DISTRIBUTIONAL GENERALIZATION: A NEW KIND OF GENERALIZATION

Abstract

We introduce a new notion of generalization-Distributional Generalizationwhich roughly states that outputs of a classifier at train and test time are close as distributions, as opposed to close in just their average error. For example, if we mislabel 30% of dogs as cats in the train set of CIFAR-10, then a ResNet trained to interpolation will in fact mislabel roughly 30% of dogs as cats on the test set as well, while leaving other classes unaffected. This behavior is not captured by classical generalization, which would only consider the average error and not the distribution of errors over the input domain. Our formal conjectures, which are much more general than this example, characterize the form of distributional generalization that can be expected in terms of problem parameters: model architecture, training procedure, number of samples, and data distribution. We give empirical evidence for these conjectures across a variety of domains in machine learning, including neural networks, kernel machines, and decision trees. Our results thus advance our understanding of interpolating classifiers.

1. INTRODUCTION

We begin with an experiment motivating the need for a notion of generalization beyond test error. Experiment 1. Consider a binary classification version of CIFAR-10, where CIFAR-10 images x have binary labels Animal/Object. Take 50K samples from this distribution as a train set, but apply the following label noise: flip the label of cats to Object with probability 30%. Now train a WideResNet f to 0 train error on this train set. How does the trained classifier behave on test samples? Options below: 1. The test error is low across all classes, since there is only 3% label noise in the train set 2. Test error is "spread" across the animal class, After all, the classifier is not explicitly told what a cat or a dog is, just that they are all animals. 3. The classifier misclassifies roughly 30% of test cats as "objects", but all other types of animals are largely unaffected. In fact, reality is closest to option (3). 



Figure 1 shows the results of this experiment with a WideResNet. The left panel shows the joint density of train inputs x with train labels Object/Animal. Since the classifier is interpolating, the classifier outputs on the train set are identical to the left panel. The right panel shows the classifier predictions f (x) on test inputs x.

Figure 1: The setup and result of Experiment 1. The CIFAR-10 train set is labeled as either Animals or Objects, with label noise affecting only cats. A WideResNet-28-10 is then trained to 0 train error on this train set, and evaluated on the test set. Full experimental details are in C.2

