Linking to the paper: https://arxiv.org/pdf/2605.01172 which is also a fantastic read, the application to deep learning is good. It does a lot of cross-mapping and highlighting a bunch of old stuff that is named differently in this paper and worth calling out for those with those backgrounds:
"Cumulative Dissipation Gramian" Ws = Observability Gramian (from Control Theory). For example the spectral cutoff is exactly the Hankel singular value truncation from model reduction.
"Signal Channel" / "Reservoir" is Controllable/Observable vs. Uncontrollable/Unobservable Subspaces. Using Adamjan-Arov-Krein (AAK) theory gives the optimal nonlinear reduced model answering the optimal compression question.
"Drift–Diffusion Separation" is Freidlin-Wentzell Large Deviation Theory. They can predict "grokking" time from the FW action.
"Population-Risk Gate" is Quantum Weak Value / Postselection (Aharonov)
So for the follow-up problems
Control theory gives the truncation error bounds for model compression. Large deviation theory gives the grokking time predictions. Quantum measurement theory gives the imaginary preconditioners. Information geometry gives the optimal continuous relaxation of the gate.
I interpreted the kernel K of this paper as the BRDF in Rendering Equation[0] and its familiar diffusion process (from light transport simulation, or really any integro-differential equation system); together with https://en.wikipedia.org/wiki/Neural_tangent_kernel I hope this paper might be accessible with some study
This is a fascinating mathematical framework, but the post title might be a bit of an overreach. I often wonder if "a theory of deep learning" could exist that could be stated succinctly and that could predict (1) scaling laws and (2) the surprising reliability of gradient descent.
Note that I said "predict" not "describe". It feels like we're still in the era of Kepler, not Newton.
I dunno... gradient descent is only really reliable with a big bag of tricks. Knowing good initializations is a starting point, but recurrent connections and batch/layer normalization go a very long way towards making it reliable.
I agree, this is the correct way to see it IMO. Instead of designing better optimizers, we designed easier parameterizations to optimize. The surprising part is that these parameterizations exist in the first place.
Gradient descent is mathematically the most efficient optimization strategy (safe for some special functions) in high dimensions. This goes so far that people nowadays even believe it has to be used in the human brain [1], if only because every other method of updating the brain would be way too energy inefficient. From that perspective, finding the right parameterization was all we ever needed to achieve AI.
Idk to me this is just redescribing what deep neural networks do without actually explaining why anything happens. I guess it "unifies" things but I am kinda over most unifying theories. Everything is Bayesian, everything is a graph or a group or some other fancy geometric structure, everything is a category. Ultimately the best framework is whatever is useful enough to explain what's happening in such a way that a practitioner can manipulate the model towards a desired outcome. In other words, where is the knob? The tool they share may be interesting and I hope to play with it to see what happens at different levels of noise applied to the labels.
> That is, if the batch signal on a parameter exceeds its leave-one-out noise, update it; if not, skip it. This is a one-line change to Adam that accelerates grokking by 5x, suppresses memorization in PINNs, and improves DPO fine-tuning, eliminating the need for validation sets entirely.
Does anyone understand the formula they expressed above this sentence? is this just the classic "skip updating parameters with high gradient/loss variance in multiple batches/samples" ?
What is classic about "skip updating parameters with high gradient/loss variance in multiple batches/samples"? Do you have a particular algorithm in mind that uses this heuristic?
Interesting read. I remember the grokking paper when it came out but I don't think I've ever seen that classic grokking loss curve in my own hands on real data. Curious if others have seen it more often in practice
To get pure grokking, you need a model large enough to easily memorize the entire training data and keep training for a long time after memorization. In practice, you'll probably use a more realistically-sized model that might grok on some subset of the data, but not so strongly that it's extremely obvious.
this landed precisely on like 3 weird bugs ive been hitting and solving in different stupid ways for dealing with things like sgd collapsing too many good answers into one bad answer, and gave me a real direction to try to fix the link missing in my own ml stuff. what timing. i have tried analytic solutions too and they're useful for like mapping prompts into memory geometry but from there ive ended up still having to use sgd. cause i think what happens is, sgd teaches the neural net both the geometry and how to navigate it. if you just teleport to the answer it doesnt learn how to walk.
This is a beautifully written way of saying “Some parts of what the network memorizes affect test behavior, and some don’t.” But that’s not a theory of deep learning, the grand unified theory would explain that.
We're given a signal channel and a reservoir. Signal lives in the channel, noise lives in the reservoir, and the reservoir supposedly doesn’t show up at test time.
Okay, but then we have: why would SGD put the right things in the right bucket?
If the answer is “because the reservoir is defined as the stuff that doesn’t transfer to test,” then this is close to circular.
The Borges/Lavoisier stuff is a tell. "We have unified the field” rhetoric should come after nontrivial predictions and results. Claiming to solve benign overfitting, double descent, grokking, implicit bias, risk of training on population, how to avoid a validation set, and last but not least, skipping training by analytically jumping to the end is 6 theory papers, 3 NeurIPS winners, and a $10B startup. Let's get some results before we tell everyone we unified the field. :) I hope you're right.
I don't know the math, but this point was clear to me and it screamed, "crank" but not being sure of that because I am not learned enough to understand the math... but even I could tell the magnitude of the claim. Even just the removing the need for validation sets would have epic consequences across many fields.
> why would SGD put the right things in the right bucket?
Think of it as a best fit curve and exceptions to that curve. The noise is essentially this set of exceptions that move points away from where they would otherwise fall on the curve.
Gradient descent wants to be able to make the smallest change that moves the most data points towards the curve. To do this it learns an arrangement where it can change, say, one parameter and have a bunch of points move at once. What does this correspond to? The big common patterns shared by many data points.
Most of the capacity gets soaked up modelling these sorts of common patterns, and after they have been learned the model starts adding exceptions that allow individual points to deviate from the curve.
Because they’re exceptions, they must not impact neighbouring points, or at least only ones within a very short distance from them. Otherwise they’re now driving the error higher by impacting more points than they should. So you end up with very narrow ranges of features that are able to trigger different sorts of noise.
How narrow they are is shaped by the training data, they’re exactly as narrow as needed not to raise the error, so assuming the total population has the same distribution, they don’t get hit. Much.
Admittedly probably some aggrandized boasting here, but I think empirical verification of that Adam modification alone would be a meaningful contribution, unless that's prior work?
A theory that skips the parameter space, and understands grokking theory, comes up with an unexplained update rule, which notably works on a per-parameter level by dropping the updates for most parameters.
I suspect there is going to be a lot of handwaving to actually go from eNTK to that new update rule.
I also doubt it helps in the non-grokking regime, given the focus of the theory, which is where all the practical applications I have ever heard from live.
Don't get me wrong, I did enjoy reading this essay. It's well written and reasonably argumented without going into details.
The handwaving required is just to assume a diagonal preconditioner, and the optimal preconditioner under that constraint corresponds to the new update rule. (See section F of the paper.) And of course a diagonal preconditioner works on the per-paramer level.
If that's the case, a way to test the theory and understanding (assuming some parts of reservoir and signal channel can be reliably identified) would be to prune the high-confidence reservoir significantly reducing the model size while still getting good predictions. I don't believe the authors mention this (though I skimmed and didn't read the full paper in detail so I may be wrong)
These are the same complaints I had. Also felt like it was high quality ai writing, possibly because of the style choices like "Benign overfitting is noise sitting in the reservoir at interpolation. XYZ is ..." and because of the similarity it has to the times I ended up with chatgpt or gemini creating very detailed and plausible reports about my own crackpot or vague-enough-to-be-useless ideas.
Nah, the softer stuff seems like valuable outreach / good science communication for people that aren't up for the math. Including probably lots of software engineers who are sick of dumb debates in forums, and starting to dip into the real literature and listen to better authorities. More people should do this really, since it's the only way to see past the marketing and hype from fully entrenched AI boosters or detractors. Neither of those groups is big on critical thinking, and they dominate most conversation.
Time/effort coming from experts who want to make things accessible is a gift! The paper is linked elsewhere in the thread if you want no-frills.
So, this is either the paper of the year, or ... definitely not the paper of the year.
https://arxiv.org/pdf/2605.01172 is the current version. The money graphs are page 8 and on where they show (some weirdly thick) line charts with loss results reached in roughly 1/5 the number of steps that Adam takes, just what the blog post mentions.
They also claim holding back test data is not needed, also with more graphs.
I'm not an ML scientist, and I did not attempt to seriously parse the math. It reads to me as something precisely in that liminal space some math papers do where there's enough new terminology that actually parsing through it all is going to take real, concerted effort, possibly with mild brain damage as a risk.
Their 3d graphs of "kernel eigenstructure" also do double duty for me as totally impenetrable and possibly part of an April fool's ML paper that's hilarious to insiders. Or maybe they show something really amazing; they definitely seem to converge into a shape...What does that shape mean??? Why??? What is an eigenstructure? Is it just 3D eigenvectors of some matrices? Is it natural to have a 3D shape representing these large matrices? If not, how and why were these projected down? And why are they different colors in the paper?? You get the feel for my level of understanding.
I think it would frankly just be easier to validate this claim than parse the whole paper. If only I could understand
> Each one-step kernel increment ηKMtSS integrates into WMS , so a sequence of one-step rate-maximizers is the greedy policy whose integral is the signal-channel content of the trajectory through G, exactly as plain SGD is the greedy step whose integral is empirical-risk descent through D. The diagonal cutoff µ2 k >σ2 k/(b−1) is the optimal first-order preconditioner for population risk on any diagonal base, and a streaming variance EMAˆst of squared gradient deviations realizes it as a one-line change to AdamW: one extra parameter-sized state vector and a per parameter gate that multiplies the standard moment update
Well enough to implement the one line update to Adam in python. I have not asked codex or claude to assist yet.
Also of note to me, they talk about grokking which I found SUUUPER fascinating when it was first reported, and have never heard about since. So I was really glad to read about it and read that there has been a little academic work on the phenomenon.
Finally, of the three models they repot results on, two are extremely tiny, the last is a DPO round on Qwen 0.5B -- if the code for that is published, I imagine it would be easy to adapt and evaluate in other regimes.
Where’s the theory of how the human brain does what it does? Maybe these high dimensional structures don’t have a nice compact “theory”. Trying to fit these systems into a nice compact theory is a very human thing, but not everything works like that.
Linking to the paper: https://arxiv.org/pdf/2605.01172 which is also a fantastic read, the application to deep learning is good. It does a lot of cross-mapping and highlighting a bunch of old stuff that is named differently in this paper and worth calling out for those with those backgrounds:
"Cumulative Dissipation Gramian" Ws = Observability Gramian (from Control Theory). For example the spectral cutoff is exactly the Hankel singular value truncation from model reduction.
"Signal Channel" / "Reservoir" is Controllable/Observable vs. Uncontrollable/Unobservable Subspaces. Using Adamjan-Arov-Krein (AAK) theory gives the optimal nonlinear reduced model answering the optimal compression question.
"Drift–Diffusion Separation" is Freidlin-Wentzell Large Deviation Theory. They can predict "grokking" time from the FW action.
"Population-Risk Gate" is Quantum Weak Value / Postselection (Aharonov)
So for the follow-up problems
Control theory gives the truncation error bounds for model compression. Large deviation theory gives the grokking time predictions. Quantum measurement theory gives the imaginary preconditioners. Information geometry gives the optimal continuous relaxation of the gate.
The relevant paper: "A Theory of Generalization in Deep Learning". https://arxiv.org/abs/2605.01172
I interpreted the kernel K of this paper as the BRDF in Rendering Equation[0] and its familiar diffusion process (from light transport simulation, or really any integro-differential equation system); together with https://en.wikipedia.org/wiki/Neural_tangent_kernel I hope this paper might be accessible with some study
[0] https://en.wikipedia.org/wiki/Rendering_equation
This is a fascinating mathematical framework, but the post title might be a bit of an overreach. I often wonder if "a theory of deep learning" could exist that could be stated succinctly and that could predict (1) scaling laws and (2) the surprising reliability of gradient descent.
Note that I said "predict" not "describe". It feels like we're still in the era of Kepler, not Newton.
I dunno... gradient descent is only really reliable with a big bag of tricks. Knowing good initializations is a starting point, but recurrent connections and batch/layer normalization go a very long way towards making it reliable.
I agree, this is the correct way to see it IMO. Instead of designing better optimizers, we designed easier parameterizations to optimize. The surprising part is that these parameterizations exist in the first place.
Gradient descent is mathematically the most efficient optimization strategy (safe for some special functions) in high dimensions. This goes so far that people nowadays even believe it has to be used in the human brain [1], if only because every other method of updating the brain would be way too energy inefficient. From that perspective, finding the right parameterization was all we ever needed to achieve AI.
[1] https://physoc.onlinelibrary.wiley.com/doi/full/10.1113/JP28...
> but the post title might be a bit of an overreach.
Really? An essay that leads off with a Borges anecdote skewed grandiose. Oh my, how unprecedented!
Idk to me this is just redescribing what deep neural networks do without actually explaining why anything happens. I guess it "unifies" things but I am kinda over most unifying theories. Everything is Bayesian, everything is a graph or a group or some other fancy geometric structure, everything is a category. Ultimately the best framework is whatever is useful enough to explain what's happening in such a way that a practitioner can manipulate the model towards a desired outcome. In other words, where is the knob? The tool they share may be interesting and I hope to play with it to see what happens at different levels of noise applied to the labels.
We're still in the era of room-sized-computers-only-scientists-understand era of the neural networks. Knobs and buttons for nerds are slowly coming.
A real theory would predict phenomena thus far unseen. We already know about this 4 part taxonomy.
[dead]
This essay seems to be related to the paper "There Will Be a Scientific Theory of Deep Learning" [1] which was discussed here recently [2].
[1] https://arxiv.org/pdf/2604.21691
[2] https://news.ycombinator.com/item?id=47893779
> That is, if the batch signal on a parameter exceeds its leave-one-out noise, update it; if not, skip it. This is a one-line change to Adam that accelerates grokking by 5x, suppresses memorization in PINNs, and improves DPO fine-tuning, eliminating the need for validation sets entirely.
Does anyone understand the formula they expressed above this sentence? is this just the classic "skip updating parameters with high gradient/loss variance in multiple batches/samples" ?
What is classic about "skip updating parameters with high gradient/loss variance in multiple batches/samples"? Do you have a particular algorithm in mind that uses this heuristic?
Interesting read. I remember the grokking paper when it came out but I don't think I've ever seen that classic grokking loss curve in my own hands on real data. Curious if others have seen it more often in practice
To get pure grokking, you need a model large enough to easily memorize the entire training data and keep training for a long time after memorization. In practice, you'll probably use a more realistically-sized model that might grok on some subset of the data, but not so strongly that it's extremely obvious.
this landed precisely on like 3 weird bugs ive been hitting and solving in different stupid ways for dealing with things like sgd collapsing too many good answers into one bad answer, and gave me a real direction to try to fix the link missing in my own ml stuff. what timing. i have tried analytic solutions too and they're useful for like mapping prompts into memory geometry but from there ive ended up still having to use sgd. cause i think what happens is, sgd teaches the neural net both the geometry and how to navigate it. if you just teleport to the answer it doesnt learn how to walk.
Does anyone happen to know what font this site is using? It looks really elegant.
It is a modified version of ET_Book called ET_Bembo:
https://github.com/DavidBarts/ET_Bembo
I love u. thanks!
apparently its the font used in Edward Tufte's books. Its on github: https://edwardtufte.github.io/et-book/
The "Quantitative Display of Information", which I just checked, is using Monotype Bembo. So still Bembo, but a different version.
Font is atrocious.
Uppercase letters have different stroke width than lowercase ones — it’s like they are *B*old *L*ike this.
Not only that: tracking, kerning is basically non-existent.
Please don’t use that open-source font
You need real paid Bembo, not that piece of shit.
What a beautifully written article. It's extremely that I favourite an article but this is one.
The Hidden Physics of LLMs: Retrieval as Thermodynamics
https://www.youtube.com/watch?v=ppCZfjLdSY8
I found this video to be illustrative as well. Simple and anyone can understand.
Very extremely. Quite a lovely presentation. I'm definitely having a Patrick Bateman-esque appreciation for that delicate cream background.
A very fascinating read.
As a fellow tufte css enjoyer, Why is user select turned off on the sidenotes? I would like to be able to copy paste them quite badly.
Layout is fine but font is atrocious.
Uppercase letters have different stroke width than lowercase ones — it’s like they are *B*old *L*ike this.
Not only that: tracking, kerning is basically non-existent.
Please don’t use that open-source font
You need real Bembo, not that piece of shit
This is a beautifully written way of saying “Some parts of what the network memorizes affect test behavior, and some don’t.” But that’s not a theory of deep learning, the grand unified theory would explain that.
We're given a signal channel and a reservoir. Signal lives in the channel, noise lives in the reservoir, and the reservoir supposedly doesn’t show up at test time.
Okay, but then we have: why would SGD put the right things in the right bucket?
If the answer is “because the reservoir is defined as the stuff that doesn’t transfer to test,” then this is close to circular.
The Borges/Lavoisier stuff is a tell. "We have unified the field” rhetoric should come after nontrivial predictions and results. Claiming to solve benign overfitting, double descent, grokking, implicit bias, risk of training on population, how to avoid a validation set, and last but not least, skipping training by analytically jumping to the end is 6 theory papers, 3 NeurIPS winners, and a $10B startup. Let's get some results before we tell everyone we unified the field. :) I hope you're right.
I don't know the math, but this point was clear to me and it screamed, "crank" but not being sure of that because I am not learned enough to understand the math... but even I could tell the magnitude of the claim. Even just the removing the need for validation sets would have epic consequences across many fields.
> why would SGD put the right things in the right bucket?
Think of it as a best fit curve and exceptions to that curve. The noise is essentially this set of exceptions that move points away from where they would otherwise fall on the curve.
Gradient descent wants to be able to make the smallest change that moves the most data points towards the curve. To do this it learns an arrangement where it can change, say, one parameter and have a bunch of points move at once. What does this correspond to? The big common patterns shared by many data points.
Most of the capacity gets soaked up modelling these sorts of common patterns, and after they have been learned the model starts adding exceptions that allow individual points to deviate from the curve.
Because they’re exceptions, they must not impact neighbouring points, or at least only ones within a very short distance from them. Otherwise they’re now driving the error higher by impacting more points than they should. So you end up with very narrow ranges of features that are able to trigger different sorts of noise.
How narrow they are is shaped by the training data, they’re exactly as narrow as needed not to raise the error, so assuming the total population has the same distribution, they don’t get hit. Much.
At least, that’s what I take away from it.
Admittedly probably some aggrandized boasting here, but I think empirical verification of that Adam modification alone would be a meaningful contribution, unless that's prior work?
A theory that skips the parameter space, and understands grokking theory, comes up with an unexplained update rule, which notably works on a per-parameter level by dropping the updates for most parameters.
I suspect there is going to be a lot of handwaving to actually go from eNTK to that new update rule.
I also doubt it helps in the non-grokking regime, given the focus of the theory, which is where all the practical applications I have ever heard from live.
Don't get me wrong, I did enjoy reading this essay. It's well written and reasonably argumented without going into details.
The handwaving required is just to assume a diagonal preconditioner, and the optimal preconditioner under that constraint corresponds to the new update rule. (See section F of the paper.) And of course a diagonal preconditioner works on the per-paramer level.
If that's the case, a way to test the theory and understanding (assuming some parts of reservoir and signal channel can be reliably identified) would be to prune the high-confidence reservoir significantly reducing the model size while still getting good predictions. I don't believe the authors mention this (though I skimmed and didn't read the full paper in detail so I may be wrong)
These are the same complaints I had. Also felt like it was high quality ai writing, possibly because of the style choices like "Benign overfitting is noise sitting in the reservoir at interpolation. XYZ is ..." and because of the similarity it has to the times I ended up with chatgpt or gemini creating very detailed and plausible reports about my own crackpot or vague-enough-to-be-useless ideas.
> The Borges/Lavoisier stuff is a tell.
Nah, the softer stuff seems like valuable outreach / good science communication for people that aren't up for the math. Including probably lots of software engineers who are sick of dumb debates in forums, and starting to dip into the real literature and listen to better authorities. More people should do this really, since it's the only way to see past the marketing and hype from fully entrenched AI boosters or detractors. Neither of those groups is big on critical thinking, and they dominate most conversation.
Time/effort coming from experts who want to make things accessible is a gift! The paper is linked elsewhere in the thread if you want no-frills.
So, this is either the paper of the year, or ... definitely not the paper of the year.
https://arxiv.org/pdf/2605.01172 is the current version. The money graphs are page 8 and on where they show (some weirdly thick) line charts with loss results reached in roughly 1/5 the number of steps that Adam takes, just what the blog post mentions.
They also claim holding back test data is not needed, also with more graphs.
I'm not an ML scientist, and I did not attempt to seriously parse the math. It reads to me as something precisely in that liminal space some math papers do where there's enough new terminology that actually parsing through it all is going to take real, concerted effort, possibly with mild brain damage as a risk.
Their 3d graphs of "kernel eigenstructure" also do double duty for me as totally impenetrable and possibly part of an April fool's ML paper that's hilarious to insiders. Or maybe they show something really amazing; they definitely seem to converge into a shape...What does that shape mean??? Why??? What is an eigenstructure? Is it just 3D eigenvectors of some matrices? Is it natural to have a 3D shape representing these large matrices? If not, how and why were these projected down? And why are they different colors in the paper?? You get the feel for my level of understanding.
I think it would frankly just be easier to validate this claim than parse the whole paper. If only I could understand
Well enough to implement the one line update to Adam in python. I have not asked codex or claude to assist yet.Also of note to me, they talk about grokking which I found SUUUPER fascinating when it was first reported, and have never heard about since. So I was really glad to read about it and read that there has been a little academic work on the phenomenon.
Finally, of the three models they repot results on, two are extremely tiny, the last is a DPO round on Qwen 0.5B -- if the code for that is published, I imagine it would be easy to adapt and evaluate in other regimes.
You don't need to understand that part of the derivation to implement it. You just need Algorithm 1 on page 33 of the paper. Or look the author's implementation: https://github.com/elonlit/PopRiskMinimization/blob/main/pop...
Thanks for the link - I did not see a GitHub.
So, your thoughts on the paper?
[flagged]
Where’s the theory of how the human brain does what it does? Maybe these high dimensional structures don’t have a nice compact “theory”. Trying to fit these systems into a nice compact theory is a very human thing, but not everything works like that.