There has been a long history of works showing that neural networks have hard time extrapolating beyond the training set. A recent study by Balestriero et al. (2021) challenges this view: defining interpolation as the state of belonging to the convex hull of the training set, they show that the test set, either in input or neural space, cannot lie for the most part in this convex hull, due to the high dimensionality of the data, invoking the well known curse of dimensionality. Neural networks are then assumed to necessarily work in extrapolative mode. We here study the neural activities of the last hidden layer of typical neural networks. Using an autoencoder to uncover the intrinsic space underlying the neural activities, we show that this space is actually low-dimensional, and that the better the model, the lower the dimensionality of this intrinsic space. In this space, most samples of the test set actually lie in the convex hull of the training set: under the convex hull definition, the models thus happen to work in interpolation regime. Moreover, we show that belonging to the convex hull does not seem to be the relevant criteria. Different measures of proximity to the training set are actually better related to performance accuracy. Thus, typical neural networks do seem to operate in interpolation regime. Good generalization performances are linked to the ability of a neural network to operate well in such a regime.
翻译:长期的工程历史表明,神经网络的外推时间比培训范围要难得多。Balestriero等人(2021年)最近的一项研究(2021年)对这一观点提出了挑战:将内推定义为属于培训组的螺旋壳状态,它们表明,无论是投入还是神经空间,测试组不能大部分地存在于这种螺旋壳中,因为数据具有高度的维度,并援引了众所周知的维度诅咒。神经网络随后被假定为必然以外推方式运作。我们在这里研究典型神经网络最后一层隐藏的神经活动。我们利用自动编码来揭示作为神经活动根基的内在空间,我们表明,这种空间实际上是低维度的,而且这种模型越好,这种内在空间的维度越低。在这个空间,测试组的大部分样本实际上都位于培训组的螺旋壳体壳中:根据Convex船体定义,模型因此会发生在内部系统的工作。此外,我们表明,在典型的神经网络中,“良好”的内置性能标准似乎与这种相互连接起来。我们表明,“正值”的内演测标准似乎与“更相”的内。