Benign interpolation and Occam's razor

Contemporary deep learning methods generalize well even when they fit their training data perfectly, a phenomenon known as benign interpolation. This phenomenon cannot be accounted for by classical statistical learning theory and has prompted a range of attempted new explanations in the statistics and machine learning literature. A common feature of these new proposals is an appeal to a simplicity preference among interpolating models, often presented as a form of Occam's razor. We clarify this debate for a philosophical audience and argue that this new appeal to simplicity creates an explanatory gap. The classical theory offers theorems which connect the simplicity of model classes to good generalization, thus underwriting methodological simplicity norms. The new accounts instead appeal to properties of individual models, which they interpret as a kind of simplicity. Lacking a provable connection to generalization, it is the name "simplicity" that does the work a theorem used to do, making a substantive and unargued assumption look like the application of a familiar methodological principle.

Hierarchical Domain Generalization

We study hierarchical domain generalization as a problem of extrapolation from finite observed regions to an entire instance space, replacing i.i.d. sampling with arbitrary domain hierarchies. We show that the central obstruction is not only the complexity of the hypothesis class, but the train/test domain partition through which evidence is revealed. In particular, no matter how small the class or how large the training size, some partition makes generalization fail for some target. These results suggest that modern generalization theory must treat domain structure as a first-class object.

Broken Ergodicity and the Violation of the Fluctuation-Dissipation Theorem Lead to Generalization Beyond Overfitting in Machine Learning

The remarkable ability of modern neural networks to generalize improves with increasing network capacity, even when the number of model parameters or effective degrees of freedom exceeds the number of training data points. This phenomenon is all the more surprising given that generalization error diverges when the number of model parameters approaches a critical value from below. Here we use dynamical mean field theory to show that this so-called "double descent" behavior is the outcome of a phase transition in the stochastic field theory describing the training process. We calculate the critical exponents and scaling function of the double descent phase transition, and show that it is marked by a breakdown of the fluctuation-dissipation theorem associated with broken ergodicity. The corresponding response function has the same functional form as the simple London model of the superconducting transition, with the rigidity of the wave function corresponding to the neural network's ability to generalize accurately.

Flatness and Generalization: Learning Multi-Index Models with Homogeneous Neural Networks

A common heuristic used to explain the generalization of first-order gradient methods on non-convex neural networks is that "flat interpolators generalize well" (Hochreiter and Schmidhuber, 1994; Keskar et al., 2017), where flatness can be measured by the trace of the Hessian of the empirical loss. However, Dinh et al. 2017) showed that, using symmetry of the network that can change flatness while keeping the population and empirical losses unchanged, any interpolator can be made sharper or flatter. This result makes the earlier heuristic statement vacuous. In this paper, we show that for learning an unknown multi-index model with $2$-layer non-convex homogeneous neural networks, there is a connection between flatness and generalization, despite the existence of symmetries. This connection pertains to the "flattest" interpolators, i.e., the interpolators that have orderwise minimum flatness among all interpolators. First, we show that there exists a natural class of non-generalizing interpolators whose flatness cannot be made closer to the flattest possible, even using symmetries. Second, we show that for data generated by a sum of single-index models, if the approximation error and label noise are low, any flattest interpolator achieves small population loss, i.e., the flattest interpolators always generalize. This establishes a direct link between flatness and generalization which applies to a large class of activations and realistic data distributions.