Reverse engineering the NTK: towards first-principles architecture design

Deep neural networks have enabled technological wonders starting from voice recognition to machine transition to protein engineering, however their design and software is nonetheless notoriously unprincipled.
The event of instruments and strategies to information this course of is likely one of the grand challenges of deep studying concept.
In Reverse Engineering the Neural Tangent Kernel, we suggest a paradigm for bringing some precept to the artwork of structure design utilizing latest theoretical breakthroughs: first design a superb kernel operate – typically a a lot simpler activity – after which “reverse-engineer” a net-kernel equivalence to translate the chosen kernel right into a neural community.
Our foremost theoretical end result permits the design of activation features from first rules, and we use it to create one activation operate that mimics deep (textrm{ReLU}) community efficiency with only one hidden layer and one other that soundly outperforms deep (textrm{ReLU}) networks on an artificial activity.

Kernels again to networks. Foundational works derived formulae that map from vast neural networks to their corresponding kernels. We get hold of an inverse mapping, allowing us to begin from a desired kernel and switch it again right into a community structure.

Neural community kernels

The sector of deep studying concept has lately been reworked by the belief that deep neural networks typically change into analytically tractable to check within the infinite-width restrict.
Take the restrict a sure means, and the community in actual fact converges to an unusual kernel methodology utilizing both the structure’s “neural tangent kernel” (NTK) or, if solely the final layer is skilled (a la random characteristic fashions), its “neural network Gaussian process” (NNGP) kernel.
Just like the central restrict theorem, these wide-network limits are sometimes surprisingly good approximations even removed from infinite width (typically holding true at widths within the tons of or 1000’s), giving a outstanding analytical deal with on the mysteries of deep studying.

From networks to kernels and again once more

The unique works exploring this net-kernel correspondence gave formulae for going from structure to kernel: given an outline of an structure (e.g. depth and activation operate), they provide the community’s two kernels.
This has allowed nice insights into the optimization and generalization of varied architectures of curiosity.
Nevertheless, if our objective just isn’t merely to know current architectures however to design new ones, then we would reasonably have the mapping within the reverse course: given a kernel we wish, can we discover an structure that offers it to us?
On this work, we derive this inverse mapping for fully-connected networks (FCNs), permitting us to design easy networks in a principled method by (a) positing a desired kernel and (b) designing an activation operate that offers it.

To see why this is sensible, let’s first visualize an NTK.
Contemplate a large FCN’s NTK (Ok(x_1,x_2)) on two enter vectors (x_1) and (x_2) (which we are going to for simplicity assume are normalized to the identical size).
For a FCN, this kernel is rotation-invariant within the sense that (Ok(x_1,x_2) = Ok(c)), the place (c) is the cosine of the angle between the inputs.
Since (Ok(c)) is a scalar operate of a scalar argument, we will merely plot it.
Fig. 2 exhibits the NTK of a four-hidden-layer (4HL) (textrm{ReLU}) FCN.

Fig 2. The NTK of a 4HL $textrm{ReLU}$ FCN as a operate of the cosine between two enter vectors $x_1$ and $x_2$.

This plot really comprises a lot details about the training habits of the corresponding vast community!
The monotonic enhance signifies that this kernel expects nearer factors to have extra correlated operate values.
The steep enhance on the finish tells us that the correlation size just isn’t too giant, and it could match difficult features.
The diverging spinoff at (c=1) tells us in regards to the smoothness of the operate we count on to get.
Importantly, none of those details are obvious from a plot of (textrm{ReLU}(z))!
We declare that, if we need to perceive the impact of selecting an activation operate (phi), then the ensuing NTK is definitely extra informative than (phi) itself.
It thus maybe is sensible to attempt to design architectures in “kernel house,” then translate them to the standard hyperparameters.

An activation operate for each kernel

Our foremost result’s a “reverse engineering theorem” that states the next:

Thm 1: For any kernel $Ok(c)$, we will assemble an activation operate $tilde{phi}$ such that, when inserted right into a single-hidden-layer FCN, its infinite-width NTK or NNGP kernel is $Ok(c)$.

We give an express components for (tilde{phi}) by way of Hermite polynomials
(although we use a unique purposeful type in apply for trainability causes).
Our proposed use of this result’s that, in issues with some identified construction, it’ll typically be potential to put in writing down a superb kernel and reverse-engineer it right into a trainable community with varied benefits over pure kernel regression, like computational effectivity and the power to be taught options.
As a proof of idea, we check this concept out on the artificial parity drawback (i.e., given a bitstring, is the sum odd and even?), instantly producing an activation operate that dramatically outperforms (textual content{ReLU}) on the duty.

One hidden layer is all you want?

Right here’s one other stunning use of our end result.
The kernel curve above is for a 4HL (textrm{ReLU}) FCN, however I claimed that we will obtain any kernel, together with that one, with only one hidden layer.
This suggests we will give you some new activation operate (tilde{phi}) that offers this “deep” NTK in a shallow community!
Fig. 3 illustrates this experiment.

Fig 3. Shallowification of a deep $textrm{ReLU}$ FCN right into a 1HL FCN with an engineered activation operate $tilde{phi}$.

Surprisingly, this “shallowfication” really works.
The left subplot of Fig. 4 under exhibits a “mimic” activation operate (tilde{phi}) that offers nearly the identical NTK as a deep (textrm{ReLU}) FCN.
The appropriate plots then present prepare + check loss + accuracy traces for 3 FCNs on a normal tabular drawback from the UCI dataset.
Be aware that, whereas the shallow and deep ReLU networks have very completely different behaviors, our engineered shallow mimic community tracks the deep community nearly precisely!

Fig 4. Left panel: our engineered “mimic” activation operate, plotted with ReLU for comparability. Proper panels: efficiency traces for 1HL ReLU, 4HL ReLU, and 1HL mimic FCNs skilled on a UCI dataset. Be aware the shut match between the 4HL ReLU and 1HL mimic networks.

That is attention-grabbing from an engineering perspective as a result of the shallow community makes use of fewer parameters than the deep community to attain the identical efficiency.
It’s additionally attention-grabbing from a theoretical perspective as a result of it raises basic questions in regards to the worth of depth.
A standard perception deep studying perception is that deeper just isn’t solely higher however qualitatively completely different: that deep networks will effectively be taught features that shallow networks merely can not.
Our shallowification end result means that, at the least for FCNs, this isn’t true: if we all know what we’re doing, then depth doesn’t purchase us something.

Conclusion

This work comes with plenty of caveats.
The most important is that our end result solely applies to FCNs, which alone are not often state-of-the-art.
Nevertheless, work on convolutional NTKs is fast progressing, and we imagine this paradigm of designing networks by designing kernels is ripe for extension in some type to those structured architectures.

Theoretical work has up to now furnished comparatively few instruments for sensible deep studying theorists.
We intention for this to be a modest step in that course.
Even and not using a science to information their design, neural networks have already enabled wonders.
Simply think about what we’ll be capable of do with them as soon as we lastly have one.

This put up is predicated on the paper “Reverse Engineering the Neural Tangent Kernel,” which is joint work with Sajant Anand and Mike DeWeese. We offer code to breed all our outcomes. We’d be delighted to subject your questions or feedback.