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

Online learning quantum states with the logarithmic loss (LL-OLQS) is a quantum generalization of online portfolio selection, a classic open problem in the field of online learning for over three decades. The problem also emerges in designing randomized optimization algorithms for maximum-likelihood quantum state tomography. Recently, Jezequel et al. (arXiv:2209.13932) proposed the VB-FTRL algorithm, the first nearly regret-optimal algorithm for OPS with moderate computational complexity. In this note, 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 $O ( d^2 \log ( d + T ) )$ for LL-OLQS. Each iteration of the algorithm consists of solving a semidefinite program that can be implemented in polynomial time by, e.g., cutting-plane methods. For comparison, the best-known regret rate for LL-OLQS is currently $O ( d^2 \log T )$, achieved by the exponential weight method. However, there is no explicit implementation 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 logarithmic barrier associated with any function to be convex and is of independent interest.