推进量子态和量子态在线学习的效率-遗憾帕累托边界
Pushing the Efficiency-Regret Pareto Frontier for Online Learning of Portfolios and Quantum States
摘要：
我们重新审视经典的在线投资组合选择问题。人们普遍认为，计算复杂性和遗憾之间的权衡是不可避免的，与封面的通用Portfolio算法，B-BAYES和ADA-BARRONS目前构成其最先进的帕累托前沿。在本文中，我们提出了第一个有效的算法，BISONS，获得多对数遗憾的内存和每一步的运行时间的要求是多项式的维度，取代ADA-BARRONS从帕累托前沿。此外，我们解决了COLT 2020公开问题，通过显示具有对数屏障正则化的某个Follow-The-Regularized-Leader算法对维度的依赖性比以前所描述的要大得多。因此，我们排除了这个算法作为帕累托边界的候选者。我们还将我们的算法和分析扩展到比在线投资组合选择更一般的问题，即具有对数损失的量子态的在线学习。这种算法，称为薛定谔的BISONS，是第一个有效的算法与多对数遗憾这个更一般的问题。
英文摘要：
We revisit the classical online portfolio selection problem. It is widely assumed that a trade-off between computational complexity and regret is unavoidable, with Cover's Universal Portfolios algorithm, SOFT-BAYES and ADA-BARRONS currently constituting its state-of-the-art Pareto frontier. In this paper, we present the first efficient algorithm, BISONS, that obtains polylogarithmic regret with memory and per-step running time requirements that are polynomial in the dimension, displacing ADA-BARRONS from the Pareto frontier. Additionally, we resolve a COLT 2020 open problem by showing that a certain Follow-The-Regularized-Leader algorithm with log-barrier regularization suffers an exponentially larger dependence on the dimension than previously conjectured. Thus, we rule out this algorithm as a candidate for the Pareto frontier. We also extend our algorithm and analysis to a more general problem than online portfolio selection, viz. online learning of quantum states with log loss. This algorithm, called SCHRODINGER'S BISONS, is the first efficient algorithm with polylogarithmic regret for this more general problem.
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