This paper presents a phenomenon in neural networks that we refer to as local elasticity. Roughly speaking, a classifier is said to be locally elastic if its prediction at a feature vector x is not significantly perturbed, after the classifier is updated via stochastic gradient descent at a (labeled) feature vector x that is dissimilar to x in a certain sense. This phenomenon is shown to persist for neural networks with nonlinear activation functions through extensive simulations on real-life and synthetic datasets, whereas this is not observed in linear classifiers. In addition, we offer a geometric interpretation of local elasticity using the neural tangent kernel (Jacot et al., 2018). Building on top of local elasticity, we obtain pairwise similarity measures between feature vectors, which can be used for clustering in conjunction with K-means. The effectiveness of the clustering algorithm on the MNIST and CIFAR-10 datasets in turn corroborates the hypothesis of local elasticity of neural networks on real-life data. Finally, we discuss some implications of local elasticity to shed light on several intriguing aspects of deep neural networks.
[Summary]
This paper proposes and studies the “local elasticity”, a quantitative measure for the ability of neural networks to only locally change its prediction (around x) after a stochastic gradient step at x. The paper verifies experimentally that nonlinear neural nets are locally elastic through showing that an elasticity-motivated similarity score can perform clustering well.
[Pros]
The notion of local elasticity is interesting and has the potential of opening up lots of further directions. The way I understand it is to relate to memorization (as the authors have indeed discussed) --- I think “local elasticity” can be viewed as some sort of “local memorization ability”, in that the NN is able to change its prediction only in a small neighborhood of x---without affecting predictions at other remote x’s---after one SGD step on x. Conceptually this is something not covered by the existing narratives in deep learning theory, yet the phenomenon itself is quite convincing and could provide a new perspective into lots of things.
[Cons]
It feels like the experimental results in the present paper is a rather indirect evidence for the local elasticity -- that the similarity score coming from the elasticity works well for a downstream clustering task. Could there be some more direct evidence about the local elasticity? How would the elasticity compare on different architectures? In the present form the experiments perhaps at most says that the similarity score makes sense, not yet that a fully quantitative characterization of the local elasticity.
I’m also a little bit concerned about the fairness of the clustering experiment, in that the elasticity-motivated clustering algorithm utilizes an auxiliary dataset whereas simple baselines such as K-means and PCA K-means are not able to use that. Is there a way of modifying the K-means and PCA K-means so that they can also use this auxiliary dataset while still giving a sensible algorithm for the primary 2-class clustering task?