Rank Collapse in Deep Learning

We can learn a lot about Why Deep Learning Works by studying the properties of the layer weight matrices of pre-trained neural networks.   And, hopefully, by doing this, we can get some insight into what a well trained DNN looks like–even without peaking at the training data.

One broad question we can ask is:

How is information concentrated in Deep Neural Network (DNNs)?  

To get a handle on this, we can run ‘experiments’ on the pre-trained DNNs available in pyTorch.

In a previous post, we formed the Singular Value Decomposition (SVD) of the weight matrices of the linear, or fully connected (FC) layers. And we saw that nearly all the FC Layers display Power Law behavior.  And, in fact, this behavior is Universal across models both ImageNet and NLP models.

But this only part of the story.  Here, we ask related question–do well trained DNNs weight matrices lose Rank ?

Matrix Rank:

Lets say \mathbf{W} is an N\times M  matrix.  We can form the Singular Value Decomposition (SVD):

\mathbf{W}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^T

\nu_{i}=\Sigma_{ii},\;\;i=1\cdots M

The Matrix Rank \mathcal{R}(\mathbf{W}) , or Hard Rank, is simply the number of non-zero singular values (\nu_{i}>0)

\mathcal{R}(\mathbf{W})=\sum\delta(\nu_{i}>0)

which express the decrease in Full Rank M.

\mathcal{R}(\mathbf{W})=M-\sum\delta(\nu_{i}=0)

Notice the Hard Rank of the rectangular matrix \mathbf{W} is the dimension of the square correlation matrix \mathbf{X}=\mathbf{W}^{T}\mathbf{W} .

In python, this can be computed using


rank = numpy.linalg.matrix_rank(W)

Of course, being a numerical method, we really mean the number of singular values above some tolerance (\nu_{i}>\text{tol})…and we can get different results depending on if we use

  • the default python tolerance
  • the numerical recipes tolerance, which is tighter

See the numpy documentation on matrix_rank for details.

Here, we will compute the rank ourselves, and use an extremely loose bound, and consider any \nu_{i}< 0.001.  As we shall see, DNNs are so good at concentrating information that it will not matter

Rank Collapse and Regularization

If all the singular values are non-zero, we say \mathbf{W} is Full Rank. If one or more \nu_{i}\sim 0, then we say \mathbf{W} is Singular.  It has lost expressiveness, and the model has undergone Rank collapse.

When a model undergoes Rank Collapse, it traditionally needs to be regularized. Say we are solving a simple linear system of equations / linear regression

\mathbf{W}\mathbf{x}=\mathbf{y}

The simple solution is to use a little linear algebra to get the optimal values for the unknown \mathbf{W}

\hat{\mathbf{x}}=(\mathbf{W}^{T}\mathbf{W})^{-1}(\mathbf{W}^{T}\mathbf{y})

But when \mathbf{W} is Singular, we can not form the matrix inverse.  To fix this, we simply add some small constant \gamma to diagonal of \mathbf{W}

\Sigma_{ii}\rightarrow\Sigma_{ii}+\gamma

\mathbf{W}^{T}\mathbf{W}\rightarrow\mathbf{W}^{T}\mathbf{W}+\gamma\gamma

So that all the singular values will now be greater than zero, and we can form a generalized pseudo-inverse, called the Moore-Penrose Inverse

\mathbf{W}^{\dagger}=(\mathbf{W}^{T}\mathbf{W}+\gamma\gamma)^{-1}

This procedure is also called Tikhonov Regularization.  The constant, or Regularizer, \gamma sets the Noise Scale for the model.    The information in \mathbf{W}  is concentrated in  the singular vectors associated with larger singular values \nu_{i}>\gamma  , and the noise is left over in the those associated with smaller singular values \nu_{i}^{2}<\gamma  :

  • Information:  vectors where  \nu_{i}>\gamma
  • Noise: vectors where \nu_{i}<\gamma

In cases where \mathbf{W}  is Singular, regularization is absolutely necessary.  But even when it is not singular, Regularization can be useful in traditional machine learning.  (Indeed, VC theory tells us that Regularization is a first class concept)

But we know that Understanding deep learning requires rethinking generalization. Which leads to the question ?

Do the weight matrices of well trained DNNs undergo Rank Collapse ?

Answer: They DO NOT — as we now see:

Analyzing Pre-Trained pyTorch Models

We can easily examine the numerous pre-trained models available in PyTorch.  We simply need to get the layer weight matrices and compute the SVD.  We then compute the minimum singular value \nu_{min} and compute a histogram of the minimums across different models.

for im, m in enumerate(model.modules()):
  if isinstance(m, torch.nn.Linear):
    W = np.array(m.weight.data.clone().cpu())
    M, N = np.min(W.shape), np.max(W.shape)
    _, svals, _ = np.linalg.svd(W)
    minsval=np.min(svals)
  ...

We do this here for numerous models trained on ImageNet and available in pyTorch, such as AlexNet, VGG16, VGG19, ResNet, DenseNet201,  etc.– as shown in this Jupyter Notebook.

We also examine the NLP models available in AllenNLP.   This is a little bit trickier; we have to install AllenNLP from source, then create an analyze.py command class, and rebuild AllenNLP. Then, to analyze, say, the AllenNLP pre-trained NER model, we run

allennlp analyze https://s3-us-west-2.amazonaws.com/allennlp/models/ner-model-2018.04.26.tar.gz

This print out the ranks (and other information, like power law fits), and then plot the results.  The code for all this is here.

Notice that many of the AllenNLP models include Attention matrices, which can be quite large and very rectangular (i.e. = \sim16,000\times 500 ), as compared to the smaller (and less rectangular) weight matrices used in the ImageNet models (i.e. \sim4,000\times 500 ),.

Note:  We restrict our analysis to rectangular layer weight matrices with an aspect ratio Q=N/M>1 , and really larger then 1.1.   This is because the Marchenko Pastur (MP) Random Matrix Theory (RMT) tells us that  \nu_{min}>0 only whenQ>1 .   We will review this in a future blog.

Minimum Singular Values of Pre-Trained Models

screen-shot-2018-09-21-at-12-10-27-am.png

For the ImageNet models, most fully connected (FC) weight matrices have a large minimum singular value  \nu_{min}\gg 0 . Only 6 of the 24 matrices looked at have \nu_{min}\sim 0 –and we have not carefully tested the numerical threshold–we are just eyeballing it here.

For the AllenNLP models, none of the FC matrices show any evidence of Rank Collapse.  All of the singular values for every linear weight matrix are non-zero.

It is conjectured that fully optimized DNNs–those with the best generalization accuracy–will not show Rank Collapse in any of their linear weight matrices.

If you are training your own model and you see Rank Collapse, you are probably over-regularizing.

Inducing Rank Collapse is easy–just over-regularize

it is, in fact, very easy to induce Rank Collapse.   We can do this in a Mini version of AlexNet, coded in Keras 2, and  available here.

Mini AlexNet
A Mini version of AlexNet, trained on CIFAR10, used to explore regularization and rank collapse in DNNs.

To induce rank collapse in our FC weight matrices, we can add large weight norm constraints to the FC1 linear layer, using the kernel_initializer=…

...
model.add(Dense(384, kernel_initializer='glorot_normal',
   bias_initializer=Constant(0.1),activation='relu', kernel_regularizer=l2(...))
...

We train this smaller MiniAlexnet model on CIFAR10 for 20 epochs, save the final weight matrix, and plot a histogram of the eigenvalues \rho(\lambda) of the weight correlation matrix

\mathbf{X}=\mathbf{W}^{T}\mathbf{W}

\mathbf{X}\mathbf{v}=\lambda\mathbf{v} .

We call \rho(\lambda) the Empirical Spectral Density (ESD).  Recall that the eigenvalues are simply the square of the singular values

\lambda=\nu^{2} .

ESD of FC1

 

Here is what happens to \rho(\lambda) when we turn up the amount of L2 Regularization from 0.0003 to 0.0005.  The \lambda_{min} decreases from 0.0414 to 0.008.

As we increase the weight norm constraints, the minimum eigenvalue approaches zero

\lambda_{min}\rightarrow 0

Note that adding too much regularization causes nearly all of the eigenvalues/singular values to collapse to zero–as well as the norm of the matrix.

We conjecture that DNNs have zero singular/eigenvalues because there is too much regularization on the layer.

And that…

Fully optimized Deep Neural Networks do not have Rank Collapse

Implications

We believe this is a unique property of DNNs, and related to how Regularization works in these models.  We will discuss this and more in an upcoming paper

Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning

by  Charles H. Martin (Calculation Consulting) and  Michael W. Mahoney (UC Berkeley).

 

And  presented at UC Berkeley this Monday at the Simons Institute

and see our long form paper

Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning

 

 

Please stay tuned and subscribe to this blog for more updates

Appendix

Rank Loss: numerical details

Here we are just looking at the distribution  \rho(\nu_{min}) to estimate the rank loss.  We could be more precise..

In the numpy.linalg.matrix_rank() funtion,  “By default, we identify singular values less than S.max() * max(M.shape) * eps as indicating rank deficiency” when using SVD

 

But there is some ambiguity here as well, since there is a different default from Numerical Recipes.  I will leave it up to the reader to select the best rank loss metric and explore further.  And I would be very interested in your findings.

40 models and 7500 matrices

We have computed the minimum singular value \nu_{min} for all the Conv2D layers in the ImageNet models deployed with pyTorch.  This covers nearly ~7500 layers across ~40 different models.

even-more-rank-loss.png

Very generously, we can say there is rank collapse with log_{10}(\nu_{min})<-8 .  Only 10%-13% of the layers show any form of rank collapse, using this simple heuristic, as easily seen on a log histogram.

Code is available here.

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