We consider the optimisation of large and shallow neural networks via gradient flow, where the output of each hidden node is scaled by some positive parameter. We focus on the case where the node scalings are non-identical, differing from the classical Neural Tangent Kernel (NTK) parameterisation. We prove that, for large neural networks, with high probability, gradient flow converges to a global minimum AND can learn features, unlike in the NTK regime. We also provide experiments on synthetic and real-world datasets illustrating our theoretical results and showing the benefit of such scaling in terms of pruning and transfer learning.

Abstract: Generalizing outside of the training distribution is an open challenge for current machine learning systems. A weak form of out-of- distribution (OoD) generalization is the ability to successfully interpolate between multiple observed distributions. One way to achieve this is through robust optimization, which seeks to minimize the worst-case risk over convex combinations of the training distributions. However, a much stronger form of OoD generalization is the ability of models to extrapolate beyond the distributions observed during training. In pursuit of strong OoD generalization, we introduce the principle of Risk Extrapolation (REx). REx can be viewed as encouraging robustness over affine combinations of training risks, by encouraging strict equality between training risks.

Generalizing outside of the training distribution is an open challenge for current machine learning systems. A weak form of out-of-distribution (OoD) generalization is the ability to successfully interpolate between multiple observed distributions. One way to achieve this is through robust optimization, which seeks to minimize the worst-case risk over convex combinations of the training distributions. However, a much stronger form of OoD generalization is the ability of models to extrapolate beyond the distributions observed during training. In pursuit of strong OoD generalization, we introduce the principle of Risk Extrapolation (REx). REx can be viewed as encouraging robustness over affine combinations of training risks, by encouraging strict equality between training risks. We show conceptually how this principle enables extrapolation, and demonstrate the effectiveness and scalability of instantiations of REx on various OoD generalization tasks. Our code can be found at https://github.com/capybaralet/REx_code_release.

Residual Neural Networks (ResNets) achieve state-of-the-art performance in many computer vision problems. Compared to plain networks without residual connections (PlnNets), ResNets train faster, generalize better, and suffer less from the so-called degradation problem. We introduce simplified (but still nonlinear) versions of ResNets and PlnNets for which these discrepancies still hold, although to a lesser degree. We establish a 1-1 mapping between simplified ResNets and simplified PlnNets, and show that they are exactly equivalent to each other in expressive power for the same computational complexity. We conjecture that ResNets generalize better because they have better noise stability, and empirically support it for both simplified and fully-fledged networks.

The question of generalization in machine learning---how algorithms are able to learn predictors from a training sample to make accurate predictions out-of-sample---is revisited in light of the recent breakthroughs in modern machine learning technology. The classical approach to understanding generalization is based on bias-variance trade-offs, where model complexity is carefully calibrated so that the fit on the training sample reflects performance out-of-sample. However, it is now common practice to fit highly complex models like deep neural networks to data with (nearly) zero training error, and yet these interpolating predictors are observed to have good out-of-sample accuracy even for noisy data. How can the classical understanding of generalization be reconciled with these observations from modern machine learning practice? In this paper, we bridge the two regimes by exhibiting a new "double descent" risk curve that extends the traditional U-shaped bias-variance curve beyond the point of interpolation. Specifically, the curve shows that as soon as the model complexity is high enough to achieve interpolation on the training sample---a point that we call the "interpolation threshold"---the risk of suitably chosen interpolating predictors from these models can, in fact, be decreasing as the model complexity increases, often below the risk achieved using non-interpolating models. The double descent risk curve is demonstrated for a broad range of models, including neural networks and random forests, and a mechanism for producing this behavior is posited.