Adversarial robustness and generalization are both crucial properties of reliable machine learning models. In this letter, we study these properties in the context of quantum machine learning based on Lipschitz bounds. We derive parameter-dependent Lipschitz bounds for quantum models with trainable encoding, showing that the norm of the data encoding has a crucial impact on the robustness against data perturbations. Further, we derive a bound on the generalization error which explicitly involves the parameters of the data encoding. Our theoretical findings give rise to a practical strategy for training robust and generalizable quantum models by regularizing the Lipschitz bound in the cost. Further, we show that, for fixed and non-trainable encodings, as those frequently employed in quantum machine learning, the Lipschitz bound cannot be influenced by tuning the parameters. Thus, trainable encodings are crucial for systematically adapting robustness and generalization during training. The practical implications of our theoretical findings are illustrated with numerical results.

翻译：鲁棒性和泛化能力是可靠机器学习模型的两个关键属性。本文基于Lipschitz界研究了量子机器学习中这些性质。我们推导了具有可训练编码的量子模型的参数相关Lipschitz界，表明数据编码的范数对数据扰动的鲁棒性具有关键影响。进一步，我们推导了显式包含数据编码参数的泛化误差界。理论发现催生了一种通过正则化损失函数中Lipschitz界来训练鲁棒且可泛化量子模型的实用策略。此外，我们证明：对于量子机器学习中常用的固定不可训练编码，参数的调整无法影响Lipschitz界。因此，可训练编码对于在训练过程中系统性地调整鲁棒性和泛化能力至关重要。数值结果验证了理论发现的实践意义。