
A review published by @FlyScholarsCS (F.A.Q.) - 21 May 2021
Paper
"Neural Anisotropy Directions" (NeurIPS 2020)
Guillermo Ortiz-Jimenez, Apostolos Modas, Seyed-Mohsen Moosavi-Dezfooli, Pascal Frossard
https://arxiv.org/abs/2006.09717
TL;DR of the review
A common hypothesis is that deep networks always find the simplest solution to a problem, a hypothesis known as the simplicity bias. This study elegantly demonstrates that sometimes, deep networks prefer complex non-linear solutions to simple linear ones, and provides a criterion to predict these occurences from the neural architecture.
Summary of the paper
In this study, the authors show that typical convolutional networks sometimes fail to generalize on the simple task of classifying two linearly separable manifolds. They first show empirically that some directions in input space are harder to linearly separate than others. They then provide a criterion on the architecture of the network to predict whether a direction will be easy or hard to learn for this architecture. They finally show that when a network is trained on a modified version of CIFAR-10 --modified to contain a linearly separable channel for the CIFAR-10 labels--, the network sometimes prefer to solve the CIFAR-10 problem instead of solving the much simpler linear problem, provided that the linear solution is aligned with a non-preferred direction of the network.
Contributions / Strengths
- It is a striking and important finding that deep networks can sometimes drastically fail to generalize on a linear problem, preferring a more complex non-linear solution.
- The study provides a criterion to predict from an architecture which directions in input space will be easy or hard for a network to linearly separate, and demonstrates both empirically and theoretically the validity of this criterion.
- By concatenating CIFAR-10 with a linearly separable channel aligned with the non-preferred directions of the network, the study elegantly shows that deep networks can sometimes prefer to optimize for a non-linear problem such as CIFAR-10 than to solve a trivial linear problem. This finding tempers the conventional wisdom that deep networks always find the solution of least complexity, as measured by the curvature of the decision boundary of the solution.
Limitations / Weaknesses
- The main limitation of this study is that the method proposed only allows the inspection of the linear inductive biases of an architecture. It is unclear what the relevance of these linear biases are for real-world problems, and/or whether the method proposed could be extended to study more sophisticated inductive biases of deep networks.
- Some theoretical aspects of the study could be more clearly exposed (see Clarity Concerns).
- Some claims could be backed up by further experiments (See Questions). In particular, I am yet to be convinced that the experiment where they rotate CIFAR-10 really shows that the linear inductive biases of convolutional networks are tailored to natural images.
Clarity Concerns
- The authors state that "a layer in the architecture can shape the bias in two ways: (1) by causing an anisotropic loss of information, or (2) by anisotropically conditioning the optimization landscape". But they do not provide an explanation as to which of these effect is at play in the networks empirically studied. It seems that (1) would not allow a perfect training accuracy, so it could be the phenomenon at play for LeNet, the only network for which the training accuracy is not perfect. If (2) is at play in the other networks, it would indicate that by training the networks for longer on more samples, the networks should eventually be able to learn to generalize, but this possibility is not discussed in the paper.
- In the experiments where the networks fail to generalize, it is not always clear whether it is because the networks found a linear separator with suboptimal margins on the training set, or whether the networks found a non-linear solution on the training set (except for the CIFAR-10 experiment where it is clear that the network preferred a non-linear solution).
Questions / Suggestions for future work
- The study claims that the linear inductive biases of ConvNets are specially adapted to natural images. To show this, a rotation of CIFAR-10 is performed, such that preferred directions of the network are swapped with non-preferred directions, which results in the network performing worse than on the non-rotated CIFAR-10. But this swapping not only destroys the linear alignement, it also destroys the translation invariant statistics of natural images. An important control would be to swap directions that have similar NAD-scores associated to them (NAD = Neural Anisotropy Directions, the preferred directions of the network). This swapping would destroy the translation invariance while preserving the linear alignement of the dataset with NADs. This experiment would control for the translation invariance inductive bias of convolutional neural networks, as opposed to the linear inductive bias studied here, and would make the claim that NADs of ConvNets are tuned to natural image statistics more convincing.
- A flurry of recent papers (Transformers, MLP-Mixer, RepMLP, ResMLP, gMLP) show that networks with fewer inductive biases can compete with convolutional networks when trained on large enough datasets. However, these networks do have some remaining structure and it is unclear whether this structure corresponds to important inductive biases or not. The method proposed here could be used to directly assess the linear inductive biases present in these architectures, and compare those to ConvNet (important inductive biases) and pure MLPs (few inductive biases).
Related Readings
Exact solutions to the nonlinear dynamics of learning in deep linear neural networks (2013)
Andrew M. Saxe, James L. McClelland, Surya Ganguli
https://arxiv.org/abs/1312.6120
NB: Saxe et al. study how the speed of learning of dense linear deep networks depends on the covariance statistics of the data. It nicely mirrors and complements this study which focuses on how the inductive biases of more complex architecture such as conv nets affect learning speed.
Understanding deep learning requires rethinking generalization (2016)
Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, Oriol Vinyals
https://arxiv.org/abs/1611.03530
NB: Zhang et al. show how neural networks can learn very complex non-linear functions such as the one obtained by randomizing the labels of the dataset. Interestingly, the present study show that networks can also learn complex functions when presented with simple linear problems.
The Pitfalls of Simplicity Bias in Neural Networks (2020)
Harshay Shah, Kaustav Tamuly, Aditi Raghunathan, Prateek Jain, Praneeth Netrapalli
https://arxiv.org/abs/2006.07710
NB: This study published in the same conference as the study reviewed (NeurIPS 2020) focuses on cases where the network does learn the simple linear solution as opposed to more complex non-linear solutions, even when the non-linear solution would in fact provide better generalization.
Why Are Convolutional Nets More Sample-Efficient than Fully-Connected Nets? (2020)
Zhiyuan Li, Yi Zhang, Sanjeev Arora
https://arxiv.org/abs/2010.08515
NB: Li et al. analytically derive how much more sample efficient convolutional neural networks are than fully-connected networks on translation invariant problems
Reviewer
~anonymous
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