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Title: Revisiting the Information Plane

Paper content:
There has recently been a heated debate (e.g. Schwartz-Ziv & Tishby (2017), Saxe et al. (2018), Noshad et al. (2018), Goldfeld et al. (2018)) about measuring the information flow in Deep Neural Networks using techniques from information theory.
It is claimed that Deep Neural Networks in general have good generalization capabilities since they not only learn how to map from an input to an output but also how to compress information about the training data input (Schwartz-Ziv & Tishby, 2017).
That is, they abstract the input information and strip down any unnecessary or over-specific information.
If so, the message compression method, Information Bottleneck (IB), could be used as a natural comparator for network performance, since this method gives an optimal information compression boundary.
This claim was then later denounced as well as reaffirmed (e.g.  Saxe et al. (2018), Achille et al. (2017), Noshad et al. (2018)), as the employed method of mutual information measuring is not actually measuring information but clustering of the internal layer representations (Goldfeld et al. (2018)).
In this paper, we will present a detailed explanation of the development in the Information Plain (IP), which is a plot-type that compares mutual information to judge compression (Schwartz-Ziv & Tishby (2017)), when noise is retroactively added (using binning estimation).
We also explain why different activation functions show different trajectories on the IP.
Further, we have looked into the effect of clustering on the network loss through early and perfect stopping using the Information Plane and how clustering can be used to help network pruning.
Deep Neural Networks (DNNs) have recently achieved promising results in many areas especially computer vision and natural language processing.
Yet, the learning process and design principles of configuring DNN architecture are under-investigated (Tishby & Zaslavsky, 2015) .
There are some recent attempts towards addressing this challenge.
From an information theoretic viewpoint, Schwartz-Ziv & Tishby (2017) have investigated the learning dynamics of DNN -how the mutual information (MI) of the layer activation with input and target develops over the course of training.
The finding is that DNNs generally first increase the MI of the layers with both, but then reduce the MI with the input.
This perceived compression has led to promising results of DNN in many applications 1 .
This compression behaviour resembles the IB-method, a constraint method which aims to retain maximum information content for given compression levels (Tishby et al. (1999) ) and these possible maxima are depicted by the IB-bound.
2 Through the similarity, the IB-bound could be used as a way to judge network architecture (Schwartz-Ziv & Tishby (2017) ).
The closer to the IB-bound the better the NN is likely to perform.
However, this finding is controversial, which has been supported by e.g. ; ; Noshad et al. (2018) and denied.
Most prominently, Saxe et al. (2018) have argued that this does not generalize for all activation functions and that compression does not necessarily lead to good generalization.
Nonetheless Alemi et al. (2016) , Kolchinsky et al. (2017) , Nguyen & Choi (2018) , Banerjee & Montúfar (2018) and Alemi et al. (2018) have tried to implement the IB-constraint as optimization parameter for DNN training leading to promising results.
Amjad & Geiger (2018) criticize these attempt claiming that they were not really sticking to the IB for their optimization process since in deterministic NNs the mutual information is either infinite or constant.
Hence, the IB cannot produce optimizeable gradients.
They therefore reason, that the results of these authors were only possible by giving up a hard IB constraint.
Recent success with fully invertible neural networks (which cannot experience any form of compression) cast doubt on the notion of compression being a necessary factor for good generalization (e.g. (Jacobsen et al., 2018) , (Chang et al., 2017) , (Ardizzone et al., 2019) , (Song et al., 2019) ).
Finally, a recent paper by Goldfeld et al. (2018) assessed that measuring MI in this scenario is actually tracking how clustered the internal layer representations of the samples are.
Building on Goldfeld et al. (2018) , this work attempts to explain the trajectories in the IP created through MI estimation using binning.
Through this we will shed more light on the investigation of the learning dynamics of DNNs through usage of the IP.
Section 2.2.1 shows that the smaller the bin size for the binning estimator, the more the layers drift towards a fixed point in the IP.
Section 2.2.2 highlights that higher layers strongly influence the shape of lower layers in the IP.
Section 2.3 explains why the IP looks the way it does.
3 Clustering is then examined as a design parameter.
This is done by investigating the connection between the loss function and clustering through the usage of early and perfect stopping in section 2.4.1.
Here, no clear connection is found.
Lastly, network pruning is attempted in section 2.5 using the IP, where slightly positive indications are found.
At first though, the experimental setup is outlined.
This paper studies the information plane as a neural network analysis tool.
We have looked into the influence of different bin sizes for binning estimation, which has led to a detailed explanation of why certain behaviour is happening in the information plane.
Thus, finding new strong evidence that the information plane only tracks clustering as Goldfeld et al. (2018) suggested.
The usage of measuring clustering has been investigated using early stopping and perfect stopping, which we have not been able to generalise the finding across different datasets.
Clustering can be used to design a NN in terms of pruning, which might be worthy of further investigation.
The information plane holds value as a measure of clustering and could potentially lead to advancements in Deep Learning.
One aspect that has not been part of the discussion so far is that in contrast to non-linearly saturating activation functions like TanH, which has no binning during the real training process, ReLU in fact has a bin.
The 0-bin could actually lead to a loss in mutual information because the injectiveness of the activation function gets lost (not invertible anymore) and mutual information is not bound to be constant or infinite.
Therefore, networks with ReLU could experience a form of compression.
ReLU does in general show better generalization capabilities than TanH, which could partially support the claim that compressed neural networks generalize better Schwartz-Ziv & Tishby (2017) .
A well known problem of ReLUs is called "dying ReLUS" which could be a case of "too high" compression.
Which would disturb the mapping between input and output.
Taking out the binning of ReLUs, like in LeakyReLUs, is almost always favorable compared to standard ReLUs in terms of generalization (Xu et al. (2015) ).
Since LeakyReLUs restore the invertibility of the activation function and therefore prevent compression, this also indicates that compression does not necessarily generalizes better in DNNs.
It remains a task for future investigations, how this can be explained in detail.

assistant: text: We give a detailed explanation of the trajectories in the information plane and investigate its usage for neural network design (pruning)