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

TL;DR

This paper develops regret bounds for online convex optimization with self-concordant barriers, improving existing results and applying to online portfolio selection and quantum state learning.

Contribution

It introduces a unified analysis of online mirror descent with self-concordant functions, leading to improved regret bounds in portfolio selection and quantum learning.

Findings

Improved regret bound for exponentiated gradient:  O(T^{2/3} d^{1/3})

Optimal regret for mirror descent with logarithmic barrier:  O(\,sqrt{T d})

Shorter per-iteration time for quantum state learning algorithms

Abstract

Consider an online convex optimization problem where the loss functions are self-concordant barriers, smooth relative to a convex function $h$, and possibly non-Lipschitz. We analyze the regret of online mirror descent with $h$. Then, based on the result, we prove the following in a unified manner. Denote by $T$ the time horizon and $d$ the parameter dimension. 1. For online portfolio selection, the regret of $\widetilde{\text{EG}}$, a variant of exponentiated gradient due to Helmbold et al., is $\tilde{O} ( T^{2/3} d^{1/3} )$ when $T > 4 d / \log d$. This improves on the original $\tilde{O} ( T^{3/4} d^{1/2} )$ regret bound for $\widetilde{\text{EG}}$. 2. For online portfolio selection, the regret of online mirror descent with the logarithmic barrier is $\tilde{O}(\sqrt{T d})$. 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 $\tilde{O} ( \sqrt{T d} )$. Its per-iteration time is shorter than all existing algorithms we know.