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

arXiv: https://arxiv.org/abs/2210.00997 · HTML: https://mlsanity.com/paper/2210.00997/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 3 Oct 2022.

## Builds on (most cited first, 13 of 13)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Online Optimization : Competing with Dynamic Comparators](https://mlsanity.com/paper/1501.06225/) (2015)
- [Improved Dynamic Regret for Non-degenerate Functions](https://mlsanity.com/paper/1608.03933/) (2016)
- [Online Learning of Quantum States](https://mlsanity.com/paper/1802.09025/) (2018)
- [Efficient Online Portfolio with Logarithmic Regret](https://mlsanity.com/paper/1805.07430/) (2018)
- [Pushing the Efficiency-Regret Pareto Frontier for Online Learning of Portfolios and Quantum States](https://mlsanity.com/paper/2202.02765/) (2022)
- [Adaptive Online Learning of Quantum States](https://mlsanity.com/paper/2206.00220/) (2022)
- [Soft-Bayes: Prod for Mixtures of Experts with Log-Loss](https://mlsanity.com/paper/1901.02230/) (2019)
- [Damped Online Newton Step for Portfolio Selection](https://mlsanity.com/paper/2202.07574/) (2022)
- [Bandit Principal Component Analysis](https://mlsanity.com/paper/1902.03035/) (2019)
- [Dynamic Regret of Online Mirror Descent for Relatively Smooth Convex Cost Functions](https://mlsanity.com/paper/2202.12843/) (2022)

## Built on by (most cited first, 7 of 7)

- [Data-Dependent Bounds for Online Portfolio Selection Without Lipschitzness and Smoothness](https://mlsanity.com/paper/2305.13946/) (2023)
- [Online Inverse Linear Optimization: Efficient Logarithmic-Regret Algorithm, Robustness to Suboptimality, and Lower Bound](https://mlsanity.com/paper/2501.14349/) (2025)
- [Generalized Exponentiated Gradient Algorithms and Their Application to On-Line Portfolio Selection](https://mlsanity.com/paper/2406.00655/) (2024)
- [Faster Stochastic First-Order Method for Maximum-Likelihood Quantum State Tomography](https://mlsanity.com/paper/2211.12880/) (2022)
- [Fast Minimization of Expected Logarithmic Loss via Stochastic Dual Averaging](https://mlsanity.com/paper/2311.02557/) (2023)
- [Online Learning Quantum States with the Logarithmic Loss via VB-FTRL](https://mlsanity.com/paper/2311.04237/) (2023)
- [Sequential Capacity of Quantum Processes with Finite Memory](https://mlsanity.com/paper/2610.02068/) (2026)

## Abstract (arXiv)

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 widetildeEG, a variant of exponentiated gradient due to Helmbold et al., is tildeO ( T2/3 d1/3 ) when T > 4 d / log d. This improves on the original tildeO ( T3/4 d1/2 ) regret bound for widetildeEG. 2. For online portfolio selection, the regret of online mirror descent with the logarithmic barrier is tildeO(√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 tildeO ( √T d ). Its per-iteration time is shorter than all existing algorithms we know.