There Will Be a Scientific Theory of Deep Learning
arxiv.org
arxiv.org
The skepticism I'm seeing in the comments really highlights how little of this work is trickling down to the public, which is very sad to see. While it can offer few mathematical mechanisms to infer optimal network design yet (mostly because just trying stuff empirically is often faster than going through the theory, so it is more common to retroactively infer things), the question "why do neural networks work better than other models?" is getting pretty close to a solid answer. Problem is, that was never the question people seem to have ever really been interested in, so the field now has to figure out what questions we ask next.
the better question is why does gradient descent work for them
Any models using an infinite dimensional Hilbert space, such as SVMs with RBF or polynomial kernels, Gaussian process regression, gradient boosted decision trees, etc. have the same property (though proven via a different theorem of course).
So the universal approximation theorem tells us nothing about why should expect neural networks to perform better than those models.
sure, that gives some relief - but it says nothing in practice unlike f.e. which side of P/NP divide the problem is on
Actually the P/NP divide is a similar case in my opinion. In practice a quadratic algorithm is sometimes unacceptably slow and an NP problem can be virtually solved. E.g. SAT problems are routinely solved at scale.
It's similar to the gap between pushdown automata and Turing machines. You can check if pushdown automata will terminate or not. You can't do it for Turing machines, but this doesn't stop you from running a pushdown automata algorithm on the turning machine with decidable termination.
If you have a tool that you don't know works when data increases (n-> infinity), then you shouldn't use it.
So practicaly, I believe it has serious implications.
Recap: 1NN says that given a query Q you choose any pair (X,Y) from your learned "model" (a finite set of (X,Y) pairs) M minimizing |Q-X|. Your output is Y.
The following kind of argument works for linear interpolation too (you can even view 1NN as 1-point interpolation), but it's ever so slightly messier since definitions vary a fair bit, you potentially need to talk about the existence of >1 discrete "nearest" or "enclosing" set of neighbors, and proving that you can get away with fewer points than 1NN or have lower error than 1NN is itself also messier.
Pick your favorite compact-domain, continuous function embedded in some Euclidean space. For any target error you'd like to hit, the uniform continuity of that function guarantees that if your samples cover the domain well enough (no point in the domain is greater than some fixed distance, needing smaller distances for lower errors, from some point in your model) then the maximum error from a 1NN strategy is bounded by the associated error given by uniform continuity (which, again, you can make as small as you'd like by increasing the sampling resolution). The compact domain means you can physically achieve those error bounds with finite sample sizes.
For a simple example, imagine fitting more and more, smaller and smaller, line segments to y=x^2 on [-1,1].
E.g. you could land perfectly on a local minima but you won’t stay the unless your step size was minute or the minima was quite substantial.
If they were all correlated with each other that does not seem far fetched.
Perhaps more important, just because it is easy to escape any local minimum does not mean that there is necessarily a trend towards a really good optimum, as it can just bounce between a bunch of really bad ones for a long time. This actually happens almost all the time if you try to design your entire architecture from scratch, e.g. highly connected networks. People who are new to the field sometimes don't seem to understand why SGD doesn't just always fix everything; this is why. You need very strong inductive biases in your architecture design to ensure that the loss (which is data-dependent so you cannot ascertain this property a priori) exhibits a global bowl-like shape (we often call this a 'funnel') to provide a general trajectory for the optimizer toward good solutions. Sometimes this only works for some optimizers and not others.
This is why architecture design is something of an art form, and explaining "why neural networks work so well" is a complex question involving a ton of parts, all of which contribute in meaningful ways. There are often plenty of counterexamples to any simpler explanation.
The only people for whom this is an open question are the academics - everyone else understands it's entirely because of the bagillions of parameters.
Data labeling is pretty big industry in some countries and I guess dropping 200 kilodollars on labeling is beyond the reach of most academics, even if they would not care about ethics of that.
The actual reason is due to complex biases that arise from the interaction of network architectures and the optimizers and persist in the regime where data scales proportionally to model size. The multiscale nature of the data induces neural scaling laws that enable better performance than any other class of models can hope to achieve.
That’s a lot of words to say that, if you encode a class of things as numbers, there’s a formula somewhere that can approximate an instance of that class. It works for linear regression and works as well for neural network. The key thing here is approximation.
I can construct a Gaussian process model (essentially fancy linear regression) that will fit _all_ of my medical image data _exactly_, but it will perform like absolute rubbish for determining tumor presence compared to if I trained a convolutional neural network on the same data and problem _and_ perfectly fit the data.
I could even train a fully connected network on the same data and problem, get any degree of fit you like, and it would still be rubbish.
Thats been my understanding of the crux of mystery.
Would love to be corrected by someone more knowledgable though
Transformers are superior "database" encodings as the hype about LLMs points out, but there have been promising ML models that were focusing on memory parts for their niche use cases, which could be promising concepts if we could make them work with attention matrixes and/or use the frequency projection idea on their neuron weights.
The way RNNs evolved to LSTMs, GRUs, and eventually DNCs was pretty interesting to me. In my own implementations and use cases I wasn't able to reproduce Deepmind's claims in the DNC memory related parts. Back at the time the "seeking heads" idea of attention matrixes wasn't there yet, maybe there's a way to build better read/write/access/etc gates now.
[1] a fairly good implementation I found: https://github.com/joergfranke/ADNC
- It's not gradient boosting per se that's good on tabular data, it's trees. Other fitting methods with trees as the model are also usually superior to NNs on tabular data.
- Trees are better on tabular data because they encode a useful inductive bias that NNs currently do not. Just like CNNs or ViTs are better on images because they encode spatial locality as an inductive bias.
As you noted, the industry has moved the goalposts to Agency and Long-horizon Persistence. The transition from building 'calculators that predict' to 'systems that endure' is a non-equilibrium thermodynamics problem. There is math/formulas and basic laws at play here that apply to AI just as much as it applies to other systems. Ironically it is the same math. The same thing that results in a signal persisting in a model will result in agents persisting.
This is my specific niche. I study how things persist. It’s honestly a bit painful watching the AI field struggle to re-learn first principles that other disciplines have already learned. I have a doc I use to help teach folks how the math works and how to apply it to their domain and it is fun giving it folks who then stop guessing and know exactly how to improve the persistence of what they are working on. Like the idea of "How many hours we can have a model work" is so cute compared to the right questions.
https://meta-r0ze.github.io/Informational-Energetics/Informa...
This is my fear with software development in general. There's a hundred-year old point of view right next door that'll solve problems and I'm too incurious to see it.
I have a relative with a focus in math education that I've been stealing ideas from, and I think we'd both appreciate a look at your doc if you don't mind.
So instead we're more likely to see navel-gazing "singularity" stories that fit with telling your investors they will become fantastically rich.
Awesome. What is holding you back? What do you need the funding for?
Many people additionally have little patience for research when the engineering is moving so quickly. Even many interpretability researchers give up far too soon if research doesn't yield immediately gratifying results.
This would be great, as from the "classical" perspective, the results of over-parametization and potentially other parts of NN architecture make no sense (to me, at least). I do accept that double-descent appears to empirically work, but it really, really shouldn't. In fact, as someone who's a big fan of Hastie et al's Elements, the bias variance tradeoff suggests that they shouldn't.
This has been bugging me (sporadically) for years, and any progress towards an answer would be incredibly useful (most probably in a philosophical sense I suppose).
As an aside, I've only read the Introduction, but this appears to be a well-written paper and a research program I can get behind. I really want this stuff to work.
I guess it's similar to bagging and boosting, which were empirically successful well before we had any theoretical understanding of why they work.
It really isn't so mysterious once you begin to examine how the rule of thumb for the bias-variance tradeoff (remember that it is the relationship with model size that is curious, not the tradeoff itself) came to be. The easiest ways to arrive at this rule are through an information criterion like the AIC or BIC, where the model size appears in the penalty term for the log-likelihood. These criteria have a bunch of assumptions, all of which are crucial, and absolutely none of which apply for neural networks. The biggest one is that the only limiting regime is in the size of the dataset, so there are vastly more data than model parameters. Neural networks have parameter counts within a constant ratio of the number of datapoints. Another is that the model has a non-singular Hessian in a neighbourhood of the optimum. Neural networks do not have this. Once you abandon the rule of thumb and actually do the math in the appropriate limiting regimes, there's no contradiction anymore.
I've found the biggest mystery for people though is the fact that performance actually _improves_ after the interpolation threshold. This seems insane if you come at it from the point of view that the model "could have done anything" if there are more parameters than data. But this isn't true at all. The fact that you have obtained _a solution_ means that you imposed some implicit bias that guided which solution you end up in. For linear regression, that is often the minimum L2 norm solution, which _literally_ minimizes the variance keeping all else fixed. If you add more parameters to play with, obviously it should be able to minimize the variance even further, right? If the bias is zero and the variance is reduced, you get better performance. If you use a different optimizer than gradient descent, you can end up at the minimum L1 norm solution (effectively LASSO), which is well-known to perform really well regardless of the number of parameters.
Of course, linear regression is not neural network regression, and the situation in deep learning is far more complicated. But the same idea applies. Every single part of the training procedure is carefully designed to bias the obtained solution toward something with minimal variance. Stochastic optimizers (even dropout) settle in wide minima which have smaller variances. Some optimizers prioritize stronger correlations in the weights. Bottlenecks in the architecture induce low-rank solutions. Data augmentation induce known invariances that reduce variance along those directions. Convolutional designs induce regularity with respect to the input space. Neural networks are not magic; they are the product of hundreds of intentional design decisions over decades. When you increase the size of the model, all of these features are exacerbated.
Quantifying all of this in the theory is difficult because there are a lot of moving parts. But if you study a simplified model and consider each mechanism individually, the picture becomes pretty clear.
But they don't give the same results at those smaller scales. People imagined, but no one could have put into practice because the hardware wasn't there yet. Simplified, LLMs is basically Transformers with the additional idea of "and a shitton of data to learn from", and for making training feasible with that amount of data, you do need some capable hardware.
In olden days, the correct way to solve a linear system of equations was to use theory of minors. With advent of computers, you suddenly had a huge theory of gaussian elimination, or Krylov spaces and what not.
After seeing AlexNet’s results, all of the major ML imaging labs switched to deep CNNs, and other approaches almost completely disappeared from SOTA imaging competitions. Over the next few years, deep neural networks took over in other ML domains as well.
The conventional wisdom is that it was the combination of (1) exponentially more compute than in earlier eras with (2) exponentially larger, high-quality datasets (e.g., the curated and hand-labeled ImageNet set) that finally allowed deep neural networks to shine.
The development of “attention” was particularly valuable in learning complex relationships among somewhat freely ordered sequential data like text, but I think most ML people now think of neural-network architectures as being, essentially, choices of tradeoffs that facilitate learning in one context or another when data and compute are in short supply, but not as being fundamental to learning. The “bitter lesson” [1] is that more compute and more data eventually beats better models that don’t scale.
Consider this: humans have on the order of 10^11 neurons in their body, dogs have 10^9, and mice have 10^7. What jumps out at me about those numbers is that they’re all big. Even a mouse needs hundreds of millions of neurons to do what a mouse does.
Intelligence, even of a limited sort, seems to emerge only after crossing a high threshold of compute capacity. Probably this has to do with the need for a lot of parameters to deal with the intrinsic complexity of a complex learning environment. (Mice and men both exist in the same physical reality.)
On the other hand, we know many simple techniques with low parameter counts that work well (or are even proved to be optimal) on simple or stylized problems. “Learning” and “intelligence”, in the way we use the words, tends to imply a complex environment, and complexity by its nature requires a large number of parameters to model.
The brain likely has more in common with Reservoir Computing (sans the actual learning algorithm) than Deep Learning.
Deep Learning relies on end to end loss optimization, something which is much more powerful than anything the brain can be doing. But the end-to-end limitation is restricting, credit assignment is a big problem.
Consider how crazy the generative diffusion models are, we generate the output in its entirety with a fixed number of steps - the complexity of the output is irrelevant. If only we could train a model to just use Photoshop directly, but we can't.
Interestingly, there are some attempts at a middle ground where a variable number of continuous variables describe an image: <https://visual-gen.github.io/semanticist/>
Predictive coding is more biologically plausible because it uses local information from neighbouring neurons only.
There's an entire line of work that goes "brain is trying to approximate backprop with local rules, poorly", with some interesting findings to back it.
Now, it seems unlikely that the brain has a single neat "loss function" that could account for all of learning behaviors across it. But that doesn't preclude deep learning either. If the brain's "loss" is an interplay of many local and global objectives of varying complexity, it can be still a deep learning system at its core. Still doing a form of gradient descent, with non-backpropagation credit assignment and all. Just not the kind of deep learning system any sane engineer would design.
It is probably coming, I get the impression - just from following the trend of the progress - that internal world models are the hardest part. I was playing with Gemma 4 and it seemed to have a remarkable amount of trouble with the idea of going from its house to another house, collecting something and returning; starting part-way through where it was already at house #2. It figured it out but it seemed to be working very hard with the concept to a degree that was really a bit comical.
It looks like that issue is solving itself as text & image models start to unify and they get more video-based data that makes the object-oriented nature of physical reality obvious. Understanding spatial layouts seems like it might be a prerequisite to being able to consistently set up a scene in Photoshop. It is a bit weird that it seems pulling an image fully formed from the aether is statistically easier than putting it together piece by piece.
What kind of sadist would wish this on an intelligent entity?
They seem to use an agentic LLM with image inputs and outputs to produce, verify, refine and compose visual artifacts. Those operations appear to be learned functions, however, not an external tool like Photoshop.
This allows for "variable depth" in practice. Composition uses previous images, which may have been generated from scratch, or from previous images.
They're obviously more general purpose but LLMs can also be used to drive external graphics programs. A relatively popular one is Blender MCP [1], which lets an LLM control Blender to build and scaffold out 3D models.
I'd thought it was some issue with training where older math didn't play nice with having too many layers.
My understanding of the development is that persistent layer-wise pretraining with RBM or autoencoder created an initiation state where the optimization could cope even for more layers, and then when it was proven that it could work, analysis of why led to some changes such as new initiation heuristics, rectified linear activation, eventually normalizations ... so that the pretraining was usually not needed any more.
One finding was that the supervised training with the old arrangement often does work on its own, if you let it run much longer than people reasonably could afford to wait around for just on speculation contrary to observations in CPU computations in the 80s--00s. It has to work its way to a reasonably optimizable state using a chain of poorly scaled gradients first though.
For a bit more context: Before 2012 most approaches were based on hand crafted features + SVMs that achieved state of the art performance on academic competitions such as Pascal VOC and neural nets were not competitive on the surface. Around 2010 Fei Fei Li of Stanford University collected a comparatively large dataset and launched the ImageNet competition. AlexNet cut the error rate by half in 2012 leading to major labs to switch to deeper neural nets. The success seems to be a combination of large enough dataset + GPUs to make training time reasonable. The architecture is a scaled version of ConvNets of Yan Lecun tying to the bitter lesson that scaling is more important than complexity.
Is this a practical viewpoint? Can you remove any of the specific architectural tricks used in Transformers and expect them to work about equally well?
I feel like you are downplaying the importance of architecture. I never read the bitter lesson, but I have always heard more as a comment on embedding knowledge into models instead of making them to just scale with data. We know algorithmic improvement is very important to scale NNs (see https://www.semanticscholar.org/paper/Measuring-the-Algorith...). You can't scale an architecture that has catastrophic forgetting embedded in it. It is not really a matter of tradeoffs, some are really worse in all aspects. What I agree is just that architectures that scale better with data and compute do better. And sure, you can say that smaller architectures are better for smaller problems, but then the framing with the bitter lesson makes less sense.
Even PID loops have a training phase separate from recitation phase.
The reason noone does this is you don’t have to and you’ll get much better results if you first fully train and then apply the best model you have to whatever problem. Biological systems don’t have that luxury.
We know a human uses roughly 100 watts. And teaching a new specific task takes only showing maybe 10 times to get to 80%.
The learning function in humans are definitely connected with both training/recitation.
I'm seeing that as the big roadblock between thinking machines and a really big autocomplete we have now.
Real intelligence deals with information over a ludicrous number of size scales. Simple models effectively blur over these scales and fail to pull them apart. However, extra compute is not enough to do this effectively, as nonparametric models have demonstrated.
The key is injecting a sensible inductive bias into the model. Nonparametric models require this to be done explicitly, but this is almost impossible unless you're God. A better way is to express the bias as a "post-hoc query" in terms of the trained model and its interaction with the data. The only way to train such a model is iteratively, as it needs to update its bias retroactively. This can only be accomplished by a nonlinear (in parameters) parametric model that is dense in function space and possesses parameter counts proportional to the data size. Every model we know of that does this is called "a neural network".
I did some ML in mid 2000s, and it was a PITA to reuse other people code (when available at all). You had some well known libraries for SVM, for HMM you had to use HTK that had a weird license, and otherwise looking at experiments required you to reimplement stuff yourself.
Late 2000s had a lot of practical innovation that democratized ML: theano and then tf/keras/pytorch for DL, scikit learn for ML, etc. That ended up being important because you need a lot of tricks to make this work on top of "textbook" implementation. E.g. if you implement EM algo for GMM, you need to do it in the log space to avoid underflow, DL as well (gorot and co initialization, etc.).
PITA - pain in the ass
SVM - support vector machines HMM - hidden Markov model EM - expectation maximization GMM - gaussian mixture model HTK - hidden Markov model tool kit
Under the very light assumption that a mouse doesn’t have neurons it doesn’t need, a mouse needs whatever number of neurons it has to do what a mouse does, so that’s not saying much.
Reading https://en.wikipedia.org/wiki/List_of_animals_by_number_of_n..., an ant has only 250k neurons and many reptiles can do with around 10 million.
That page also says 71 million for the house mouse. So what is it that a mouse does that reptiles do not do that requires them to have that much larger a brain? Caring for their children?
I daresay I don't think animals actually need some number or neurons. There's probably just a trade off between more giving better results versus being heavier and more energy consuming.
I also think you might be discounting exactly how much compute is used to train these monsters. A single 1ghz processor would take about 100,000,000 years to train something in this class. Even with on the order of 25k GPUs training GPT3 size models takes a couple months. The anemic RAM on GPUs a decade ago (I think we had k80 GPUs with 12GB vs 100’s of GBs on H100/H200 today) and it was actually completely impossible to train a large transformer model prior to the early 2020s.
I’m even reminded how much gamers complained in the late 2010s about GPU prices skyrocketing because of ML use.
It could have been done in the early 1970s -- see "Paper tape is all you need" at https://github.com/dbrll/ATTN-11 and the various C-64 projects that have been posted on HN -- but the problem was that Marvin Minsky "proved" that there was no way a perceptron-based network could do anything interesting. Funding dried up in a hurry after that.
What result are you referring to?
I'm sure it's an oversimplification to blame the entire 1970s AI winter on Minsky, considering they couldn't have gotten much further than the proof-of-concept stage due to lack of hardware. But his voice was a loud, widely-respected one in academia, and it did have a negative effect on the field.
I agree with your larger point but dismissed is rather too strong. They were considered fiddly to train, prone to local minima, long training time, no clear guidelines about what the number of hidden layers and number of nodes ought to be. But for homework (toy) exercises they were still ok.
In comparison, kernel methods gave a better experience over all for large but not super large data sets. Most models had easily obtainable global minimum. Fewer moving parts and very good performance.
It turns out, however, that if you have several orders of magnitude more data, the usual kernels are too simple -- (i) they cannot take advantage of more data after a point and start twiddling the 10th place of decimal of some parameters and (ii) are expensive to train for very large data sets. So bit of a double whammy. Well, there was a third, no hardware acceleration that can compare with GPUs.
Kernels may make a comeback though, you never know. We need to find a way to compose kernels in a user friendly way to increase their modeling capacity. We had a few ways of doing just that but they weren't great. We need a breakthrough to scale them to GPT sized data sets.
In a way DNNs are "design your own kernels using data" whereas kernels came in any color you liked provided it was black (yes there were many types, but it was still a fairly limited catalogue. The killer was that there was no good way of composing them to increase modeling capacity that yielded efficiently trainable kernel machines)
Cut to 2008-9,and I started to see smartphones, grid (then cloud) computing and social networks emerging. My MBA dissertation, finished in 2011, was about how that would change the world, because the requirements for meaningful AI were coming along - data and compute. The theory was already there, Hinton, LeCun, Schmidhuber,etc.
That got me back into the Data Science field, after years working in Data Engineering. Too bad I lived in Brazil back then and couldn't find a way to join the emerging scene in California and other top places. I'd be rich now...
Or perhaps a world where it happened later. I think a big part of what enabled the AI boom was the concentration of money and compute around the crypto boom.
at that time all the compute resources in the world would not have been enough to train the models from even the last ~6 years or so, probably more.
The incentive to design something new - which became the Transformer - came from language model researchers who had been working with recurrent models such as LSTMs, whose recurrent nature made them inefficient to train (needing BPPT), and wanted to come up with a new seq-2-seq/language model that could take advantage of the parallel hardware that now existed and (since AlexNet) was now being used to good effect for other types of model.
As I understand it, the inspiration for the concept of what would become the Transformer came from Attention paper co-author Jakob Uzkoreit who realized that language, while superficially appearing sequential (hence a good match for RNNs) was in fact really parallel + hierarchical as can be seen by linguist's sentence parse trees where different branches of the tree reflect parallel analysis of different parts of the sentence, which are then combined at higher levels of the hierarchical parse tree. This insight gave rise to the idea of a language model that mirrored this analytical structure with hierarchical layers of parallel processing, with the parallel processing being the whole point since this could be accelerated by GPUs. While the concept was Uzkoreit's, it took another researcher, Noam Shazeer, to take the concept and realize it as a performant architecture - the Transformer.
Without the fast parallel hardware already pre-existing, there would not have been any incentive to design a new type of language model to take advantage of it!
The other point is that while the Transformer is a very powerful general purpose and scalable type of model, it only really comes into it's own at scale. If a Transformer had somehow been designed in the pre-GPU-compute era, before the compute power to scale it up to massive size existed it, then it would likely not have appeared so promising/interesting.
The other aspect to the history is that neural networks, of various types, have evolved in complexity and sophistication over time. RNNs and LSTMs came first, then Bahdanau attention as a way to improve their context focus and performance. Attention was now seen to be a valuable part of language and seq-2-seq modelling, so when GPUs motivated the Transformer, attention was retained, recurrence ditched, and hence "Attention is all you need".
The time was right for the Transformer to appear when it did, designed to take advantage of recent GPU advances, building on top of this new attention architecture, and now with the compute power and dataset size available that it started to really shine when scaled from GPT-1 to GPT-2 size, and beyond.
https://youtu.be/glWvwvhZkQ8?si=-HGtfd_KHYfatEQ
Although it’s focused on Ilya, some great history is covered.
It might lead to understanding how to measure when a deep learning system is making stuff up or hallucinating. That would have a huge payoff. Until we get that, deep learning systems are limited to tasks where the consequences of outputting bullshit are low.
That's a great problem to solve! (Maybe biased, because this is my primary research direction). One popular approach is OOD detection, but this always seemed ill-posed to me. My colleagues and I have been approaching this from a more fundamental direction using measures of model misspecification, but this is admittedly niche because it is very computationally expensive. Could still be a while before a breakthrough comes from any direction.
It would be valuable enough that getting significant funding to work on it is probably possible. Especially with all the money being thrown at AI.
There is so much to digest here but it's fascinating seeing it all put together!
https://arxiv.org/abs/2510.12269
https://www.mdpi.com/1099-4300/28/3/332
I am also interested in connection with fuzzy logic - it seems that NNs can reason in a fuzzy way, but what they are doing, formally? For years, people have been trying to formalize fuzzy reasoning but it looks like we don't care anymore.
I feel like NNs (and transformers) are the OOP (object-oriented programming) of ML. Really popular, works pretty well in practice, but nobody understands the fundamentals; there is a feeling it is a made up new language to express things expressible before, but hard to pinpoint where exactly it helps.
Funny statement to be found in the discussion about... research results on the fundamentals.
Neural networks in general are Turing models. Human brains are in the abstract Turing complete as well, as a simple example. LLMs being run iteratively in an unbounded loop may be "effectively Turing complete" for this simple reason, as well.
Regardless, any theory purporting to be foundational ought to explicitly address this demarcation. Unless practitioners think computability and formal complexity are not scientific foundations for CS.
You need a memory element the network can interact with, just like an ALU by itself is not TC, but a barebones stateful CPU (ALU + registers) is.
Wait ... they created something they do not understand and can not yet describe?
And this is now called ... science?
Remember that they "borrowed" words from biology, in particular neurobiology, decades ago already. Monkey see, monkey do copy paste.
My fear is that this is as hopeless right now as explaining why humans or other animals can learn certain things from their huge amount of input data. We'll gain better empirical understanding, but it won't ever be fundamental computer science again, because the giga-datasets are the fundamental complexity not the architecture.
That would be amazing, but personally I’m skeptical.
What makes statistical mechanics so brilliant is that it takes first principle ideas (particle energies + ensemble) to derive macroscopic thermodynamic rules, all of which were originally derived from observation.
What the OP is proposing is a mathematical analysis of SGD + generic deep learning architectures might be able to derive the rules we have empirically derived from experiments in model training.
And we may find that biology also exploits structures like deep nets