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Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States

Abstract

Consider an online convex optimization problem where the loss functions are self-concordant barriers, smooth relative to a convex function hh, and possibly non-Lipschitz. We analyze the regret of online mirror descent with hh. Then, based on the result, we prove the following in a unified manner. Denote by TT the time horizon and dd the parameter dimension. 1. For online portfolio selection, the regret of EG~\widetilde{\text{EG}}, a variant of exponentiated gradient due to Helmbold et al., is O~(T2/3d1/3)\tilde{O} ( T^{2/3} d^{1/3} ) when T>4d/logdT > 4 d / \log d. This improves on the original O~(T3/4d1/2)\tilde{O} ( T^{3/4} d^{1/2} ) regret bound for EG~\widetilde{\text{EG}}. 2. For online portfolio selection, the regret of online mirror descent with the logarithmic barrier is O~(Td)\tilde{O}(\sqrt{T d}). The regret bound is the same as that of Soft-Bayes due to Orseau et al. up to logarithmic terms. 3. For online learning quantum states with the logarithmic loss, the regret of online mirror descent with the log-determinant function is also O~(Td)\tilde{O} ( \sqrt{T d} ). Its per-iteration time is shorter than all existing algorithms we know.

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