In practice it is often found that large over-parameterized neural networks generalize better than their smaller counterparts, an observation that appears to conflict with classical notions of function complexity, which typically favor smaller models. In this work, we investigate this tension between complexity and generalization through an extensive empirical exploration of two natural metrics of complexity related to sensitivity to input perturbations. Our experiments survey thousands of models with different architectures, optimizers, and other hyper-parameters, as well as four different image classification datasets.

 We find that trained neural networks are more robust to input perturbations in the vicinity of the training data manifold, as measured by the input-output Jacobian of the network, and that this correlates well with generalization. We further establish that factors associated with poor generalization -- such as full-batch training or using random labels -- correspond to higher sensitivity, while factors associated with good generalization  -- such as data augmentation and ReLU non-linearities -- give rise to more robust functions. Finally, we demonstrate how the input-output Jacobian norm can be predictive of generalization at the level of individual test points. The empirical success of deep learning has thus far eluded interpretation through existing lenses of computational complexity BID2 , numerical optimization BID4 BID8 BID5 and classical statistical learning theory (Zhang et al., 2016) : neural networks are highly non-convex models with extreme capacity that train fast and generalize well. In fact, not only do large networks demonstrate good test performance, but larger networks often generalize better, counter to what would be expected from classical measures, such as VC dimension. This phenomenon has been observed in targeted experiments BID29 , historical trends of Deep Learning competitions BID3 , and in the course of this work ( Figure 1 ).This observation is at odds with Occam's razor, the principle of parsimony, as applied to the intuitive notion of function complexity (see §A.2 for extended discussion). One resolution of the apparent contradiction is to examine complexity of functions in conjunction with the input domain. f ( x) = x 3 sin(x) may seem decisively more complex than g(x) = x. But restrained to a narrow input domain of [−0.01, 0 .01] they appear differently: g remains a linear function of the input, while f (x) = O x 4 resembles a constant 0. In this work we find that such intuition applies to neural networks, that behave very differently close to the data manifold than away from it ( §4.1).We therefore analyze the complexity of models through their capacity to distinguish different inputs in the neighborhood of datapoints, or, in other words, their sensitivity. We study two simple metrics presented in §3 and find that one of them, the norm of the input-output Jacobian, correlates with generalization in a very wide variety of scenarios. Train loss Figure 1 : 2160 networks trained to 100% training accuracy on CIFAR10 (see §A.5.5 for experimental details). Left: while increasing capacity of the model allows for overfitting (top), very few models do, and a model with the maximum parameter count yields the best generalization (bottom right). Right : train loss does not correlate well with generalization, and the best model (minimum along the y-axis) has training loss many orders of magnitude higher than models that generalize worse (left). This observation rules out underfitting as the reason for poor generalization in low-capacity models. See BID29 for similar findings in the case of achievable 0 training loss.This work considers sensitivity only in the context of image classification tasks. We interpret the observed correlation with generalization as an expression of a universal prior on (natural) image classification functions that favor robustness (see §A.2 for details). While we expect a similar prior to exist in many other perceptual settings, care should be taken when extrapolating our findings to tasks where such a prior may not be justified (e.g. weather forecasting). We have investigated sensitivity of trained neural networks through the input-output Jacobian norm and linear regions counting in the context of image classification tasks. We have presented extensive experimental evidence indicating that the local geometry of the trained function as captured by the input-output Jacobian can be predictive of generalization in many different contexts, and that it varies drastically depending on how close to the training data manifold the function is evaluated. We further established a connection between the cross-entropy loss and the Jacobian norm, indicating that it can remain informative of generalization even at the level of individual test points. Interesting directions for future work include extending our investigation to more complex architectures and other machine learning tasks.
We perform massive experimental studies characterizing the relationships between Jacobian norms, linear regions, and generalization.
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