# Online Learning Quantum States with the Logarithmic Loss via VB-FTRL

arXiv: https://arxiv.org/abs/2311.04237 · HTML: https://mlsanity.com/paper/2311.04237/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 25 references that are in the map.
- First posted on arXiv 6 Nov 2023.

## Builds on (most cited first, 9 of 9)

- [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)
- [Online Learning of Quantum States](https://mlsanity.com/paper/1802.09025/) (2018)
- [Efficient and Near-Optimal Online Portfolio Selection](https://mlsanity.com/paper/2209.13932/) (2022)
- [Adaptive Online Learning of Quantum States](https://mlsanity.com/paper/2206.00220/) (2022)
- [Data-Dependent Bounds for Online Portfolio Selection Without Lipschitzness and Smoothness](https://mlsanity.com/paper/2305.13946/) (2023)
- [Soft-Bayes: Prod for Mixtures of Experts with Log-Loss](https://mlsanity.com/paper/1901.02230/) (2019)
- [Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States](https://mlsanity.com/paper/2210.00997/) (2022)
- [Fast Minimization of Expected Logarithmic Loss via Stochastic Dual Averaging](https://mlsanity.com/paper/2311.02557/) (2023)

## Built on by (most cited first, 1 of 1)

- [Online learning of a panoply of quantum objects](https://mlsanity.com/paper/2406.04245/) (2024)

## Abstract (arXiv)

Online learning of quantum states with the logarithmic loss (LL-OLQS) is a quantum generalization of online portfolio selection (OPS), a classic open problem in online learning for over three decades. This problem also emerges in designing stochastic optimization algorithms for maximum-likelihood quantum state tomography. Recently, Jezequel et al. (arXiv:2209.13932) proposed the VB-FTRL algorithm, the first regret-optimal algorithm for OPS with moderate computational complexity. In this paper, we generalize VB-FTRL for LL-OLQS. Let d denote the dimension and T the number of rounds. The generalized algorithm achieves a regret rate of O ( d2 log ( d + T ) ) for LL-OLQS. Each iteration of the algorithm consists of solving a semidefinite program that can be implemented in polynomial time by, for example, cutting-plane methods. For comparison, the best-known regret rate for LL-OLQS is currently O ( d2 log T ), achieved by an exponential weight method. However, no explicit implementation is available for the exponential weight method for LL-OLQS. To facilitate the generalization, we introduce the notion of VB-convexity. VB-convexity is a sufficient condition for the volumetric barrier associated with any function to be convex and is of independent interest.