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Title: All Neural Networks are Created Equal

Paper content:
One of the unresolved questions in deep learning is the nature of the solutions that are being discovered.
We investigate the collection of solutions reached by the same network architecture, with different random initialization of weights and random mini-batches.
These solutions are shown to be rather similar - more often than not, each train and test example is either classified correctly by all the networks, or by none at all.
Surprisingly, all the network instances seem to share the same learning dynamics, whereby initially the same train and test examples are correctly recognized by the learned model, followed by other examples which are learned in roughly the same order.
When extending the investigation to heterogeneous collections of neural network architectures, once again examples are seen to be learned in the same order irrespective of architecture, although the more powerful architecture may continue to learn and thus achieve higher accuracy.
This pattern of results remains true even when the composition of classes in the test set is unrelated to the train set, for example, when using out of sample natural images or even artificial images.
To show the robustness of these phenomena we provide an extensive summary of our empirical study, which includes hundreds of graphs describing tens of thousands of networks with varying NN architectures, hyper-parameters and domains.
We also discuss cases where this pattern of similarity breaks down, which show that the reported similarity is not an artifact of optimization by gradient descent.
Rather, the observed pattern of similarity is characteristic of learning complex problems with big networks.
Finally, we show that this pattern of similarity seems to be strongly correlated with effective generalization.
The recent success of deep networks in solving a variety of classification problems effectively, in some cases reaching human-level precision, is not well understood.
One baffling result is the incredible robustness of the learned models: using variants of Stochastic Gradient Descent (SGD), with random weight initialization and random sampling of mini-batches, different solutions are obtained.
While these solutions typically correspond to different parameter values and possibly different local minima of the loss function, nevertheless they demonstrate similar performance reliably.
To advance our understating of this issue, we are required to compare different network instances.
Most comparison approaches (briefly reviewed in Appendix A) are based on deciphering the internal representations of the learned models (see Lenc & Vedaldi, 2015; Alain & Bengio, 2016; Li et al., 2016; Raghu et al., 2017; Wang et al., 2018) .
We propose a simpler and more direct approachcomparing networks by their classifications of the data.
To this end, we represent each network instance by 2 binary vectors which capture the train and test classification accuracy.
Each vector's dimension corresponds to the size of the train/test dataset; each element is assigned 1 if the network classifies the corresponding data point correctly, and 0 otherwise.
Recall the aforementioned empirical observation -different neural network instances, obtained by repeatedly training the same architecture with SGD while randomly sampling its initial weights, achieve similar accuracy.
At the very least, this observation predicts that the test-based vector representation of different networks should have similar L 1 /L 2 norms.
But there is more: it has been recently shown that features of deep networks capture perceptual similarity reliably and consistently, similarly across different instances and different architectures (Zhang et al., 2018) .
These results seem to suggest that our proposed representation vectors may not only have a similar norm, but should also be quite similar as individual vectors.
But similar in what way?
In this paper, we analyze collections of deep neural networks classifiers, where the only constraint is that the instances are trained on the same classification problem, and investigate the similarity between them.
Using the representation discussed above, we measure this similarity by two scores, consistency score and consensus score, as defined in §2.
Like other comparison approaches (see Appendix A), our analysis reveals a high level of similarity between trained networks.
Interestingly, it reveals a stronger sense of similarity than previously appreciated: not only is the accuracy of all the networks in the collection similar, but so is the pattern of classification.
Specifically, at each time point during the learning process (or in each epoch), most of the data points in both the train and test sets are either classified correctly by all the networks, or by none at all.
As shown in §3, these results are independent of choices such as optimization method, hyperparameter values, the detailed architecture, or the particular dataset.
They can be replicated for a fixed test set even when each instance in the collection sees a different train set, as long as the training data is sampled from the same distribution.
Moreover, the same pattern of similarity is observed for a wide range of test data, including out-of-sample images of new classes, randomly generated images, or even artificial images generated by StyleGAN (Karras et al., 2019) .
These results are also reproduce-able across domains, and were reproduced using BiLSTM (Hochreiter & Schmidhuber, 1997) with attention (Bahdanau et al., 2014) for text classification.
We may therefore conclude that different network instances compute similar classification functions, even when being trained with different training samples.
It is in the dynamic of learning, where the results of our analysis seem to go significantly beyond what has been shown before, revealing an even more intriguing pattern of similarity between trained NN instances.
Since deep NNs are almost always trained using gradient descent, each network can be represented by a time series of train-based and test-based representation vectors, one per epoch.
We find that network instances in the collection do not only show the same pattern of classification at the end of the training, but they also evolve in the same way across time and epochs, gradually learning to correctly or incorrectly classify the same examples in the same order.
When considering bigger classification problems such as the classification of ImageNet with big modern CNN architectures, a more intricate pattern of dynamics is evident: to begin with, all networks wrongly classify most of the examples, and correctly classify a minority of the examples.
The learning process is revealed by examples moving from one end (100% false classification) to the other end (100% correct classification), which implies two things:
(i) the networks learn to correctly classify examples in the same order;
(ii) the networks agree on the examples they misclassify throughout.
As shown in §4, these results hold regardless of the network' architecture.
To drive this point home we compare a variety of public domain architectures such as VGG19 (Simonyan & Zisserman, 2014) , AlexNet (Krizhevsky et al., 2012) , DenseNet (Huang et al., 2017) and ResNet-50 (He et al., 2016) .
In all cases, different architectures may learn at a different pace and achieve different generalization accuracy, but they still learn in the same order.
Thus all networks start by learning roughly the same examples, but the more powerful networks may continue to learn additional examples as learning proceeds.
A related phenomenon is observed when extending the analysis to simpler learning paradigms, such as deep linear networks, SVM, and KNN classifiers.
Our empirical study extends to cases where these robust patterns of similarity break down, see §5.
For example, when randomly shuffling the labels in a known benchmark (Zhang et al., 2016) , the agreement between different classifiers disappear.
This stands in agreement with (Morcos et al., 2018) , where it is shown that networks that generalize are more similar than those that memorize.
Nevertheless, the similarity in learning dynamic is not an artifact of learnability, or the fact that the networks have converged to solutions with similar accuracy.
To see this we constructed a test case where shallow CNNs are trained to discriminate an artificial dataset of images of Gabor patches (see Appendix C).
Here it is no longer true that different network instances learn in the same order; rather, each network instance follows its own path while converging to the final model.
The similarity in learning dynamic is likewise not an artifact of using gradient descent.
To see this we use SGD to train linear classifiers to discriminate vectors sampled from two largely overlapping Gaussian distributions.
Once again, each classifier follows its own path while converging to the same optimal solution.
We empirically show that neural networks learn similar classification functions.
More surprisingly with respect to earlier work, the learning dynamics is also similar, as they seem to learn similar functions also in intermediate stages of learning, before convergence.
This is true for a variety of architectures, including different CNN architectures and LSTMs, irrespective of size and other hyper-parameters of the learning algorithms.
We have verified this pattern of results using many different CNN architectures, including most of those readily available in the public domain, and many of the datasets of natural images which are in common use when evaluating deep learning.
The similarity of network instances is measured in the way they classify examples, including known (train) and new examples.
Typically, the similarity over test data is as pronounced as it is over train data, as long as the train and test examples are sampled from the same distribution.
We show that this similarity extends also to out of sample test data, but it seems to decrease as the gap between the distribution of the train data and the test data is increased.
This pattern of similarity crosses architectural borders: while different architectures may learn at a different speed, the data is learned in the same order.
Thus all architectures which reach a certain error rate seem to classify, for the most part, the same examples in the same manner.
We also see that stronger architectures, which reach a lower generalization error, seem to start by first learning the examples that weaker architectures classify correctly, followed by the learning of some more difficult examples.
This may suggest that the order in which data is learned is an internal property of the data.
We also discuss cases where this similarity breaks down, indicating that the observed similarity is not an artifact of using stochastic gradient descent.
Rather, the observed pattern of similarity seems to characterize the learning of complex problems with big networks.
Curiously, the deeper the network is and the more non-linearities it has, and even though the model has more learning parameters, the progress of learning in different network instances becomes more similar to each other.
Un-intuitively, this suggests that in a sense the number of degrees of freedom in the learning process is reduced, and that there are fewer ways to learn the data.
This effect seems to force different networks, as long they are deep enough, to learn the dataset in the same way.
This counter-intuitive result joins other non-intuitive results, like the theoretical result that a deeper linear neural network converges faster to the global optimum than a shallow network (Arora et al., 2018) .
We also show that the observed pattern of similarity is strongly correlated with effective generalization.
What does it tell us about the generalization of neural networks, a question which is considered by many to be poorly understood?
Neural networks can memorize an almost limitless number of examples, it would seem.
To achieve generalization, most training protocols employ some regularization mechanism which does not allow for unlimited data memorization.
As a result, the network fits only the train and test examples it would normally learn first, which are, based on our analysis, also the "easier" (or more typical) examples.
We hypothesize that this may explain why a regularized network discovers robust solutions, with little variability among its likely instances.
(Cybenko, 1989) , and they can learn any arbitrary complex function (Hornik et al., 1989) .
This extended capacity can indeed be reached, and neural networks can memorize datasets with randomly assigned labels (Zhang et al., 2016) .
Nevertheless, the dominant hypothesis today is that in natural datasets they "prefer" to learn an easier hypothesis that fits the data rather than memorize it all (Zhang et al., 2016; Arpit et al., 2017) .
Our work is consistent with a hypothesis which requires fewer assumptions, see Section 6.
The direct comparison of neural representations is regarded to be a hard problem, due to a large number of parameters and the many underlying symmetries.
Many non-direct approaches are available in the literature: (Li et al., 2016; Wang et al., 2018) compare subsets of similar features across multiple networks, which span similar low dimensional spaces, and show that while single neurons can vary drastically, some features are reliably learned across networks.
(Raghu et al., 2017) proposed the SVCCA method, which can compare layers and networks efficiently, with an amalgamation of SVD and CCA.
They showed that multiple instances of the same converged network are similar to each other and that networks converge in a bottom-up way, from earlier layers to deeper ones.
Morcos et al. (2018) builds off the results of (Raghu et al., 2017) , further showing that networks which generalize are more similar than ones which memorize, and that similarity grows with the width of the network.
In various machine learning methods such as curriculum learning (Bengio et al., 2009 ), self-paced learning (Kumar et al., 2010) and active learning (Schein & Ungar, 2007) , examples are presented to the learner in a specific order (Hacohen & Weinshall, 2019; Jiang et al., 2017) .
Although conceptually similar, here we analyze the order in which examples are learned, while the aforementioned methods seek ways to alter it.
Likewise, the design of effective initialization methods is a striving research area (Erhan et al., 2010; Glorot & Bengio, 2010; Rumelhart et al., 1988 ).
Here we do not seek to improve these methods, but rather analyze the properties of a collection of network instances generated by the same initialization methodology.

assistant: text: Most neural networks approximate the same classification function, even across architectures, through all stages of learning.