In this work, we bound a machine's ability to learn based on computational limitations implied by physicality. We start by considering the information processing capacity (IPC), a normalized measure of the expected squared error of a collection of signals to a complete basis of functions. We use the IPC to measure the degradation under noise of the performance of reservoir computers, a particular kind of recurrent network, when constrained by physical considerations. First, we show that the IPC is at most a polynomial in the system size $n$, even when considering the collection of $2^n$ possible pointwise products of the $n$ output signals. Next, we argue that this degradation implies that the family of functions represented by the reservoir requires an exponential number of samples to learn in the presence of the reservoir's noise. Finally, we conclude with a discussion of the performance of the same collection of $2^n$ functions without noise when being used for binary classification.

翻译：在本工作中，我们根据物理性所隐含的计算局限性，界定了机器从数据中学习的能力。我们首先考虑信息处理容量（IPC），这是对一个信号集合相对于完备函数基的期望平方误差进行归一化后的度量。我们利用IPC来衡量在物理约束下，储层计算机（一种特殊的循环网络）性能在噪声影响下的退化程度。首先，我们证明了即使考虑$n$个输出信号的所有$2^n$个可能的分量乘积集合，IPC最多也只是系统规模$n$的多项式函数。其次，我们论证这种退化意味着，在储层噪声存在的情况下，表示储层所对应函数族所需的样本数量呈指数级增长。最后，我们讨论了同一$2^n$个函数集合在无噪声条件下用于二分类时的性能表现。