Compressing information through the information bottleneck during deep learning

Read an article in Quanta Magazine (New theory cracks open the black box of deep learning) about a talk (see 18: Information Theory of Deep Learning, YouTube video) done a month or so ago given by Professor Naftali (Tali) Tishby on his theory that all deep learning convolutional neural networks (CNN) exhibit an “information bottleneck” during deep learning. This information bottleneck results in compressing the information present, in for example, an image and only working with the relevant information.

The Professor and his researchers used a simple AI problem (like recognizing a dog) and trained a deep learning CNN to perform this task. At the start of the training process the CNN nodes at the top were all connected to the next layer, and those were all connected to the next layer and so on until you got to the output layer.

Essentially, the researchers found that during the deep learning process, the CNN went from recognizing all features of an image to over time just recognizing (processing?) only the relevant features of an image when successfully trained.

Limits of deep learning CNNs

In his talk the Professor identifies two modes of operations of a deep learning CNN: the encoder layers and decoder layers. The encoder function identifies relevant information in the input and the decoder function takes this relevant information and maps this to an output.

This view results in two statistics that can characterize any deep learning CNN:

  • Sample complexity which refers to the the mutual information inside the last hidden layer of the encoder function, and
  • Accuracy or generalization error, which refers to the mutual information inside the last hidden layer of the decoder function.

Where mutual information is defined as how much of the uncertainty of an input is removed when you have an output that is based on that input. (See the talk for a more formal explanation).

The professor states that any complex deep learning CNN can be characterized by these two statistics where sample complexity determines the number of samples required and accuracy determines the precision by which the deep learning CNN can properly interpret those samples. The deep black line in the chart represents the limits of accuracy achievable at some number of training events, with some number of hidden layers and some sample set.

What happens during deep learning

Moreover, the professor shows an interesting characteristic of all CNNs is that they converge over time in accuracy and that convergence differs based mostly on the number of layers, sample size and training count used.

In the chart, the top row show 3 CNNs with different amounts of training data (5%, 40% and 80% of total). The chart shows the end result and trace of learning within the CNN over the same number of epochs (training cycles). More training data generates more accurate results.

The Professor views those epochs after the farthest right traces (where the trace essentially starts moving up and to the left in the chart), the compression phase of deep learning.

Statistics of deep learning process

The professor goes on to characterize the deep learning  process by calculating the mean and variance of each layers connection weights.

In the chart he shows an standard “eiffel tower” neural network, with 6 hidden layers, each with less neurons (nodes)  than the previous layer (12 nodes, 10 nodes, 7 nodes, etc.). And what he plots is the average weights and variance between layers (red lines are average and variance of the weights for arcs[connections] between nodes in layer 1 to nodes in layer 2, blue lines the mean and variance of weights for arcs between layer 2 and 3, purple lines the mean and variance of weights for arcs between layer 3 and 4, etc.).

He shows that at the start of training the (randomly assigned) weights for each layer have a normalized mean which is higher than its normalized variance. He calls this phase as high signal to noise (I would say the opposite, its low signal to noise, more noise than signal). But as training proceeds (over more epochs), there comes a point where the layer mean drops below its variance and the signal to noise ratio changes dramatically. After that point the mean weights and variance of the group of layers start to diverge or move apart.

The phase (epochs) after the line where the weights means are lower than its variance, he calls the Compression phase of the deep layer CNN training.

The Professor suggests that every complex deep learning CNN looks the same during training if you perform the calculations. The professor shows charts like this for other deep learning CNNs used on different problems and they all exhibit some point where their means are lower than their weights after which means and variances between layers starts to differentiate.

Do layer counts and sample size matter?

It turns out that the more hidden layers you have, the sooner (less training) you need to begin the compression phase. This chart shows the same problem, with different hidden layer counts. One can see in the traces, that not only is accuracy improved with more layers but it also more quickly reaches the compression phase.

Using his sample complexity and accuracy statistics, the Professor has also shown that their are limits to the amount of accuracy to any deep learning CNN based on the function of layer counts, sample size and training event counts.


As far as I know, The Professor and his team are the first to try to characterize and understand what happens during deep learning. In doing so, he has shown that the number of layers and the number of samples can be used to predict the speed of learning. And ultimately how accurate any deep learning CNN can be.