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 sequential 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 batch setting.

翻译：我们考虑在线设置下，在两个无限维希尔伯特空间之间学习平方损失下的线性算子问题。我们证明，对于任意 $p \in [1, \infty)$，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，我们通过证明在算子范数意义下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线性算子类，展示了序列一致收敛性与在线可学习性之间的分离。最后，我们证明了在批量设置下，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。