A Mean Field Theory Learning Algorithm for Neural Networks
just a couple years after Hinton’s seminal 1985 paper , “A Learning Algorithm for Boltzmann Machines“.
What I really like is how we see the foundations of deep learning arose from statistical physics and theoretical chemistry. My top 10 favorite take-a-ways are:
Happy New Year everyone!
They are basically a solved problem, and while of academic interest, not really used in complex modeling problems. They were, upto 10 years ago, used for pretraining deep supervised nets. Today, we can train very deep, supervised nets directly.
RBMs are the foundation of unsupervised deep learning–
an unsolved problem.
RBMs appear to outperform Variational Auto Encoders (VAEs) on simple data sets like the Omniglot set–a data set developed for one shot learning, and used in deep learning research.
RBM research continues in areas like semi-supervised learning with deep hybrid architectures, Temperature dependence, infinitely deep RBMs, etc.
Many of basic concepts of Deep Learning are found in RBMs.
Sometimes clients ask, “how is Physical Chemistry related to Deep Learning ?”
In this post, I am going to discuss a recent advanced in RBM theory based on ideas from theoretical condensed matter physics and physical chemistry,
the Extended Mean Field Restricted Boltzmann Machine: EMF_RBM
(see: Training Restricted Boltzmann Machines via the Thouless-Anderson-Palmer Free Energy )
[Along the way, we will encounter several Nobel Laureates, including the physicists David J Thouless (2016) and Philip W. Anderson (1977), and the physical chemist Lars Onsager (1968).]
RBMs are pretty simple, and easily implemented from scratch. The original EMF_RBM is in Julia; I have ported EMF_RBM to python, in the style of the scikit-learn BernoulliRBM package.
https://github.com/charlesmartin14/emf-rbm/blob/master/EMF_RBM_Test.ipynb
We examined RBMs in the last post on Cheap Learning: Partition Functions and RBMs. I will build upon that here, within the context of statistical mechanics.
RBMs are defined by the Energy function
To train an RBM, we minimize the log likelihood ,
the sum of the clamped and (actual) Free Energies, where
and
The sums range over a space of ,
which is intractable in most cases.
Training an RBM requires computing the log Free Energy; this is hard.
When training an RBM, we
We don’t include label information, although a trained RBM can provide features for a down-stream classifier.
The Extended Mean Field (EMF) RBM is a straightforward application of known statistical mechanics theories.
There are, literally, thousands of papers on spin glasses.
The EMF RBM is a great example of how to operationalize spin glass theory.
The Restricted Boltzmann Machine has a very simple Energy function, which makes it very easy to factorize the partition function Z , explained by Hugo Larochelle, to obtain the conditional probabilities
The conditional probabilities let us apply Gibbs Sampling, which is simply
In statistical mechanics, this is called a mean field theory. This means that the Free Energy (in ) can be written as a simple linear average over the hidden units
.
where is the mean field of the hidden units.
At high Temp., for a spin glass, a mean field model seems very sensible because the spins (i.e. activations) become uncorrelated.
Theoreticians use mean field models like the p-spin spherical spin glass to study deep learning because of their simplicity. Computationally, we frequently need more.
How we can go beyond mean field theory ?
Onsager was awarded 1968 the Nobel Prize in Chemistry for the development of the Onsager Reciprocal Relations, sometimes called the ‘4th law of Thermodynamics’
The Onsager relations provides the theory to treat thermodynamic systems that are in a quasi-stationary, local equilibrium.
Onsager was the first to show how to relate the correlations in the fluctuations to the linear response. And by tying a sequence of quasi-stationary systems together, we can describe an irreversible process…
..like learning. And this is exactly what we need to train an RBM.
In an RBM, the fluctuations are variations in the hidden and visible nodes.
In a BernoulliRBM, the activations can be 0 or 1, so the fluctuation vectors are
The simplest correction to the mean field Free Energy, at each step in training, are the correlations in these fluctuations:
where W is the Energy weight matrix.
The corrections make sense under the stationarity constraints, that the Extended Mean Field RBM Free Energy () is at a critical point
,
That is, small changes in the activations do not change the Free Energy.
We will show that we can write
as a Taylor series in , the inverse Temperature, where is the Entropy
,
is the standard, mean field RBM Free energy
,
and is the Onsager correction
.
Given the expressions for the Free Energy, we must now evaluate it.
The Taylor series above is a result of the TAP theory — the Thouless-Anderson-Palmer approach developed for spin glasses.
The TAP theory is outlined in the Appendix; here it is noted that
Thouless just shared the 2016 Nobel Prize in Physics (for his work in topological phase transitions)
Being a series in inverse Temperature , the theory applies at low , or high Temperature. For fixed , this also corresponds to small weights W.
Specifically, the expansion applies at Temperatures above the glass transition–a concept which I describe in a recent video blog.
Here, to implement the EMF_RBM, we set
,
and, instead, apply weight decay to keep the weights W from exploding
where may be an L1 or L2 norm.
Weight Decay acts to keep the Temperature high.
Early RBM computational models were formulated using statistical mechanics (see the Appendix) language, and so included a Temperature parameter, and were solved using techniques like simulated annealing and the (mean field) TAP equations (described below).
Adding Temperature allowed the system to ‘jump’ out of the spurious local minima. So any usable model required a non-zero Temp, and/or some scheme to avoid local minima that generalized poorly. (See: Learning Deep Architectures for AI, by Bengio)
These older approaches did not work well –then — so Hinton proposed the Contrastive Divergence (CD) algorithm. Note that researchers struggled for some years to ‘explain’ what optimization problem CD actually solves.
More that recent work on Temperature Bases RBMs also suggests that higher T solutions perform better, and that
“temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layer.”
Standard RBM training approximates the (unconstrained) Free Energy, F=ln Z, in the mean field approximation, using (one or more steps of) Gibbs Sampling. This is usually implemented as Contrastive Divergence (CD), or Persistent Contrastive Divergence (PCD).
Using techniques of statistical mechanics, however, it is possible to train an RBM directly, without sampling, by solving a set of deterministic fixed point equations.
Indeed, this approach clarifies how to view an RBM as solving a (determinisitic) fixed point equation of the form
Consider each step, at at fixed (), as a Quasi-Stationary system, which is close to equilibrium, but we don’t need to evaluate ln Z(v,h) exactly.
We can use the stationary conditions to derive a pair of coupled, non-linear equations
They extend the standard formula of sigmoid linear activations with additional, non-linear, inter-layer interactions.
They differs significantly from (simple) Deep Learning activation functions because the activation for each layer explicitly includes information from other layers.
This extension couples the mean () and total fluctuations () between layers. Higher order correlations could also be included, even to infinite order, using techniques from field theory.
We can not satisfy both equations simultaneously, but we can satisfy each condition individually, letting us write a set of recursion relations
These fixed point equations converge to the stationary solution, leading to a local equilibrium. Like Gibbs Sampling, however, we only need a few iterations (say t=3 to 5). Unlike Sampling, however, the EMF RBM is deterministic.
https://github.com/charlesmartin14/emf-rbm/
If there is enough interest, I can do a pull request on sklearn to include it.
The next blog post will demonstrate how the python code in action.
Most older physicists will remember the Hopfield model. They peaked in 1986, although interesting work continued into the late 90s (when I was a post doc).
Originally, Boltzmann machines were introduced as a way to avoid spurious local minima while including ‘hidden’ features into Hopfield Associative Memories (HAM).
Hopfield himself was a theoretical chemist, and his simple model HAMs were of great interest to theoretical chemists and physicists.
Hinton explains Hopfield nets in his on-line lectures on Deep Learning.
The Hopfield Model is a kind of spin glass, which acts like a ‘memory’ that can recognize ‘stored patterns’. It was originally developed as a quasi-stationary solution of more complex, dynamical models of neuronal firing patterns (see the Cowan-Wilson model).
Early theoretical work on HAMs studied analytic approximations to ln Z to compute their capacity (), and their phase diagram. The capacity is simply the number of patterns a network of size N can memorize without getting confused.
The Hopfield model was traditionally run at T=0.
Looking at the T=0 line, at extremely low capacity, the system has stable mixed states that correspond to ‘frozen’ memories. But this is very low capacity, and generally unusable. Also, when the capacity too large, , (which is really not that large), the system abruptly breaks down completely.
There is a small window of capacity, ,with stable pattern equilibria, dominated by frozen out, spin glass states. The problem is, for any realistic system, with correlated data, the system is dominated by spurious local minima which look like low energy spin glass states.
So the Hopfield model suggested that
glass states can be useful minima, but
we want to avoid low energy (spurious) glassy states.
One can try to derive direct mapping between Hopfield Nets and RBMs (under reasonable assumptions). Then the RBM capacity is proportional to the number of Hidden nodes. After that, the analogies stop.
The intuition about RBMs is different since (effectively) they operate at a non-zero Temperature. Additionally, it is unclear to this blogger if the proper description of deep learning is a mean field spin glass, with many useful local minima, or a strongly correlated system, which may behave very differently, and more like a funneled energy landscape.
Thouless-Anderson-Palmer Theory of Spin Glasses
The TAP theory is one of the classic analytic tools used to study spin glasses and even the Hopfield Model.
We will derive the EMF RBM method following the Thouless-Anderson-Palmer (TAP) approach to spin glasses.
On a side note he TAP method introduces us to 2 more Nobel Laureates:
The TAP theory, published in 1977, presented a formal approach to study the thermodynamics of mean field spin glasses. In particular, the TAP theory provides an expression for the average spin
where is like an activation function, C is a constant, and the MeanField and Onsager terms are like our terms above.
In 1977, they argued that the TAP approach would hold for all Temperatures (and external fields), although it was only proven until 25 years later by Talagrand. It is these relatively new, rigorous approaches that are cited by Deep Learning researchers like LeCun , Chaudhari, etc. But many of the cited results have been suggested using the TAP approach. In particular, the structure of the Energy Landscape, has been understood looking at the stationary points of the TAP free energy.
More importantly, the TAP approach can be operationalized, as a new RBM solver.
We start with the RBM Free Energy .
Introduce an ‘external field’ which couples to the spins, adding a linear term to the Free Energy
Physically, would be an external magnetic field which drives the system out-of-equilibrium.
As is standard in statistical mechanics, we take the Legendre Transform, in terms of a set of conjugate variables . These are the magnetizations of each spin under the applied field, and describe how the spins behave outside of equilibrium
The transform which effectively defines a new interaction ensemble . We now set , and note
Define an interaction Free Energy to describe the interaction ensemble
which equals the original Free Energy when .
Note that because we have visible and hidden spins (or nodes), we will identify magnetizations for each
Now, recall we want to avoid the glassy phase; this means we keep the Temperature high. Or low.
We form a low Taylor series expansion in the new ensemble
which, at low order in , we expect to be reasonably accurate even away from equilibrium, at least at high Temp.
This leads to an order-by-order expansion for the Free Energy . The first order () correction is the mean field term. The second order term () is the Onsager correction.
Upto , we have
or
Rather than assume equilibrium, we assume that, at each step during inference –at fixed (–the system satisfies a quasi-stationary condition. Each step reaches a a local saddle point in phase space, s.t.
where, for each layer
Applying the stationary conditions lets us write coupled equations for the individual magnetizations that effectively define the (second order), high Temp, quasi-equilibrium states
Notice that the resemble the RBM conditional probabilities; in fact, at first order in , they are the same.
gives a mean field theory, and
And in the late 90’s, mean field TAP theory was attempted, unsuccessfully, to create an RBM solver.
At second order, the magnetizations are coupled through the Onsager corrections. To solve them, we can write down the fixed point equations, shown above.
We can include higher order correction the Free Energy by including more terms in the Taylor Series. This is called a Plefka expansion. The terms can be represented using diagrams
Plefka derived these terms in 1982, although it appears he only published up to the Onsager correction; a recent paper shows how to obtain all high order terms.
The Diagrammatic expansion appears not to have been fully worked out, and is only sketched above.
I can think of at least 3 ways to include these higher terms:
This is similar, in some sense, to the infinite RBM by Larochelle, which uses an resummation trick to include an infinite number of Hidden nodes.
, where
,
Obviously there are lots of interesting things to try.
The current python EMF_RBM only treats binary data, just like the scikit learn BernoulliRBM. So for say MNIST, we have to use the binarized MNIST.
There is some advantage to using Binarized Neural Networks on GPUs.
Still, a non-binary RBM may be useful. Tremel et. al. have suggested how to use real-valued data in the EMF_RBM, although in the context of Compressed Sensing Using Generalized Boltzmann Machines.
This is the question posed by a recent article. Deep Learning seems to require knowing the Partition Function–at least in old fashioned Restricted Boltzmann Machines (RBMs).
Here, I will discuss some aspects of this paper, in the context of RBMs.
We can use RBMs for unsupervised learning, as a clustering algorithm, for pretraining larger nets, and for generating sample data. Mostly, however, RBMs are an older, toy model useful for understanding unsupervised Deep Learning.
We define an RBM with an Energy Function
and it’s associated Partition Function
The joint probability is then
and the probability of the visible units is computed by marginalizing over the hidden units
Note we also mean the probability of observing the data X={v}, given the weights W.
The Likelihood is just the log of the probability
We can break this into 2 parts:
is just the standard Free Energy
We call the clamped Free Energy
because it is like a Free Energy, but with the visible units clamped to the data X.
The clamped is easy to evaluate in the RBM formalism, whereas is computationally intractable.
Knowing the is ‘like’ knowing the equilibrium distribution function, and methods like RBMs appear to approximate in some form or another.
Training an RBM proceeds iteratively by approximating the Free Energies at each step,
and then updating W with a gradient step
RBMs are usually trained via Contrastive Divergence (CD or PCD). The Energy function, being quadratic, lets us readily factor Z using a mean field approximation, leading to simple expressions for the conditional probabilities
and the weight update rule
RBM codes may use the terminology of positive and negative phases:
: The expectation is evaluated, or clamped, on the data.
: The expectation is to be evaluated on the prior distribution . We also say is evaluated in the limit of infinite sampling, at the so-called equilibrium distribution. But we don’t take the infinite limit.
CD approximates –effectively evaluating the (mean field) Free Energy — by running only 1 (or more) steps of Gibbs Sampling.
So we may see
or, more generally, and in some code bases, something effectively like
Initialize the positive and negative
Run N iterations of:
Run 1 Step of Gibbs Sampling to get the negative :
sample the hiddens given the (current) visibles:
sample the visibles given the hiddens (above):
Calculate the weight gradient:
Apply Weight decay or other regularization (optional):
Apply a momentum (optional):
Update the Weights:
What is Cheap about learning ? A technical proof in the Appendix notes that
knowing the Partition function is not the same as knowing the underlying distribution .
This is because the Energy can be rescaled, or renormalized, in many different ways, without changing .
This is a also key idea in Statistical Mechanics.
The Partition function is a generating function; we can write all the macroscopic, observable thermodynamic quantities as partial derivatives of . And we can do this without knowing the exact distribution functions or energies–just their renormalized forms.
Of course, our W update rule is a derivative of
The proof is technically straightforward, albeit a bit odd at first.
Let’s start with the visible units . Write
We now introduce the hidden units, , into the model, so that we have a new, joint probability distribution
and a new, Renormalized , partition function
Where RG means Renormalization Group. We have already discussed that the general RBM approach resembles the Kadanoff Variational Renormalization Group (VRG) method, circa 1975. This new paper points out a small but important technical oversight made in the ML literature, namely that
having does not imply
That is, just because we can estimate the Partition function well does not mean we know the probability distributions.
Why? Define an arbitrary non-constant function and write
.
K is for Kadanoff RG Transform, and ln K is the normalization.
We can now write an joint Energy with the same Partition function as our RBM , but with completely different joint probability distributions. Let
Notice what we are actually doing. We use the K matrix to define the RBM joint Energy function. In RBM theory, we restrict to a quadratic form, and use variational procedure to learn the weights , thereby learning K.
In a VRG approach, we have the additional constraint that we restrict the form of K to satisfy constraints on it’s partition function, or, really, how the Energy function is normalized. Hence the name ‘Renormalization.‘ This is similar, in spirit, but not necessarily in form, to how the RBM training regularizes the weights (above).
Write the total, or renormalized, Z as
Expanding the Energy function explicitly, we have
where the Kadanoff normalization factor appears now the denominator.
We can can break the double sum into sums over v and h
Identify in the numerator
which factors out, giving a very simple expression in h
In the technical proof in the paper, the idea is that since h is just a dummy variable, we can replace h with v. We have to be careful here since this seems to only applies to the case where we have the same number of hidden and visible units–a rare case. In an earlier post on VRG, I explain more clearly how to construct an RG transform for RBMs. Still, the paper is presenting a counterargument for arguments sake, so, following the argument in the paper, let’s say
This is like saying we constrain the Free Energy at each layer to be the same.
This is also another kind of Layer Normalization–a very popular method for modern Deep Learning methods these days.
So, by construction, the renormalized and data Partition functions are identical
The goal of Renormalization Group theory is to redefine the Energy function on a difference scale, while retaining the macroscopic observables.
But , and apparently this has been misstated in some ML papers and books, the marginalized probabilities can be different !
To get the marginals, let’s integrate out only the h variables
Looking above, we can write this in terms of K and its normalization
which implies
RBMs let us represent data using a smaller set of hidden features. This is, effectively, Variational Renormalization Group algorithm, in which we approximate the Partition function, at each step in the RBM learning procedure, without having to learn the underlying joining probability distribution. And this is easier. Cheaper.
In other words, Deep Learning is not Statistics. It is more like Statistical Mechanics.
And the hope is that we can learn from this old scientific field — which is foundational to chemistry and physics — to improve our deep learning models.
Shortly after this paper came out, Comment on “Why does deep and cheap learning work so well?”that the proof in the Appended is indeed wrong–as I suspected and pointed out above.
It is noted that the point of the RG theory is to preserve the Free Energy form one layer to another, and, in VRG, this is expressed as a trace condition on the Transfer operator
where
It is, however, technically possible to preserve the Free Energy and not preserve the trace condition. Indeed, because is not-constant, thereby violating the trace condition.
From this bloggers perspective, the idea of preserving Free Energy, via either a trace condition, or, say, by layer normalization, is the import point. And this may mean to only approximately satisfy the trace condition.
In Quantum Chemistry, there is a similar requirement, referred to as a Size-Consistency and/ or Size-Extensivity condition. And these requirements proven essential to obtaining highly accurate, ab initio solutions of the molecular electronic Schrodinger equation–whether implemented exactly or approximately.
And, I suspect, a similar argument, at least in spirit if not in proof, is at play in Deep Learning.
Please chime in our my YouTube Community Channel
see also: https://m.reddit.com/r/MachineLearning/comments/4zbr2k/what_is_your_opinion_why_is_the_concept_of/)
Stay tuned for a video link, and a blog post to accompany this.
Comments, questions, and bug fixes are welcome.
The motivation: to understand how to build very deep networks and why they do (or don’t) work.
There are several papers that caught my eye, starting with
Unifying Distillation and Privileged Information (2015)
These papers set the foundation for looking at much larger, deeper networks such as
FractalNet’s are particularly interesting since they suggest that very deep networks do not need student-teacher learning, and, instead, can be self similar. (which is related to very recent work on the Statistical Physics of Deep Learning, and the Renormalization Group analogy).
IMHO, it is not enough just to implement the code; the results have to be excellent as well. I am not impressed with the results I have seen so far, and I would like to flush out what is really going on.
The 2010 paper still appears to be 1 of the top 10 results on MNIST:
http://rodrigob.github.io/are_we_there_yet/build/classification_datasets_results.html
The idea is simple. They claim to get state-of-the-art accuracy on MNIST using a 5-layer MLP, but running a large number of epochs with just SGD, a decaying learning rate, and an augmented data set.
The key idea is that the augmented data set can provide, in practice, an infinite amount of training data. And having infinite data means that we never have to worry about overtraining because we have too many adjustable parameters, and therefore any reasonable size network will do the trick if we just run it long enough.
In other words, there is no convolution gap, no need for early stopping, or really no regularization at all.
This sounds dubious to me, but I wanted to see for myself. Also, perhaps I am missing some subtle detail. Did they clip gradients somewhere ? Is the activation function central ? Do we need to tune the learning rate decay ?
I have initial notebooks on github, and would welcome feedback and contributions, plus ideas for other papers to reproduce.
I am trying to repeat this experiment using Tensorflow and 2 kinds of augmented data sets:
(and let me say a special personal thanks to Søren Hauberg for providing this recent data set)
I would like to try other methods, such as the Keras Data Augmentation library (see below), or even the recent data generation library coming out of OpenAI.
Current results are up for
The initial results indicate that AlignMNIST is much better that InfiMNIST for this simple MLP, although I still do not see the extremely high, top-10 accuracy reported.
Furthermore, the 5-Layer InfiMNIST actually diverges after ~100 epochs. So we still need early stopping, even with an infinite amount of data.
It may be interesting try using the Keras ImageDataGenerator class, described in this related blog on “building powerful image classification models using very little data”
Also note that the OpenAI group as released a new paper and code for creating data used in generative adversarial networks (GANs).
I will periodically update this blog as new data comes in, and I have the time to implement these newer techniques.
Next, we will check in the log files and discuss the tensorboard results.
Comments, criticisms, and contributions are very welcome.
Please ask questions in the comments section.
I will try to answer them in the blog over the weekend of the next week or so.
Thanks in advance.
I have a client who is using Naive Bayes pretty successfully, and the subject has come up as to ‘why do we need fancy machine learning methods?’ A typical question I am asked is:
Why do we need Regularization ?
What happens if we just turn the regularizer off ?
Can we just set it to a very small, default value ?
After all, we can just measure what customers are doing exactly. Why not just reshow the most popular items, ads, etc to them. Can we just run a Regression? Or Naive Bayes ? Or just use the raw empirical estimates?
Do we Need Hundreds of Classifiers to Solve Real World Classification Problems?
Sure, if you are simply estimating historical frequencies, n-gram statistics, etc, then a simple method like Naive Bayes is fine. But to predict the future..
the Reason we need Regularization is for Generalization.
After all, we don’t just want to make the same recommendations over and over.
Unless you are just in love with ad re-targeting (sic). We want to predict on things we have never seen.
And we need to correct for presentation bias when collecting data on things we have seen.
Most importantly, we need methods that don’t break down unexpectedly.
In this post, we will see why Regularization is subtle and non-trivial even in seemingly simple linear models like the classic Tikhonov Regularization. And something that is almost never discussed in modern classes.
We demonstrate that ‘strong’ overtraining is accompanied by a Phase Transition–and optimal solutions lies just below this.
The example is actually motivated by something I saw recently in my consulting practice–and it was not obvious at first.
Still, please realize–we are about to discuss a pathological case hidden in the dark corners of Tikhonov Regularization–a curiosity for mathematicians.
See this Quora post on SVD in NLP for some additional motivations.
Let’s begin when Vapnik [1], Smale [2], and the other great minds of the field begin:
(Regularized) Linear Regression:
The most basic method in all statistics is linear regression
.
It is attributed to Gauss, one of the greatest mathematicians ever.
One solution is to assume Gaussian noise and minimize the error.
The solution can also be constructed using the Moore-Penrose PseudoInverse.
If we multiply by to the left on both sides, we obtain
The formal solution is
which is only valid when we can actually invert the data covariance matrix .
We say the problem is well-posed when
Otherwise we say it is singular, or ill-posed [1].
In nearly all practical methods, we would not compute the inverse directly, but use some iterative technique to solve for . In nearly all practical cases, even when the matrix seems invertible, or even well posed, it may have numerical instabilities, either due to the data or the numerical solver. Vapnik refers to these as stochastically ill posed problems [4]
Frequently X^{T}X can not be inverted..or should not be inverted..directly.
A trivial solution is to simply ‘regularize’ the matrix by adding a small, non-zero value to the diagonal:
.
Naively, this seems like it would dampen instabilities and allow for a robust numerical solution. And it does…in most cases.
If you want to sound like you are doing some fancy math, give it a Russian name; we call this Tikhonov Regularization [4]:
This is (also) called Ridge Regression in many common packages such as scikit learn.
The parameter is the regularization parameter values, usually found with Cross Validation (CV). While a data science best practice, CV can fail !
Grid searching , we can obtain the infamous L-curve:
The L-curve is a log-log plot of the the norm of the regularized solution vs. the norm of the residual. The best lies somewhere along the bottom of the L, but we can’t really tell where (even with cross validation).
This challenge of regularization is absent from basic machine learning classes?
The optimal is actually hard to find. Cross Validation (CV) only works in ideal cases. And we need a CV metric, like , which also only works in ideal cases.
We might naively imagine can just be very small–in effect, turning off the regularizer. This is seen in the curve above, where the solution norm diverges.
The regularization fails in these 2 notable cases, when
Today we will examine this curious, spurious behavior in case 2 (and look briefly at 1)
As von Neumann said once, “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk”
To understand where the regularization can fail, we need to distinguish between the cases in which Ridge Regression is commonly used.
Say we have features, and instances. We distinguish between the High Bias and High Variance regimes.
This regime is overdetermined, complicated models that are subject to overtraining. We find multiple solutions which satisfy our constraints.
Most big data machine learning problems lie here. We might have 100M documents and 100K words. Or 1M images and simply all pixels as the (base) features.
Regularization lets us pick the best of solution which is the most smooth (L2 norm), or the most sparse (L1 norm), or maybe even the one with the least presentation bias (i.e. using Variance regularization to implement Counterfactural Risk Minimization)
This is the Underdetermined regime, and any solution we find, regularized or not, will most likely generalize poorly.
When we have more features than instances, there is no solution at all (let alone a unique one)
Still, we can pick a regularizer, and the effect is similar to dimensional reduction. Tikhonov Regularization is similar truncated SVD (explained below)
Any solution we find may work, but the predictions will be strongly biased towards the training data.
But it is not only just sampling variability that can lead to poor predictions.
In between these 2 limits is an seemingly harmless case, however, this is really a
This is a rare case where the number of features = the number of instances.
This does come up in practical problems, such as classifying a large number of small text phrases. (Something I have been working on with one of my larger clients)
The number of phrases ~ the number of unique words.
This is a dangerous case … not only because it seems so simple … but because the general theory breaks down. Fortunately, it is only pathologicial in
We examine how these different regimes behave for small values of
Let’s formalize and consider the how the predicted accuracy behaves.
We analyze our regression under Gaussian noise . Let
where is a Gaussian random variable with unit mean and variance
This simple model lets us analyze predicted Accuracy analytically.
Let
be the optimal Estimate, and
be the Ground Truth, and
be expected (mean).
We would like define, and the decompose, the Generalization Accuracy into Bias and Variance contributions.
Second, to derive the Generalization Error, we need to work out how the estimator behaves as a random variable. Frequently, in Machine Learning research, one examines the worst case scenario. Alternatively, we use the average case.
Define the Estimation Error
Notice that by Generalization Error, we usually we want to know how our model performs on a hold set or test point :
In the Appendix, we work out the generalization bounds for the worst case and the average case. From here on, however, when we refer to Generalization Error, we mean the average case Estimation Error (above).
For Ridge Regression, the mean estimator is
and its variation is
We can now decompose the Estimation Error into 2 terms:
, where
is the Bias, and
is the Variance,
and examine each regime in the limit
The Bias is just the error of the average estimator:
and the Variance is the trace of the Covariance of the estimator
We can examine the limiting behavior of these statistics by looking at the leading terms in a series expansion for each. Consider the Singular Value Decomposition (SVD) of
.
Let be the positive singular values, and be the right singular vectors.
We can write the Bias as
With no regularization, and the number of features exceeds the number of instances, there is a strong bias, determined by the ‘extra’ singular values
Otherwise, the Bias vanishes.
So the Bias appears to only matter in the underdetermined case.
(Actually, this is misleading; in the overdetermined case, we can introduce a bias when tuning the regularization parameter too high)
In contrast, the Variance
can diverge if the minimal singular value is small:
That is, if the singular values decrease too fast, the variance can explode.
And this can happen in the danger zone
When the variance can be infinite!
Which also means the Central Limit Theorem (CLT) breaks down…at least how we normally use it.
Traditionally, the (classical) CLT says that the infinite sum of i.i.d. random variables converges to a Gaussian distribution–when the variance of the is finite. We can, however, generalize the CLT to show that, when the variance is infinite, the sum converges to a Levy or -stable, power-law distribution.
These power-law distributions are quite interesting–and arise frequently in chemistry and physics during a phase transition. But first, let’s see the divergent behavior actually arise.
Ryota Tomioka [3] has worked some excellent examples using standard data sets
[I will work up some ipython notebook examples soon]
The Spambase dataset is a great example. It has features, and instances. We run a Ridge Regression, with , with .
As the data set size increases, the accuracy sharps to zero at , and then increases sharply again, saturating at $latex N_{f}\sim 10^{3} $.
Many other datasets show this anomalous behavior at , as predicted by our analysis above.
Where Ridge Regression fails, Logistic Regression bounces back. Below we show some examples of RR vs LR in the danger zone
Logistic Regression still has problems, but it no where near as pathological as RR.
The reason for this is subtle.
Traditional theory (very losely) bounds for quantities like the generalization error and (therefore) the variance.
We work out some of examples in the Appendix.
Clearly these bounds don’t work in all cases. So what do they really tell us? What’s the point?
These theories gives us some confidence that we can actually apply a learning algorithm–but only when the regularizer is not too small and the noise is not too large.
They essentially try to get us far away from the pathological phase transition that arises when the variance diverges.
In practice, we would never just cross our fingers set .
We pick a hold out set and run Generalized Cross Validation (GCV) . Yet this can fail when
Hansen [5, 7] has discussed these issue in detail…and he provides an alternative method, the L-curve, which attempt to balance the size of the regularized solution and the residuals.
For example, we can sketch the relative errors for the L-curve and GVC method (see 7). Above we see that the L-curve relative errors are more Gaussian than GCV.
In other words, while most packages provide a small choice of regression metrics; as data scientists, the defaults like may not be good enough. Even using the 2-norm may not represent the errors well. As data scientists, we may need to develop our own metrics to get a good for a regression. (or maybe just use an SVM regression).
Hansen claims that,”experiments confirm that whenever GCV finds a good regularization parameter, the corresponding solution is located at the corner of the L-curve.” [7]
This means that the optimal regularized solution lives ‘just below’ the phase transition, where the norm diverges.
What do we mean by a phase transition?
The simplest thing is to we plot a graph and look a steep cliff or sharp change.
Here, generalization accuracies drops off suddenly as
In a physical phase transition,the fluctuations (i.e. variances and norms) in the system approach infinity.
Imagine watching a pot boil
We see bubbles of all sizes, small and large. The variation in bubble size is all over the map. This is characteristic of a phase transition: i.e. water to gas.
When we cook, however, we frequently simmer our foods–we keep the water just below the boiling point. This is as hot as the water can get before it changes to gas.
Likewise, it appears that in Ridge Regression, we frequently operate at a point just below the phase transition–where the solution norm explodes. And this is quite interesting. And, I suspect, may be important generally.
In chemical physics, we need special techniques to treat this regime, such as the Renormalization Group. Amazingly, Deep Learning / Restricted Boltzmann Machines look very similar to the Variational Renormalization Group method.
In our next post, we will examine this idea further, by looking at the phase diagram of the traditional Hopfield Neural Networ, and the idea of Replica Symmetry Breaking in the Statistical Mechanics of Generalization. Stay tuned !
1. Vapnik and Izmailov, V -Matrix Method of Solving Statistical Inference Problems, JMLR 2015
2. Smale, On the Mathematical Foundations of Learning, 2002
3. https://github.com/ryotat
4. http://ttic.uchicago.edu/~gregory/courses/LargeScaleLearning/lectures/proj_learn1.pdf
5. The L-curve and its use in the numerical treatment of inverse problems
6. Discrete Ill-Posed and Rank Deficient Problems
7. The use of the L-curve in the regularization of discrete ill-posed problems, 1993
,
when ,
then
Let
be the data covariance matrix. And , for convenience, let
.
The Expected value of the optimal estimator, assuming Gaussian noise, is given using the Penrose Pseudo-Inverse
The Covariance is
The regularized Covariance matrix arises so frequently that we will assign it a symbol
We can now write down the mean and covariance
[more work needed here]
Consider the Generalization Error for a test point
We can now obtain some very simple bounds on the Generalization Accuracy (just like a bona-a-fide ML researcher)
,
where we assume the ‘covariance*’ is
We can use the SVD approach to make stronger statements about our ability to solve the inversion problem
.
I will sketch an idea called the “Discrete Picard Condition” and provide some intuition
Write as a sum over singular vectors
.
Introduce an abstract PseudoInverse, defined by
Express the formal solution as
Imagine the SVD of the PseudoInverse is
.
and use it to obtain
.
For this expression to be meaningful, the SVD coefficients must decay, on average, faster than the corresponding singular values .
Consequently, the only coefficients that carry any information are larger than the noise level. The small coefficients have small singular values and are dominated by noise. As the problem becomes more difficult to solve uniquely, the singular values decay more rapidly.
If we discard the singular components, we obtain a regularized, unsupervised solution called
.
which is similar in spirit to spectral clustering.
Recently it has been shown that [1]
Unsupervised Deep Learning implements the Kadanoff Real Space Variational Renormalization Group (1975)
This means the success of Deep Learning is intimately related to some very deep and subtle ideas from Theoretical Physics. In this post we examine this.
An AutoEncoder is a Unsupervised Deep Learning algorithm that learns how to represent an complex image or other data structure . There are several kinds of AutoEncoders; we care about so-called Neural Encoders–those using Deep Learning techniques to reconstruct the data:
The simplest Neural Encoder is a Restricted Boltzman Machine (RBM). An RBM is non-linear, recursive, lossy function that maps the data from visible nodes into hidden nodes :
The RBM is learned by selecting the optimal parameters that minimize some measure of the reconstruction error (see Training RBMs, below)
RBMs and other Deep Learning algos are formulated using classical Statistical Mechanics. And that is where it gets interesting!
In machine learning (ML), we map (visible) data into (hidden) features
The hidden units discover features at a coarser grain level of scale
With RBMs, when features are complex, we may stack them into a Deep Belief Network (DBM), so that we can learn at different levels of scale
and leads to multi-scale features in each layer
Deep Belief Networks are a Theory of Unsupervised MultiScale Feature Learning
We call a flow map
eIf we apply the flow map to the data repeatedly, (we hope) it converges to a fixed point
Notice that we usually expect to apply the same map each time , however, for a computational theory we may need more flexibility.
The simplest example of a flow map is the simple linear map
so that
where C is a non-negative, low rank matrix
We have seen this before: this leads to a Convex form of NonNegative Matrix Factorization NMF
Convex NMF applies when we can specify the feature space and where the data naturally clusters. Here, there are a few instances that are archetypes that define the convex hull of the data.
Amazingly, many clustering problems are provably convex–but that’s a story for another post.
Near a fixed point, we commonly approximate the flow map by a linear operator
This lets us capture the structure of the true data manifold, and is usually described by the low lying eigen-spectra of
.
In the same spirit, Semi & Unsupervised Manifold Learning, we model the data using a Laplacian operator , usually parameterized by a single scale parameter .
These methods include Spectral Clustering, Manifold Regularization , Laplacian SVM, etc. Note that manifold learning methods, like the Manifold Tanget Classifier, employ Contractive Auto Encoders and use several scale parameters to capture the local structure of the data manifold.
In chemistry and physics, we frequently encounter problems that require a multi-scale description. We need this for critical points and phase transitions, for natural crashes like earthquakes and avalanches, for polymers and other macromolecules, for strongly correlated electronic systems, for quantum field theory, and, now, for Deep Learning.
A unifying idea across these systems is the Renormalization Group (RG) Theory.
Renormalization Group Theory is both a conceptual framework on how to think about physics on multiple scales as well as a technical & computational problem solving tool.
Ken Wilson won the 1982 Nobel Prize in Physics for the development and application of his Momentum Space RG theory to phase transitions.
We used RG theory to model the recent BitCoin crash as a phase transition.
Wilson invented modern multi-scale modeling; the so-called Wilson Basis was an early form of Wavelets. Wilson was also a big advocate of using supercomputers for solving problems. Being a Nobel Laureate, he had great success promoting scientific computing. It was thanks to him I had access to a Cray Y-MP when I was in high school because he was a professor at my undergrad, The Ohio State University.
Here is the idea. Consider a feature map which transforms the data to a different, more coarse grain scale
The RG theory requires that the Free Energy is rescaled, to reflect that
the Free Energy is both Size-Extensive and Scale Invariant near a Critical Point
This is not obvious — but it is essential to both having a conceptual understanding of complex, multi scale phenomena, and it is necessary to obtain very highly accurate numerical calculations. In fact, being size extensive and/or size consistent is absolutely necessary for highly accurate quantum chemistry calculations of strongly correlated systems. So it is pretty amazing but perhaps not surprising that this is necessary for large scale deep learning calculations also!
If we (can) apply the same map, , repeatedly, we obtain a RG recursion relation, which is the starting point for most analytic work in theoretical physics. It is usually difficult to obtain an exact solution to the RGE (although it is illuminating when possible [20]).
Many RG formulations both approximate the exact RGE and/or only include relevant variables. To describe a multiscale system, it is essential to distinguish between these relevant and irrelevant variables.
Let’s say the feature map is a simple linear rescaling
We can obtain a very elegant, approximate RG solution where F(x) obeys a complex (or log-periodic) power law.
This behavior is thought to characterize Per-Bak style Self-Organized Criticality (SOC), which appears in many natural systems–and perhaps even in the brain itself. Which leads to the argument that perhaps Deep Learning and Real Learning work so well because they operate like a system just near a phase transition–also known as the Sand Pile Model--operating at a state between order and chaos.
Leo Kadanoff, now at the University of Chicago, invented some of the early ideas in Renormalization Group. He is most famous for the Real Space formulation of RG, sometimes called the Block Spin approach. He also developed an alternative approach, called the Variational Renormalization Group (VRG, 1975), which is, remarkably, what Unsupervised RBMs are implementing!
Let’s consider a traditional Neural Network–a Hopfield Associative Memory (HAM). This is also known as an Ising model or a Spin Glass in statistical physics.
An HAM consists of only visible units; it stores memories explicitly and directly in them:
We specify the Energy — called the Hamiltonian — for the nodes. Note that all the nodes are visible. We write
The Hopfield model has only single and pair-wise interactions.
A general Hamiltonian might have many-body, multi-scale interactions:
The Partition Function is given as
And the Free Energy is
The idea was to mimic how our neurons were thought to store memories–although perhaps our neurons do not even do this.
Either way, Hopfield Neural Networks have many problems; most notably they may learn spurious patterns that never appeared in the training set. So they are pretty bad memories.
Hinton created the modern RBM to overcome the problems of the Hopfield model. He used hidden units to represent the features in the data–not to memorize the data examples directly.
An RBM is specified Energy function for both the visible and hidden units
This also defines joint probability of simultaenously observing a configuration of hidden and visible spins
which is learned variationally, by minimizing the reconstruction error…or the cross entropy (KL divergence), plus some regularization (Dropout), using Greedy layer-wise unsupervised training, with the Contrastive Divergence (CD or PCD) algo, …
The specific details of an RBM Energy are not addressed by these general concepts; these details do not affect these arguments–although clearly they matter in practice !
It turns out that
Introducing Hidden Units in a Neural Network is a Scale Renormalization.
When changing scale, we obtain an Effective Hamiltonian that acts on a the new feature space (i.e the hidden units)
or, in operator form
This Effective Hamiltonian is not specified explicitly, but we know it can take the general form (of a spin funnel, actually)
The RG transform preservers the Free Energy (when properly rescaled):
where
The RG theory provides a way to iteratively update, or renormalize, the system Hamiltonian. Each time we add a layer of hidden units (h1, h2, …), we have
We imagine that the flow map is attracted to a Critical Trajectory which naturally leads the algorithm to the fixed point. At each step, when we apply another RG transform, we obtain a new, Renormalized Manifold, each one closer to the optimal data manifold.
Conceptually, the RG flow map is most useful when applied to critical phenomena–physical systems and/or simple models that undergo a phase transition. And, as importantly, the small changes in the data should ‘wash away’ as noise and not affect the macroscopic / critical phenomena. Many systems–but not all–display this.
Where Hopfield Nets fail to be useful here, RBMs and Deep Learning systems shine.
We now show that these RG transformations are achieved by stacking RBMs and solving the RBM inference problem!
Kadanoff’s Variational Renormalization Group
As in many physics problems, we break the modeling problem into two parts: one we know how to solve, and one we need to guess.
The joint Hamiltonian, or Energy function, is then given by
The Correlation V(v,h) is defined so that the partition function is not changed
This gives us
(Sometimes the Correlation V is called a Transfer Operator T, where V(v,h)=-T(v,h) )
We may now define a renormalized effective Hamilonian that acts only on the hidden nodes
so that we may write
Because the partition function does not change, the Exact RGE preserves the Free Energy (up to a scale change, we we subsume into
We generally can not solve the exact RGE–but we can try to minimize this Free Energy difference.
What Kadanoff showed, way back in 1975, is that we can accurately approximate the Exact Renormalization Group Equation by finding a lower bound using this formalism
Deep learning appears to be a real-space variational RG technique, specifically applicable to very complex, inhomogenous systems where the detailed scale transformations have to be learned from the data
We will now show how to express RBMs using the VRG formalism and provide some intuition
In an RBM, we simply want to learn the Energy function directly; we don’t specify the Hamiltonian for the visible or hidden units explicitly, like we would in physics. The RBM Energy is just
We identify the Hamiltonian for the hidden units as the Renormalized Effective Hamiltonian from the VRG theory
To obtain RBM Hamiltonians for just the visible or hidden nodes, we need to integrate out the other nodes; that is, we need to find the marginal probabilities.
and
To train an RBM, we apply Contrastive Divergence (CD), or, perhaps today, Persistent Contrastive Divergence (PCD). We can kindof think of this as slowly approximating
In practice, however, RBM training minimizes the associated Free Energy difference … or something akin to this…to avoid overfitting.
In the “Practical Guide to Training Restricted Boltzmann Machines”, Hinton explains how to train an RBM (circa 2011). Section 6 addresses “Monitoring the overfitting”
“it is possible to directly monitor the overfitting by comparing the free energies of training data and held out validation data…If the model is not overfitting at all, the average free energy should be about the same on training and validation data”
Modern variants of Real Space VRG are not “‘forced’ to minimize the global free energy” and have attempted other approaches such as Tensor-SVD Renormalization. Likeswise, some RBM / DBM approaches do likewise may minimize a different objective.
In some methods, we minimize the KL Divergence; this has a very natural analog in VRG language [1].
The Renormalization Group Theory provides new insights as to why Deep Learning works so amazingly well. It is not, however, a complete theory. Rather, it is framework for beginning to understand what is an incredibly powerful, modern, applied tool. Enjoy!
[1] An exact mapping between the Variational Renormalization Group and Deep Learning, 2014
[2] Variational Approximations for Renormalization Group Transformations, 1976
[3] A Common Logic to Seeing Cats and Cosmos
[4] WHY DOES UNSUPERVISED DEEP LEARNING WORK? – A PERSPECTIVE FROM GROUP THEORY, 2015
[5] A Fundamental Theory to Model the Mind
[6] A Practical Guide to Training Restricted Boltzmann Machines, 2010
[7] On the importance of initialization and momentum in deep learning, 2013
[8] Dropout: A Simple Way to Prevent Neural Networks from Overfitting, 2014
[9] Ken Wilson, A Scientific Appreciation
[10] http://www-math.unice.fr/~patras/CargeseConference/ACQFT09_JZinnJustin.pdf
[11] Training Products of Experts by Minimizing Contrastive Divergence, 2002
[12] Training Restricted Boltzmann Machines using Approximations to the Likelihood Gradient, 2008
[13] http://www.nonlin-processes-geophys.net/3/102/1996/npg-3-102-1996.pdf
[14] THE RENORMALIZATION GROUP AND CRITICAL PHENOMENA, Ken Wilson Nobel Prize Lecture
[15] Scaling, universality, and renormalization: Three pillars of modern critical phenomena
[16] The Potential Energy of an Autoencoder, 2014
[17] Contractive Auto-Encoders: Explicit Invariance During Feature Extraction, 2011
[19] Quora: What is Renormalization group theory?
[20] Renormalization group and critical localization, 1977
This is the big question on everyone’s mind these days. C’mon we all know the answer already:
“the long-term behavior of certain neural network models are governed by the statistical mechanism of infinite-range Ising spin-glass Hamiltonians” [1] In other words,
Multilayer Neural Networks are just Spin Glasses? Right?
This is kinda true–depending on what you mean by a spin glass.
In a recent paper by LeCun, he attempts to extend our understanding of training neural networks by studying the SGD approach to solving the multilayer Neural Network optimization problem [1]. Furthermore, he claims
None of these works however make the attempt to explain the paradigm of optimizing the highly non-convex neural network objective function through the prism of spin-glass theory and thus in this respect our approach is very novel. And, again, this is kinda true
But here’s the thing…we already have a good idea of what the Energy Landscape of multiscale spin glass models* look like–from early theoretical protein folding work by Peter Wolynes to modern work by Ken Dill, etc [2,3,4]). In fact, here is a typical surface:
*[technically these are Ising spin models with multi-spin interactions]
Let us consider the nodes, which above represent partially folded states, as nodes in a multiscale spin glass–or , say, a multilayer neural network. Immediately we see the analogy and the appearance of the ‘Energy funnel’ In fact, researchers have studied these ‘folding funnels’ of spin glass models over 20 years ago [2,3,4]. And we knew then that
as we increase the network size, the funnel gets sharper
Note: the Wolynes protein-folding spin-glass model is significantly different from the p-spin Hopfield model that LeCun discusses because it contains multi-scale, multi-spin interactions. These details matter.
Spin glasses and spin funnels are quite different. Spin glasses are highly non-convex with lots of local minima, saddle points, etc. Spin funnels, however, are designed to find the spin glass of minimal-frustration, and have a convex, funnel shaped, energy landscape.
This seemed to be necessary to resolve one of the great mysteries of protein folding: Levinthal’s paradox [5]. If nature just used statistical sampling to fold a protein, it would take longer than the ‘known’ lifetime of the Universe. It is why Machine Learning is not just statistics.
Deep Learning Networks are (probably) Spin Funnels
So with a surface like this, it is not so surprising that an SGD method might be able to find the Energy minima (called the Native State in protein folding theory). We just need to jump around until we reach the top of the funnel, and then it is a straight shot down. This, in fact, defines a so-called ‘folding funnel’ [4]
So is not surprising at all that SGD may work.
Recent research at Google and Stanford confirms that the Deep Learning Energy Landscapes appear to be roughly convex [6], as does LeCuns work on spin glasses.
Note that a real theory of protein folding, which would actually be able to fold a protein correctly (i.e. Freed’s approach [7]), would be a lot more detailed than a simple spin glass model. Likewise, real Deep Learning systems are going to have a lot more engineering details–to avoid overtraining (Dropout, Pooling, Momentum) than a theoretical spin funnel.
[Indeed, what is unclear is what happens at the bottom of the funnel. Does the system exhibit a spin glass transition (with full blown Replica Symmetry Breaking, as LeCun suggests), or is the optimal solution more like a simple native state defined by but a few low-energy configurations ? Do DL systems display a phase transition, and is it first order like protein folding? We will address these details in a future post.] In any case case,
It is not that Deep Learning is non-convex–is that we need to avoid over-training
And it seems modern systems can do this using a few simple tricks, like Rectified Linear Units, Dropout, etc.
Hopefully we can learn something using the techniques developed to study the energy landscape of multi-scale spin glass/ spin funnels models. [8,9], thereby utilizing methods theoretical chemistry and condensed matter physics.
In a future post, we will look in detail at the folding funnel including details such as misfolding, the spin glass transition in the funnel, and the relationship to symmetry breaking, overtraining, and LeCun’s very recent analysis. These are non-trivial issues.
I believe this is the first conjecture that Supervised Deep Learning is related to a Spin Funnel. In the next post, I will examine the relationship between Unsupervised Deep Learning and the Variational Renormalization Group [10].
I discuss more of this on my YouTube channel : MMDS 2016 Talk
Learn more about us at: http://calculationconsulting.com
[1] LeCun et. al., The Loss Surfaces of Multilayer Networks, 2015
[2] Spin glasses and the statistical mechanics of protein folding, PNAS, 1987
[3] THEORY OF PROTEIN FOLDING: The Energy Landscape Perspective, Annu. Rev. Phys. Chem. 1997
[4] ENERGY LANDSCAPES, SUPERGRAPHS, AND “FOLDING FUNNELS” IN SPIN SYSTEMS, 1999
[5] From Levinthal to pathways to funnels, Nature, 1997
[6] QUALITATIVELY CHARACTERIZING NEURAL NETWORK OPTIMIZATION PROBLEMS, Google Research (2015)
[7] Mimicking the folding pathway to improve homology-free protein structure prediction, 2008
[8] Funnels in Energy Landscapes, 2007
[9] Landscape Statistics of the Low Autocorrelated Binary String Problem, 2007
[10] A Common Logic to Seeing Cats and Cosmos, 2014