### Mathematical Aspects of Spin Glasses and Neural Networks (Progress in Probability)

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Information via backpropagation flows more efficiently backwards into the network, but it can not jump as far in each iteration as the shortcut network. The mutual information between two distributions X and Y is defined as:. Mutual information is the amount of uncertainty, in bits, reduced in a distribution X by knowing Y. These properties make mutual information useful for quantifying the similarity between two nonlinearly different layers. It will capture the information lost by sending information through the network, but, unlike traditional correlation measures, it does not require a purely affine relationship between X and Y to be maximized.

We calculate the mutual information between the features of two layers by using the Kraskov method Kraskov et al. In particular, we take an input image and evaluate the activations at each layer. We then calculate the mutual information between the activations of the first layer and the last layer, using the entire validation set as an ensemble. To ensure that the mutual information between the first and last layer is not trivial, we make the first and last layers twice as wide, to force the network to discard information between the first and last layer.

As shown in Figures 4A,B , as the nets train, they progressively move toward an apparent optimum mutual information between the first and last layers.

Traditional MLPs follow a trend of systematically increasing the mutual information. On the other hand, MLPs with shortcuts start with higher mutual information which then decreases toward the optimum. This may be interpreted as the shortcut helping the network to first find a low dimensional manifold, and then progressively exploring larger and larger volumes of state-space without losing accuracy.

We should note that the purpose of this study is not to present the state of the art results e. Figure 4. Comparison of performance for nets with A various layer widths and B various numbers of hidden layers.

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Each trace represents a different random weight initialization. Test error is the proportion of validation examples the network incorrectly labels. In Figures 5A,B we compare the performance of different ResNets widths and the effects of adding residual skip-connects, shortcuts, or both respectively. As ResNets train, they start with low mutual information between weights. The MI gradually increases as it trains, maximizes and begins to decrease again see Figure 5A. The lack of mutual information in the final trained networks shows that a well trained network does not learn identity transforms.

The objective of Figure 5B is twofold: i to show that the shortcut improves upon the traditional MLP and ii that both the shortcut and traditional MLP benefit from the additional introduction of residuals. Note that the main improvement over the traditional MLP comes from the shortcut as can be seen from the green crosses and the blue diamonds.

The residuals add an extra mild improvement for both the traditional MLP and the shortcut as can be seen from the red and turqoise circles. Figure 5. Comparison of performance for A various ResNet widths without any shortcuts. Each color trace is a single training run and B various combinations of architectures. In this plot, as neural networks train, they start at high error and progressively decrease error after each epoch represented by each point.

In Figure 5A we see evidence that high mutual information is not a necessary condition for accuracy. However, high mutual information allows the weights to lie upon a low-dimensional manifold that speeds training. In Figure 5A , we see that high mutual information produces rapid decrease in test error: The points that represent the outcome of each epoch of training show a high slope and decrease in error at high mutual information, and a low slope at low mutual information Figure 5B , notice that the x-axis has a different scale.

### Mathematical Psychology

This behavior agrees with the analysis in Schwartz-Ziv and Tishby, which identifies two phases in the training process: i a drift phase where the error decreases fast while the successive layers are highly correlated and ii a diffusion phase where the error decreases slowly if at all and the representation becomes more efficient. The training progress of networks both MLP and ResNets with shortcut connections, indicated by the larger turquoise circles and green crosses, starts with such a high mutual information that the networks are largely trained within a single epoch.

Successive layers which enjoy high mutual information obviously learn features that cannot be far from the previous layer in the space of possible features. However, mutual information alone cannot tell us what these features are. In other words, while we see that the deep net must be learning slowly we cannot use solely mutual information to say what it is that it learns first, second, third etc. This is particularly evident in our observation that training first correlates features in different layers, and then the mutual information steadily decreases as the network fine-tunes to its final accuracy.

Thus, we see that high mutual information between layers particularly between the first and last layer allows the neural network to quickly find a low dimensional manifold of much smaller effective dimension than the total number of free parameters. Gradually, the network begins to explore away from that manifold as it fine tunes to its final level of accuracy.

The gathered experience by us and others about the difficulty of training deep nets over shallow nets points to the fact that the first features learned have to be simple ones. If not, if the complicated features were the ones learned through the first few layers, then the deeper layers would not make much difference.

Another way to think of this is that the depth of the deep net allows one to morph a representation of the input space from a rudimentary one to a sophisticated one. This makes mathematical, physical and evolutionary sense too see also the analysis in Schwartz-Ziv and Tishby, This point of view agrees with the success of the recently proposed ResNets. ResNets enforce the gradual learning of features by strongly coupling successive layers.

In particular, in vRNG one proceeds to estimate the conditional probability distribution of one layer conditioned on the previous one. This task is made simpler IF the two successive layers are closely related.

In machine learning parlance, this means that the two successive layers are coupled so that the features learned by one layer do not differ a lot from those learned by the previous one. This also chimes with the recent mathematical analysis about deep convolutional networks Mallat, In particular, tracking the evolution of mutual information and the associated test error with the number of iterations helps us delineate which architectures will find the optimal mutual information manifold, something one should keep in mind when fiddling with the myriads of possible architecture variants.

However, mutual information alone is not enough, because it can help evaluate a given architecture but cannot propose suggest a new architecture. An adaptive scheme which can create hybrids between different architectures is some kind of remedy but of course does not solve the problem in its generality.

This is a well-known problem in artificial intelligence and for some cases it may be addressed through techniques like reinforcement learning Sutton and Barto, Overall, the successful training of a deep net points to the successful discovery of a low-dimensional manifold in the huge space of features and using it as a starting point for further excursions in the space of features. Also, this low-dimensional manifold in the space of features constrains the weights to also lie in a low-dimensional manifold.

In this way, one avoids being lost in unrewarding areas and thus leads to robust training of the deep net. Introducing long-range correlations appears to be an effective way to enable training of extremely large neural networks. Interestingly, it seems that maximizing mutual information does not directly produce maximum accuracy, but finding a high-MI manifold and from there evolving toward a low-MI manifold allows training to unfold more efficiently.

When the output of two layers is highly correlated, many of the potential degrees of freedom collapse into a lower dimensional manifold due to the redundancy between features. Thus, high mutual information between the first and last layer enables effective training of deep nets by exponentially reducing the size of the potential training state-space. Despite having millions of free parameters, deep neural networks can be effectively trained.

## Replica bounds for diluted non-Poissonian spin systems

We showed that significant inter-layer correlation mutual information reduces the effective state-space size, making it feasible to train such nets. By encouraging the correlation with shortcuts, we reduce the effective size of the training space, and we speed training and increase accuracy. Hence, we observe that long range correlation effectively pulls systems onto a low-dimensional manifold, greatly increasing tractability of the training process.

Once the system has found this low-dimensional manifold, it then tends to gradually leave the manifold as it finds better training configurations. Thus, high correlation followed by de-correlation appears to be a promising method for finding optimal configurations of high-dimensional systems.

## Open Access Journals

By experimenting with artificial neural networks, we can begin to gain insight into the developmental processes of biological neural networks, as well as protein folding Dill and Chan, Even when batch normalization is used to help eliminate vanishing gradients, deep MLPs remain difficult to train. As we see in Figure 4B , beyond 5—10 layers, adding depth to a MLP slows training and converges to a lower accuracy. This has also been demonstrated in other applications with other types of neural networks Srivastava et al. Our measures of mutual information also show that deeper networks reduce mutual information between the first and last layer, increasing the difficulty for the training to find a low-dimensional manifold to begin fine tuning.

The present results imply that the power of residual networks lies in their ability to efficiently correlate features via backpropagation, not simply in their ability to easily learn identity transforms or unit Jacobians. The shortcut architecture we describe here is easy to implement using deep learning software tools, such as Keras or TensorFlow. Despite adding no new free parameters, the shortcut conditions the network's gradients in a way that increases correlation between layers. This follows from the nature of the backpropagation algorithm: error in the final output of the neural network is translated into weight updates via the derivative chain rule.

Adding a shortcut connection causes the gradients in the first layer and final layer to be summed together, forcing their updates to be highly correlated. Adding the skip connection increases coupling between the first and final layer, which constrains the variation of weights in the intervening layers, driving the space of possible weight configurations onto a lower dimensional manifold.

Thus, a contribution of understanding that the neural networks train more effectively when they start on a low dimensional manifold includes demonstrating how long range shortcuts improve network trainability. Two of these were previously established as methodologies, namely mathematical theory and experimentation on real systems, although the types of studies discussed above have led to many new approaches within these areas. But it has also seen the birth of and helped to drive as an equally important partner in the quest another mode of investigation that was previously little represented.

For example, one can effectively perform experiments on systems whose microscopic properties are known exactly, typically corresponding to the same minimalist models as studied theoretically, without the complications of real nature and with the possibility to vary from real nature e. There remain many further opportunities for both scientific understanding and practical application. Some can already be anticipated, but others are yet to be thought of; the developments of the last three decades have led to so many remarkable and unanticipated discoveries that it seems inevitable that many more will arise.

The next few decades offer the prospect of much richness for the scientific explorer and technological applier alike. I shall use the physics convention; the mapping between them is just a minus sign. National Center for Biotechnology Information , U.

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Author information Copyright and License information Disclaimer. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article has been cited by other articles in PMC. Abstract This paper is concerned with complex macroscopic behaviour arising in many-body systems through the combinations of competitive interactions and disorder, even with simple ingredients at the microscopic level.

Keywords: complex systems, spin glasses, NP-completeness, hard optimization, econophysics, neural networks. Neural networks and proteins Multi-valley landscapes are not always a nuisance—in fact, they can be very valuable. Econophysics More interesting examples of complex systems are found in economic and financial systems, the topic of the new science of econophysics. Magnitudes As emphasized earlier, our interest is in systems of a large number of individuals.

## Mathematical Aspects of Spin Glasses and Neural Networks

Conclusions In this brief perspective, I have attempted to give an impression of the conceptual, mathematical, experimental and simulational challenges and novel discoveries that the combination of disorder and frustration in many-body systems has yielded, together with hints of some of the application opportunities their recognition has offered and continues to offer. References Amit D. Modeling brain function. More is different. Graph bipartitoning and statistical mechanics.

Theory of superconductivity. Topological phase transition in complex networks. Binder K. Spin glasses: experimental facts, theoretical concepts, and open questions.