We consider the problem of learning linear operators under squared loss between two infinite-dimensional Hilbert spaces in the online setting. We show that the class of linear operators with uniformly bounded $p$-Schatten norm is online learnable for any $p \in [1, \infty)$. On the other hand, we prove an impossibility result by showing that the class of uniformly bounded linear operators with respect to the operator norm is \textit{not} online learnable. Moreover, we show a separation between online uniform convergence and online learnability by identifying a class of bounded linear operators that is online learnable but uniform convergence does not hold. Finally, we prove that the impossibility result and the separation between uniform convergence and learnability also hold in the agnostic PAC setting.

翻译：我们考虑在在线设置下，于两个无限维希尔伯特空间之间学习具有平方损失的线性算子的问题。我们证明，对于任何 $p \in [1, \infty)$，具有一致有界 $p$-Schatten 范数的线性算子类是在线可学习的。另一方面，我们通过证明相对于算子范数一致有界的线性算子类 \textit{不} 是在线可学习的，给出了一个不可能性结果。此外，我们通过识别一个在线可学习但一致收敛不成立的有界线性算子类，展示了在线一致收敛与在线可学习性之间的分离。最后，我们证明，在不可知 PAC 设置下，不可能性结果以及一致收敛与可学习性之间的分离同样成立。