This study investigates leveraging stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We propose weak and strong regularity conditions for the target operator to depict its intrinsic structure and complexity. Under these conditions, we establish upper bounds for convergence rates of the SGD algorithm and conduct a minimax lower bound analysis, further illustrating that our convergence analysis and regularity conditions quantitatively characterize the tractability of solving operator learning problems using the SGD algorithm. It is crucial to highlight that our convergence analysis is still valid for nonlinear operator learning. We show that the SGD estimator will converge to the best linear approximation of the nonlinear target operator. Moreover, applying our analysis to operator learning problems based on vector-valued and real-valued reproducing kernel Hilbert spaces yields new convergence results, thereby refining the conclusions of existing literature.

翻译：本研究探讨了利用随机梯度下降（SGD）学习一般希尔伯特空间之间算子的方法。我们提出了目标算子的弱正则性和强正则性条件，以刻画其内在结构与复杂性。在这些条件下，我们建立了SGD算法收敛速率的上界，并进行了极小极大下界分析，进一步表明我们的收敛性分析与正则性条件从定量角度刻画了使用SGD算法求解算子学习问题的可处理性。需要重点指出的是，我们的收敛性分析对于非线性算子学习仍然成立。我们证明了SGD估计量将收敛到非线性目标算子的最佳线性逼近。此外，将我们的分析应用于基于向量值及实值再生核希尔伯特空间的算子学习问题，得到了新的收敛性结果，从而完善了现有文献的结论。