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.

翻译：对数损失下的在线学习量子态（LL-OLQS）是经典在线投资组合选择问题的量子推广，后者是三十多年来在线学习领域的一个经典开放问题。该问题还出现在为最大似然量子态层析成像设计随机优化算法的过程中。最近，Jezequel等人（arXiv:2209.13932）提出了VB-FTRL算法，这是首个在中等计算复杂度下实现近乎最优遗憾率的在线投资组合选择算法。在本研究中，我们将VB-FTRL推广至LL-OLQS问题。设$d$表示维度，$T$表示轮数。该推广算法在LL-OLQS问题中达到了$O ( d^2 \log ( d + T ) )$的遗憾率。算法的每次迭代需要求解一个半定规划，该问题可通过切平面法等方法在多项式时间内实现。相比之下，目前LL-OLQS问题已知的最佳遗憾率为$O ( d^2 \log T )$，由指数加权方法实现。然而，针对LL-OLQS问题，尚不存在指数加权方法的显式实现方案。为便于推广，我们引入VB-凸性的概念。VB-凸性是任意函数相关的对数障碍函数为凸的充分条件，该性质具有独立的研究价值。