Why does Deep Learning work?

Why does Deep Learning work?

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

3D Energy Landscape of the Folding Funnel of a Spin Glass

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.

Spin Funnels have been characterized and cataloged by theoretical chemists and can take many different convex shapes

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.

Spin Funnels and Spin Glasses — Spin Funnels (DFM) vs Spin Glasses (SG) [4]

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

Update: Jan 2018

We have observed what I conjectured 3 years ago on my blog and at MMDS 2016 –the Spin Glass of Minimal Frustration (left) appearing in Deep Networks (right). Stay tuned for some very exciting results into theory of Deep Learning.

Update: Sep 2018: Predictions can come true

This paper was just brought to my attention on “VISUALIZING THE LOSS LANDSCAPE OF NEURAL NETS” which exactly verifies the above predictions…3 years later

• “We observe that, when networks become sufficiently deep, neural loss landscapes quickly transition from being nearly convex to being highly chaotic. This transition from convex to chaotic behavior coincides with a dramatic drop in generalization error, and ultimately to a lack of trainability.”

• “We show that skip connections promote flat minimizers and prevent the transition to chaotic behavior, which helps explain why skip connections are necessary for training extremely deep networks”