用量子黎尼发散推导出更优的量子学习泛化界。
Generalization Bounds for Quantum Learning via Rényi Divergences
- 基于量子与经典黎尼发散，提出新型泛化误差上界。
- 新提出的修正夹层量子黎尼发散使界更紧致，数值验证更优。
- 适用于研究量子机器学习理论性能的研究者。
本文通过引入Caro等人（2024）提出的框架和新的期望真实损失定义，推进了对量子学习的理论理解。主要贡献是利用变分法评估量子黎尼发散，包括Petz发散和一种新提出的修正夹层量子黎尼发散，推导出期望泛化误差的上界。分析与数值实验表明，基于修正夹层量子黎尼发散的界优于基于Petz发散的界。此外，我们还采用两种不同技术提供了概率性泛化误差界：一种结合修正夹层量子黎尼发散与经典黎尼发散，另一种使用平滑最大黎尼发散。
原文摘要 · Abstract (English)
This work advances the theoretical understanding of quantum learning by establishing a new family of upper bounds on the expected generalization error of quantum learning algorithms, leveraging the framework introduced by Caro et al. (2024) and a new definition for the expected true loss. Our primary contribution is the derivation of these bounds in terms of quantum and classical Rényi divergences, utilizing a variational approach for evaluating quantum Rényi divergences, specifically the Petz and a newly introduced modified sandwich quantum Rényi divergence. Analytically and numerically, we demonstrate the superior performance of the bounds derived using the modified sandwich quantum Rényi divergence compared to those based on the Petz divergence. Furthermore, we provide probabilistic generalization error bounds using two distinct techniques: one based on the modified sandwich quantum Rényi divergence and classical Rényi divergence, and another employing smooth max Rényi divergence.
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