A jamming transition from under- to over-parametrization affects loss landscape and generalization

We argue that in fully-connected networks a phase transition delimits the over- and under-parametrized regimes where fitting can or cannot be achieved. Under some general conditions, we show that this transition is sharp for the hinge loss. In the whole over-parametrized regime, poor minima of the loss are not encountered during training since the number of constraints to satisfy is too small to hamper minimization. Our findings support a link between this transition and the generalization properties of the network: as we increase the number of parameters of a given model, starting from an under-parametrized network, we observe that the generalization error displays three phases: (i) initial decay, (ii) increase until the transition point --- where it displays a cusp --- and (iii) slow decay toward a constant for the rest of the over-parametrized regime. Thereby we identify the region where the classical phenomenon of over-fitting takes place, and the region where the model keeps improving, in line with previous empirical observations for modern neural networks.Comment: arXiv admin note: text overlap with arXiv:1809.0934

A Modern Take on the Bias-Variance Tradeoff in Neural Networks

The bias-variance tradeoff tells us that as model complexity increases, bias falls and variances increases, leading to a U-shaped test error curve. However, recent empirical results with over-parameterized neural networks are marked by a striking absence of the classic U-shaped test error curve: test error keeps decreasing in wider networks. This suggests that there might not be a bias-variance tradeoff in neural networks with respect to network width, unlike was originally claimed by, e.g., Geman et al. (1992). Motivated by the shaky evidence used to support this claim in neural networks, we measure bias and variance in the modern setting. We find that both bias and variance can decrease as the number of parameters grows. To better understand this, we introduce a new decomposition of the variance to disentangle the effects of optimization and data sampling. We also provide theoretical analysis in a simplified setting that is consistent with our empirical findings

Representation mitosis in wide neural networks

Deep neural networks (DNNs) defy the classical bias-variance trade-off: adding parameters to a DNN that interpolates its training data will typically improve its generalization performance. Explaining the mechanism behind this ``benign overfitting'' in deep networks remains an outstanding challenge. Here, we study the last hidden layer representations of various state-of-the-art convolutional neural networks and find evidence for an underlying mechanism that we call "representation mitosis": if the last hidden representation is wide enough, its neurons tend to split into groups which carry identical information, and differ from each other only by a statistically independent noise. Like in a mitosis process, the number of such groups, or ``clones'', increases linearly with the width of the layer, but only if the width is above a critical value. We show that a key ingredient to activate mitosis is continuing the training process until the training error is zero