMARK [2][-]{subsection.0.C.6}{Gamma with fixed shape}{section.0.C}% 20 \BOOKMARK [2][-]{subsection.0.C.7}{Weibull with fixed shape}{section.0.C}% 21 \BOOKMARK [2][-]{subsection.0.C.8}{Pareto with fixed scale}{section.0.C}% 22 \BOOKMARK [2][-]{subsection.0.C.9}{Chi-square}{section.0.C}% 23 \BOOKMARK [2][-]{subsection.0.C.10}{Poisson}{section.0.C}% 24 \BOOKMARK [2][-]{subsection.0.C.11}{Summary table}{section.0.C}% 25 \BOOKMARK [1][-]{section.0.D}{Empirical Comparison with Existing Time-uniform Confidence Sequences}{}% 26 \BOOKMARK [2][-]{subsection.0.D.1}{Gaussian with unknown mean and variance}{section.0.D}% 27 \BOOKMARK [2][-]{subsection.0.D.2}{Bernoulli}{section.0.D}% 28 \BOOKMARK [2][-]{subsection.0.D.3}{Gaussian}{section.0.D}% 29 \BOOKMARK [2][-]{subsection.0.D.4}{Exponential}{section.0.D}% 30 \BOOKMARK [2][-]{subsection.0.D.5}{Chi-square}{section.0.D}% 31 \BOOKMARK [2][-]{subsection.0.D.6}{Key Observations}{section.0.D}% 32 \BOOKMARK [2][-]{subsection.0.D.7}{Tuning of The Regularization Parameter c}{section.0.D}% 33 \BOOKMARK [1][-]{section.0.E}{Application: Generalized Likelihood Ratio Test in Exponential Families}{}% 34 \BOOKMARK [2][-]{subsection.0.E.1}{Doubly-time uniform concentration inequalities for exponential families}{section.0.E}% 35 \BOOKMARK [2][-]{subsection.0.E.2}{Computing the inflating factor g\(t\)}{section.0.E}% 36 \BOOKMARK [2][-]{subsection.0.E.3}{Experiments on change point detection for Gaussian with unknown variance}{section.0.E}% 37 \BOOKMARK [1][-]{section.0.F}{Application: Linear Contextual Bandits}{}% 38