分析高斯初始化下量子生成模型的可训练性问题
Trainability of IQP Quantum Circuit Born Machines Under Gaussian Initialization
- 用斯坦因引理和高斯浓度不等式推导梯度方差下界
- 发现高斯初始化可能引发梯度指数集中，导致训练失效
- 为避免梯度消失提供参数初始化策略，适合量子机器学习研究者
量子电路玻恩机（QCBM）通过利用玻恩规则自然地实现生成建模。近期工作提出使用即时量子多项式（IQP）电路结合最大均值差异（MMD）损失来经典训练QCBM。尽管类比采样在经典计算上难以实现，但其期望值可被经典计算，从而支持训练。然而，量子机器学习模型存在多种挑战，包括由指数集中或平坦山谷引起的可训练性问题。此前研究多针对均匀分布初始化，缺乏对任意高斯初始化的严格分析。本文利用斯坦因引理与高斯随机变量的利普希茨浓度界，提供了梯度方差的解析下界及梯度偏离均值的概率浓度界限。讨论了避免或诱发指数集中的策略，并明确了平坦山谷更易出现的条件。
原文摘要 · Abstract (English)
Quantum Circuit Born Machines (QCBMs) offer a natural approach to generative machine learning by leveraging the Born rule. Recent work has provided a method to classically train QCBMs with Instantaneous Quantum Polynomial (IQP) circuits via the Maximum Mean Discrepancy (MMD) loss. Despite the assumed intractability of sampling from IQP circuits classically, their expectation values can be computed classically, enabling training of these IQP QCBMs. However, quantum machine learning (QML) models have various other challenges, including trainability issues caused by exponential concentration or barren plateaus. While these issues have been explored for parameters sampled from a uniform distribution, little work has been done to rigorously treat the use of arbitrary Gaussian initialization schemes. This work leverages Stein's lemma and Lipschitz concentration bounds for Gaussian random variables to provide an analytical lower bound of the variance of the gradient and a probabilistic concentration bound of the deviation of the gradient from its mean. It discusses strategies to either avoid or encourage exponential concentration, as well as the conditions under which barren plateaus are more likely to occur.
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