Identity Crisis: Memorization and Generalization under Extreme Overparameterization

We study the interplay between memorization and generalization of overparameterized networks in the extreme case of a single training example and the task of an input pattern mapping to itself on the output. We examine fully-connected and convolutional networks (FCN and CNN), both linear and nonlinear, initialized randomly and then trained to minimize the reconstruction error. The trained networks stereotypically take one of the two forms: the constant function (memorization) and the identity function (generalization). We show that different architectures exhibit strikingly different inductive biases, which are sensitive to many architectural decisions. For example, CNNs of up to 10 layers are able to generalize from a single example, whereas FCNs cannot learn the identity function reliably from 60k examples. Deeper CNNs often fail, but nonetheless do astonishing work to memorize the training output: because CNN biases are location invariant, the model must progressively grow an output pattern from the image boundaries via the coordination of many layers. Our work helps to quantify and visualize inductive biases due to architectural choices such as depth, kernel width, and number of channels.