The Matrix Multiplicative Weight Update (MMWU) is a seminal online learning algorithm with numerous applications. Applied to the matrix version of the Learning from Expert Advice (LEA) problem on the $d$-dimensional spectraplex, it is well known that MMWU achieves the minimax-optimal regret bound of $O(\sqrt{T\log d})$, where $T$ is the time horizon. In this paper, we present an improved algorithm achieving the instance-optimal regret bound of $O(\sqrt{T\cdot S(X||d^{-1}I_d)})$, where $X$ is the comparator in the regret, $I_d$ is the identity matrix, and $S(\cdot||\cdot)$ denotes the quantum relative entropy. Furthermore, our algorithm has the same computational complexity as MMWU, indicating that the improvement in the regret bound is ``free''. Technically, we first develop a general potential-based framework for matrix LEA, with MMWU being its special case induced by the standard exponential potential. Then, the crux of our analysis is a new ``one-sided'' Jensen's trace inequality built on a Laplace transform technique, which allows the application of general potential functions beyond exponential to matrix LEA. Our algorithm is finally induced by an optimal potential function from the vector LEA problem, based on the imaginary error function. Complementing the above, we provide a memory lower bound for matrix LEA, and explore the applications of our algorithm in quantum learning theory. We show that it outperforms the state of the art for learning quantum states corrupted by depolarization noise, random quantum states, and Gibbs states. In addition, applying our algorithm to linearized convex losses enables predicting nonlinear quantum properties, such as purity, quantum virtual cooling, and R\'{e}nyi-$2$ correlation.

翻译：矩阵乘性权重更新（MMWU）是一种具有广泛应用的经典在线学习算法。将其应用于$d$维谱单纯形上的"专家建议学习"问题的矩阵版本时，已知MMWU可实现极小极大最优的遗憾界$O(\sqrt{T\log d})$，其中$T$为时间跨度。本文提出一种改进算法，实现了实例最优的遗憾界$O(\sqrt{T\cdot S(X||d^{-1}I_d)})$，其中$X$为遗憾中的比较器，$I_d$为单位矩阵，$S(\cdot||\cdot)$表示量子相对熵。此外，我们的算法与MMWU具有相同的计算复杂度，表明遗憾界的改进是"免费"的。技术层面，我们首先为矩阵专家建议学习建立了一个通用的基于势函数的框架，MMWU是其由标准指数势函数导出的特例。随后，我们分析的核心是建立在拉普拉斯变换技术上的新型"单边"詹森迹不等式，这使得超越指数函数的广义势函数可应用于矩阵专家建议学习问题。我们的算法最终由基于虚误差函数的向量专家建议学习问题的最优势函数导出。作为补充，我们给出了矩阵专家建议学习的存储下界，并探讨了算法在量子学习理论中的应用。研究表明，该算法在受去极化噪声污染的量子态学习、随机量子态学习以及吉布斯态学习方面均优于现有技术。此外，将算法应用于线性化凸损失函数可实现非线性量子性质的预测，例如纯度、量子虚拟冷却以及R\'{e}nyi-$2$关联。