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Diagnostic Tool for Deep Neural Networks

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Weight Watcher

Current Version / Release: 0.5.1

WeightWatcher (WW): is an open-source, diagnostic tool for analyzing Deep Neural Networks (DNN), without needing access to training or even test data. It can be used to:

  • analyze pre/trained pyTorch, Keras, DNN models (Conv2D and Dense layers)
  • monitor models, and the model layers, to see if they are over-trained or over-parameterized
  • predict test accuracies across different models, with or without training data
  • detect potential problems when compressing or fine-tuning pretrained models
  • layer warning labels: over-trained; under-trained

ad well several new experimental model transformations, including:

  • SVDSmoothing: builds a model that can be used to predict test accuracies, but only with the training data.
  • SVDSharpness: removes Correlation Traps, which arise from sub-optimal regularization pre-trained models.

Experimental / Most Recent version 0.5.5

You may install the latest / Trunk from testpypi

python3 -m pip install --index-url --extra-index-url weightwatcher

The testpypi version usually has the most recent updates, including experimental methods qnd bug fixes

From Research to Production

WeightWatcher is based on theoretical research (done injoint with UC Berkeley) into Why Deep Learning Works, based on our Theory of Heavy Tailed Self-Regularization (HT-SR). It uses ideas from Random Matrix Theory (RMT), Statistical Mechanics, and Strongly Correlated Systems.

More details and demos can be found on the Calculated Content Blog

Reproducing Old Results

We strive to make all of our results 100% reproducible; this is not easy.

To reproduce some older results, such as the Nature paper (which is actually 2 years old), use the ww2x option

watcher.analyze(..., ww2x=True, ...)

If you are unable to reproduce the results, please file a bug and I will try to address it.


pip install  weightwatcher


import weightwatcher as ww
import torchvision.models as models

model = models.vgg19_bn(pretrained=True)
watcher = ww.WeightWatcher(model=model)
details = watcher.analyze()
summary = watcher.get_summary(details)

It is as easy to run and generates a pandas dataframe with details (and plots) for each layer

Sample Details Dataframe

and summary dict of generalization metrics

    {'log_norm': 2.11,
      'alpha': 3.06,
      'alpha_weighted': 2.78,
      'log_alpha_norm': 3.21,
      'log_spectral_norm': 0.89,
      'stable_rank': 20.90,
      'mp_softrank': 0.52}]

Tips for First Time Users (SEE BELOW)

Layer Details:

WW computes several Scale and Shape metrics for each layer Weight matrix W, as described in our papers (see below)

These are reported in a details dataframe, including:

Generalization Metrics

The goal of the WeightWatcher project is find generalization metrics that most accurately reflect observed test accuracies, across many different models and architectures, and both pre-trained and during training.

Our HTSR theory says that well trained, well correlated layers should be signficantly different from the MP random bulk, and, even more specifically, be heavy tailed. There are different layer metrics in weightwatcher for this, including:

  • rand_distance: the distance in distribution from the randomized layer
  • alpha: the slope of the tail of the ESD, on a log-log scale
  • alpha-hat: a scale-adjusted form of alpha (similar to the alpha-shatten-Norm)
  • stable-rank: a norm-adjusted measure of the scale of the ESD
  • num_spikes: the number of spikes outside the MP bulk region
  • etc

All of these attempt to measure how on-random and/or non-heavy-tailed the layer ESDs are.

Scale Metrics

  • log Frobenius norm:

  • log Spectral norm:

  • Stable Rank:

  • MP Soft Rank:

Shape Metrics

  • PL exponent alpha:

Scale-adjusted Shape Metrics

  • weighted alpha:
  • log alpha norm (Shatten norm):

Direct Correlation Metrics

The rand_distance metrics is a new, non-parameteric approach that appears to work well in early testing. See this recent blog post

  • rand_distance:

Misc Details

  • N, M: Matrix or Tensor Slice Dimensions
  • D: Quality of the (Truncated) Power law fit (D is the Kolmogorov Smirnov Distance metric)
  • num_spikes: number of spikes outside the bulk region of the ESD, when fit to an MP distribution

Summary Statistics:

The layer metrics are be averaged in the summary statistics:

Get the average metrics, as a summary (dict), from the given (or current) details dataframe

details = watcher.analyze(model=model)
summary = watcher.get_summary(model)

or just

summary = watcher.get_summary()

The summary statistics can be used to gauge the test error of a series of pre/trained models, without needing access to training or test data.

  • average alpha can be used to compare one or more DNN models with different hyperparemeter settings θ, but of the same depth.
  • average log spectral norm is useful to compare models of different depths L
  • average weighted alpha and log alpha norm are suitable for DNNs of differing hyperparemeters θ and depths L simultaneously.

Advanced Usage

The watcher object has several functions and analyze features described below

analyze( model=None, layers=[], min_evals=0, max_evals=None,
	 plot=True, randomize=True, mp_fit=True, ww2x=False, savefig=True):
describe(self, model=None, layers=[], min_evals=0, max_evals=None,
         plot=True, randomize=True, mp_fit=True, ww2x=False):
get_summary(details) or get_summary()
distances(model_1, model_2)

Ploting and Fitting the Empirical Spectral Density (ESD)

WW creates plots for each layer weight matrix to observe how well the power law fits work

details = watcher.analyze(plot=True)

For each layer, Weightwatcher plots the ESD--a histogram of the eigenvalues of the layer correlation matrix X=WTW. It then fits the tail of ESD to a (Truncated) Power Law, and plots these fits on different axes. The metrics (above) characterize the Shape and Scale of each ESD.


Detecting OverTraining

Weightwatcher can detect the signatures of overtraining in specific layers of a pre/trained Deep Neural Networks.

Early stopping

The weightwatcher alpha metric can be used to detect when to apply early stopping. When the average alpha (summary statistic) drops below 2.0, this indicates that the model may be overtrained and early stopping is necesary.

Below is an example of this, showing training loss and test lost curves for a small Transformer model, trained from scratch, along with the average alpha summary statistic.

Early Stopping

We can see that as the training and test losses decrease, so does alpha. But when the test loss saturates and then starts to increase, alpha drops below 2.0.

Correlation Traps

The randomize option compares the ESD of the layer weight matrix (W) to the ESD of the randomized W matrix. This is good way to visualize the correlations in the true ESD.

details = watcher.analyze(randomize=True, plot=True)

Fig (a) is well trained; Fig (b) may be over-trained. That orange spike on the far right is the tell-tale clue; it's caled a Correlation Trap.

A Correlation Trap is characterized by Fig (b); here the actual (green) and random (red) ESDs look almost identical, except for a small shelf of correlation (just right of 0). And for the random (red) ESD, the largest eigenvalue (orange) is far to the right of and seperated from the bulk of the ESD. Correlation Traps

Weightwatcher will analyze your model, layer-by-layer, and show you where these kind of problems may be lurking.

Predicting the Generalization Error

WeightWatcher (WW)can be used to compare the test error for a series of models, trained on the similar dataset, but with different hyperparameters, or even different but related architectures.

Our Theory of HT-SR predicts that models with smaller PL exponents alpha , on average, correspond to models that generalize better.

The WW summary metric alpha (α) can predict the generalization Δ error when varying the model hyperparmeters θ (like batch size, learning rate, momentum, etc)

  • PL exponent alpha:

whereas the summary metric weighed alpha can predict the generalization error Δ when varying hyperparmeters θ and depth L

  • weighted alpha:

Here is an example of the Weighted Alpha capacity metric for all the current pretrained VGG models.

alt text

This can be reppduced with the Demo Notebook

Notice: we did not peek at the ImageNet test data to build this plot.

See also the recent rand_distance metric.

SVDSmoothing and SVDSharpness Transforms

As descibed in our latest paper

Smoothed models can be used to predict test accuracies, by evaluating the training accuracy on the smoothed model.

smoothed_model = watcher.SVDSmoothing(model=...)

Sharpned models can be used when fine-tuning pre-trained models that have not been fully optimized yet.

sharpemed_model = watcher.SVDSharpness(model=...)

Sample notebooks are provided for each new feature

Additional Features

filter by layer types




filter by ids or name


minimum, maximum number of eigenvalues of the layer weight matrix

Sets the minimum and maximum size of the weight matrices analyzed. Setting max is useful for a quick debugging.

details = watcher.analyze(min_evals=50, max_evals=500)

fit ESDs to a Marchenko-Pastur (MP) distrbution

The mp_fit option tells WW to fit each layer ESD as a Random Matrix as a Marchenko-Pastur (MP) distribution, as described in our papers on HT-SR.

details = watcher.analyze(mp_fit=True, plot=True)

and reports the

num_spikes, mp_sigma, and mp_sofrank

Also works for randomized ESD and reports

rand_num_spikes, rand_mp_sigma, and rand_mp_sofrank

get the ESD for a specific layer, for visualization or further analysis

esd = watcher.get_ESD()

describe a model

Describe a model and report the details dataframe, without analyzing it

details = watcher.describe(model=model)

compare 2 models

The new distances method reports the distances between 2 models, such as the norm between the initial weight matrices and the final, trained weight matrices

details = watcher.distances(initial_model, trained_model)

compatability with version 0.2x

The new 0.4 version of weightwatcher treats each layer as a single, unified set of eigenvalues. In contrast, the 0.2x versions split the Conv2D layers into n slices, 1 for each receptive field. The ww2x option provides results which are back-compatable with the 0.2x version of weightwatcher, with details provide for each slice for each layer.

details = watcher.analyze(ww2x=True)

Save figures

Saves the layer ESD plots for each layer


generating 4 files per layer


Frameworks supported

  • Tensorflow 2.x / Keras
  • PyTorch
  • HuggingFace

Layers supported

  • Dense / Linear / Fully Connected (and Conv1D)
  • Conv2D

Known issues

  • rankloss is currently not working , may be always set to 0

  • the embedded powerlaw packages may show warning messages; you can ignore these

   /home/xander/anaconda3/envs/my_model/lib/python3.7/site-packages/ RuntimeWarning: divide by zero encountered in true_divide
  (Theoretical_CDF * (1 - Theoretical_CDF))

Tips for First Time Users

On using weightwatcher for the first time. I recommend selecting at least 1 trained model, and running weightwatcher with all analyze options on, including the plots, to see

  • if the layers ESDs are well formed and heavy tailed
  • if any layers are nearly random, indicating they are not well trained
  • if all the power law a fits look reasonable, and xmin is small enough that the fit captures a good part of the tail of the ESD

Moreover, the Power Laws fits, and the alpha fit, only work well when the ESDs are both heavy tailed, and( can be easily fit to a single power law. But sometimes the power law / alpha fits don't work. This happens when

  • the ESD is random, not heavy tailed. Here, alpha > 8 or larger.
  • the ESD is multimodal (rare, but does occur)
  • the ESD is heavy tailed, but not well described by a single power law. In these cases , sometimes alpha only fits the the very last part of the tail, and is too large. This is easily seen on the Lin-Lin plots

In any of these cases, I usually throw away alphas > 8 because they are spurious./. If you suspect your layers are undertrained, you have to look both at alpha and a plot of the ESD itself (to see if it is heavy tailed or just random-like)

Demo Notebooks

Basic Usage

Analyzing the VGG series

Using the ww2x option

How to Release

Publishing to the PyPI repository:

# 1. Check in the latest code with the correct revision number (__version__ in
vi weightwatcher/ # Increse release number, remove -dev to revision number
git commit
# 2. Check out latest version from the repo in a fresh directory
cd ~/temp/
git clone
cd WeightWatcher/
# 3. Use the latest version of the tools
python -m pip install --upgrade setuptools wheel twine
# 4. Create the package
python sdist bdist_wheel
# 5. Test the package
twine check dist/*
# 6. Upload the package to PyPI
twine upload dist/*
# 7. Tag/Release in github by creating a new release (


Apache License 2.0

Academic Presentations and Media Appearances

This tool is based on state-of-the-art research done in collaboration with UC Berkeley:

Latest papers and talks

and has been presented at Stanford, UC Berkeley, etc:

KDD2019 Workshop

KDD 2019 Workshop: Statistical Mechanics Methods for Discovering Knowledge from Production-Scale Neural Networks

KDD 2019 Workshop: Slides

Popular Popdcasts and Blogs

and has been the subject many popular podcasts

2021 Short Presentations

MLC Research Jam March 2021

PyTorch2021 Poster April 2021

Recent talk(s) by Mike Mahoney, UC Berekely

IARAI, the Institute for Advanced Research in Artificial Intelligence

Slack Channel

We have a slack channel for the tool if you need help For an invite, please send an email to


Charles H Martin, PhD Calculation Consulting

Serena Peng

Consulting Practice

Calculation Consulting homepage

Calculated Content Blog

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