1. calculation | consulting
weightwatcher - a diagnostic tool
for deep neural networks
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charles@calculationconsulting.com

2. calculation|consulting
weightwatcher - a diagnostic tool
for deep neural networks
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charles@calculationconsulting.com

3. calculation | consulting why deep learning works
Who Are We?
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Dr. Charles H. Martin, PhD
University of Chicago, Chemical Physics
NSF Fellow in Theoretical Chemistry, UIUC
Over 15 years experience in applied Machine Learning and AI
ML algos for: Aardvark, acquired by Google (2010)
Demand Media (eHow); first $1B IPO since Google
Wall Street: BlackRock
Fortune 500: Roche, France Telecom
BigTech: eBay, Aardvark (Google), GoDaddy
Private Equity: Griffin Advisors
Alt. Energy: Anthropocene Institute (Page Family)
www.calculationconsulting.com
charles@calculationconsulting.com
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Michael W. Mahoney
ICSI, RISELab, Dept. of Statistics UC Berkeley
Algorithmic and statistical aspects of modern large-scale data analysis.
large-scale machine learning | randomized linear algebra
geometric network analysis | scalable implicit regularization
PhD, Yale University, computational chemical physics
SAMSI National Advisory Committee
NRC Committee on the Analysis of Massive Data
Simons Institute Fall 2013 and 2018 program on the Foundations of Data
Biennial MMDS Workshops on Algorithms for Modern Massive Data Sets
NSF/TRIPODS-funded Foundations of Data Analysis Institute at UC Berkeley
https://www.stat.berkeley.edu/~mmahoney/
mmahoney@stat.berkeley.edu
Who Are We?

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calculation | consulting why deep learning works
Understanding deep learning requires rethinking generalization
Motivations: WeightWatcher Theory
The weightwatcher theory is a Semi-Empirical theory based on:
the Statistical Mechanics of Generalization,
Random MatrixTheory, and
the theory of Strongly Correlated Systems
Great nerdy stuff, however, I will be discussing the weightwatcher tool
and what it can do for you

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calculation | consulting why deep learning works
Open source tool: weightwatcher

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Open source tool: weightwatcher
pip install weightwatcher
…
import weightwatcher as ww
watcher = ww.WeightWatcher(model=model)
results = watcher.analyze()
watcher.get_summary()
watcher.print_results()
https://github.com/CalculatedContent/WeightWatcher

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WeightWatcher: A diagnostic tool
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Analyze pre/trained pyTorch, TF/Keras, and ONNX models
Inspect models that are difficult to train
Gauge improvements in model performance
Predict test accuracies across different models
Detect problems when compressing or fine-tuning pretrained models
pip install weightwatcher

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Research: Implicit Self-Regularization in Deep Learning
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• Implicit Self-Regularization in Deep Neural Networks: Evidence from
Random Matrix Theory and Implications for Learning.
• Heavy-Tailed Universality Predicts Trends in Test Accuracies for Very Large
Pre-Trained Deep Neural Networks
• Workshop: Statistical Mechanics Methods for Discovering Knowledge from
Production-Scale Neural Networks
• Predicting trends in the quality of state-of-the-art neural networks without
access to training or testing data
• (more submitted and in progress today)
(JMLR 2021)
(ICML 2019)
(KDD 2020)
(Nature Communications 2021)
Selected publications

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WeightWatcher: analyzes the ESD
(eigenvalues) of the layer weight matrices
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The tail of the ESD contains the information

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ESD of DNNs: detailed insight into W
Empirical Spectral Density (ESD: eigenvalues of X)
import keras
import numpy as np
import matplotlib.pyplot as plt
…
W = model.layers[i].get_weights()[0]
N,M = W.shape()
…
X = np.dot(W.T, W)/N
evals = np.linalg.eigvals(X)
plt.hist(evals, bin=100, density=True)

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WeightWatcher: analyzes the ESD
(eigenvalues) of the layer weight matrices
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details_df = watcher.analyze(model=your_model)
The tool provides various plots, quality metrics, and transforms

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WeightWatcher: analyzes the ESD
(eigenvalues) of the layer weight matrices
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watcher.analyze(…, plot=True, …)

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WeightWatcher: analyzes the ESD
(eigenvalues) of the layer weight matrices
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Well trained laters are heavy-tailed and well shaped
GPT-2 Fits a Power Law
(or Truncated Power Law)
alpha in [2, 6]
watcher.analyze(plot=True)
Good quality of fit (D is small)

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WeightWatcher: analyzes the ESD
(eigenvalues) of the layer weight matrices
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Better trained laters are more heavy-tailed and better shaped
GPT GPT-2

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Heavy Tailed Metrics: GPT vs GPT2
The original GPT is poorly trained on purpose; GPT2 is well trained
alpha for every layer
smaller alpha is better
large alpha are bad fits

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Power Law Universality: ImageNet
All ImageNet models display remarkable Heavy Tailed Universality
500 matrices
~50 architectures
Linear layers &
Conv2D feature maps
80-90% < 4

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Random Matrix Theory: detailed insight into WL
DNN training induces breakdown of Gaussian random structure
and the onset of a new kind of heavy tailed self-regularization
Gaussian
random
matrix
Bulk+
Spikes
Heavy
Tailed
Small, older NNs
Large, modern DNNs
and/or
Small batch sizes

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Random Matrix Theory: Marcenko Pastur
plus Tracy-Widom fluctuations
very crisp edges
Q
RMT says if W is a simple random Gaussian matrix,
then the ESD will have a very simple , known form
Shape depends on Q=N/M
(and variance ~ 1)
Eigenvalues tightly bounded
a few spikes may appear

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RMT: AlexNet
Marchenko-Pastur Bulk-decay | Heavy Tailed
FC1
zoomed in
FC2
zoomed in

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Random Matrix Theory: Heavy Tailed
But if W is heavy tailed, the ESD will also have heavy tails
(i.e. its all spikes, bulk vanishes)
If W is strongly correlated , then the ESD can be modeled as if W is drawn
from a heavy tailed distribution
Nearly all pre-trained DNNs display heavy tails…as shall soon see

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AlexNet,
VGG,
ResNet,
Inception,
DenseNet,
…
Heavy Tailed RMT: Scale Free ESD
All large, well trained, modern DNNs exhibit heavy tailed self-regularization
scale free

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HT-SR Theory: 5+1 Phases of Training
Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning
Charles H. Martin, Michael W. Mahoney; JMLR 22(165):1−73, 2021.

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Heavy Tailed RMT: Universality Classes
The familiar Wigner/MP Gaussian class is not the only Universality class in RMT

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WeightWatcher: predict trends in generalization
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Predict test accuracies across variations in hyper-parameters
The average Power Law exponent alpha
predicts generalization—at fixed depth
Smaller average-alpha is better
Better models are easier to treat

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WeightWatcher: Shape vs Scale metrics
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Purely norm-based (scale) metrics (from SLT) can be correlated with depth
but anti-correlated with hyper-parameter changes

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WeightWatcher: treat architecture changes
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Predict test accuracies across variations in hyper-parameters and depth
The alpha-hat metric combines
shape and scale metrics
and corrects
for different depths (grey line)
can be derived from theory…

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WeightWatcher: predict test accuracies
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alpha-hat works for 100s of different CV and NLP models
(Nature Communications 2021)
We do not have access to
The training or test data
But we can still predict
trends in the generalization

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Predicting test accuracies: Heavy-tailed shape metrics
The heavy tailed metrics perform best
Weighted Alpha Alpha (Shatten) Norm

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WeightWatcher: predict test accuracies
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ResNet, DenseNet, etc.
(Nature Communications 2021)

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Experiments: just apply to pre-trained Models
LeNet5 (1998)
AlexNet (2012)
InceptionV3 (2014)
ResNet (2015)
…
DenseNet201 (2018)
https://medium.com/@siddharthdas_32104/
cnns-architectures-lenet-alexnet-vgg-googlenet-resnet-and-more-666091488df5
Conv2D MaxPool Conv2D MaxPool FC FC

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Predicting test accuracies: 100 pretrained models
The heavy tailed metrics perform best
https://github.com/osmr/imgclsmob
From an open source sandbox of
nearly 500 pretrained CV models
(picked >=5 models / regression)

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Correlation Flow: CV Models
We can study correlation flow looking at vs. depth
VGG ResNet DenseNet

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WeightWatcher: detect overfitting
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Provide a data-independent criteria for early-stopping

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WeightWatcher: global and local convexity metrics
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Smaller alpha corresponds to more convex energy landscapes
Transformers (alpha ~ 3-4 or more)
alpha 2-3 (or less)
Rational Decisions, Random Matrices and Spin Glasses" (1998)
by Galluccio, Bouchaud, and Potters:

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WeightWatcher: global and local convexity metrics
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When the layer alpha < 2, we think this means the layer is overfit
We suspect that the early layers
of some Convolutional Nets
may be slightly overtrained
Some alpha < 2
This is predicted from our HTSR theory

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WeightWatcher: scale and shape anomalies
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We can detect problems in layers not detectable otherwise

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Detect potential signatures of over-fitting
WeightWatcher: Correlation Traps
watcher.analyze(plot=True, randomize=True)

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WeightWatcher: SVDSharpness transform
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Remove potential signatures of over-fitting
Like PAC-bounds Sharpness transform
Clips bad elements in W using RMT
clip
smoothed_model =
watcher.SVDSharpness(model=your_model)

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WeightWatcher: RMT-based shape metrics
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ww also includes predictive, non-parametric shape metrics
rand_distance =
jensen_shannon_distance(original_esd, random_esd)

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WeightWatcher: RMT-based shape metrics
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the layer rand_distance and alpha metrics are correlated

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WeightWatcher: interpreting shapes
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very high accuracy requires advanced methods
hard
easy
X
:)

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WeightWatcher: more Power Law shape metrics
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watcher.analyze(…, fit=‘TPL’)
Truncated Power Law fits
fit=‘E_TPL’)
weightwatcher provides several shape (and scale) metrics
plus several more unpublished experimental options

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WeightWatcher: E_TPL shape metric
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the E_TPL (and rand_distance)
shape metrics track the
learning curve epoch-by-epoch
Training MT transformers
from scratch to SOTA
Extended Truncated Power Law
highly accurate results leverage the advanced shape metrics
Here, (Lambda) is the shape metric

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WeightWatcher: why Power Law fits ?
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Spiking (i.e real) neurons exhibit power law behavior
weightwatcher supports several PL fits
from experimental neuroscience
plus totally new shape metrics
we have invented (and published)

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WeightWatcher: why Power Law fits ?
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Spiking (i.e real) neurons exhibit (truncated) power law behavior
The Critical Brain Hypothesis
Evidence of Self-Organized Criticality (SOC)
Per Bak (How Nature Works)
As neural systems become more complex
they exhibit power law behavior
and then truncated power law behavior
We see exactly this behavior in DNNs
and it is predictive of learning capacity

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WeightWatcher: open-source, open-science
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We are looking for early adopters and collaborators
github.com/CalculatedContent/WeightWatcher
We have a Slack channel to support the tool
Please file issues
Ping me to join

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Statistical Mechanics
derivation of the alpha-hat metric

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Classic Set Up: Student-Teacher model
Statistical Mechanics of Learning Engle &Van den Broeck (2001)
Generalization error ~ phase space volume
Average error ~ overlap between T and J

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Classic Set Up: Student-Teacher model
Statistical Mechanics of Learning Engle &Van den Broeck (2001)
Standard approach:
• Teacher (T) and Student (J) random Perceptron vectors
• Treat data as an external random Gaussian field
• Apply Hubbard–Stratonovich to get mean-field result
• Assume continuous or discrete J
• Solve for as a function of load (# data points / # parameters)

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New Set Up: Matrix-generalized Student-Teacher
x
Continuous perceptron
Ising Perceptron
Uninteresting
Replica theory, shows phase behavior,
entropy collapse, etc

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New Set Up: Matrix-generalized Student-Teacher
“Towards a new theory…” Martin, Milletari, & Mahoney (in preparation)
real DNN matrices:
NxM
Strongly correlated
Heavy tailed
correlation matrices
Solve for total integrated phase-space volume

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New approach: HCIZ Matrix Integrals
Write the overlap as
Fix a Teacher. The integral is now over all random students J that overlap w/T
Use the following result in RMT
“Asymptotics of HCZI integrals …” Tanaka (2008)

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RMT: Annealed vs Quenched averages
“A First Course in Random Matrix Theory” Potters and Bouchaud (2020)
good outside spin-glass phases
where system is trained well
We imagine averaging over all (random) students DNNs
with (correlations that) look like the teacher DNN

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New interpretation: HCIZ Matrix Integrals
Generating functional
R-Transform (inverse Green’s function, via Contour Integral)
in terms of Teacher's eigenvalues , and Student’s cumulants

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Some basic RMT: Greens functions
The Green’s function is the Stieltjes transform of eigenvalue distribution
Given the empirical spectral density (average eigenvalue density)
and using:

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Some basic RMT: Moment generating functions
The Green’s function has poles
at the actual eigenvalues
But is analytic in the complex plane
up and away from the real axis
z
Expand in a series around z = ∞
moment generating function

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Some basic RMT: R-Transforms
Which gives a (similar) moment generating function
The free-cumulant-generating function (R-transform)
is related to the Green function as
Gaussian random and very Heavy Tailed (Levy) random matrices
but which takes a simple form for both

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Results: Gaussian Random Weight Matrices
“Random Matrix Theory (book)” Bouchaud and Potters (2020)
Recover the Frobenius Norm (squared) as the metric

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Results: (very) Heavy Tailed Weight Matrices
“Heavy-tailed random matrices” Burda and Jukiewicz (2009)
Recover a Shatten Norm, in terms of the Heavy Tailed exponent

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Application to: Heavy Tailed Weight Matrices
Some reasonable approximations give the weighted alpha metric
Q.E.D.

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charles@calculationconsulting.com