[altairprime]

For those trying to understand the most important parts of the paper, here's what I think is the most significant two statements, subquoted out of two (consecutive) paragraphs midway through the paper:

> we selected five additional, previously unseen pretrained ViT models for which we had access to evaluation data. These models, considered out-of-domain relative to the initial set, had all their weights reconstructed by projecting onto the identified 16-dimensional universal subspace. We then assessed their classification accuracy and found no significant drop in performance

> we can replace these 500 ViT models with a single Universal Subspace model. Ignoring the task-variable first and last layer [...] we observe a requirement of 100 × less memory, and these savings are prone to increase as the number of trained models increases. We note that we are, to the best of our knowledge, the first work, to be able to merge 500 (and theoretically more) Vision Transformer into a single universal subspace model. This result implies that hundreds of ViTs can be represented using a single subspace model

So, they found an underlying commonality among the post-training structures in 50 LLaMA3-8B models, 177 GPT-2 models, and 8 Flan-T5 models; and, they demonstrated that the commonality could in every case be substituted for those in the original models with no loss of function; and noted that they seem to be the first to discover this.

For a tech analogy, imagine if you found a bzip2 dictionary that reduced the size of every file compressed by 99%, because that dictionary turns out to be uniformly helpful for all files. You would immediately open a pull request to bzip2 to have the dictionary built-in, because it would save everyone billions of CPU hours. [*]

[*] Except instead of 'bzip2 dictionary' (strings of bytes), they use the term 'weight subspace' (analogy not included here[**]) — and, 'file compression' hours becomes 'model training' hours. It's just an analogy.

[**] 'Hilbert subspaces' is just incorrect enough to be worth appending as a footnote[***].

[***] As a second footnote.

[tsurba]

Edit: actually this paper is the canonical reference (?): https://arxiv.org/abs/2007.00810 models converge to same space up to a linear transformation. Makes sense that a linear transformation (like PCA) would be able to undo that transformation.

You can show for example that siamese encoders for time-series, with MSE loss on similarity, without a decoder, will converge to the the same latent space up to orthogonal transformations (as MSE is kinda like gaussian prior which doesn’t distinguish between different rotations).

Similarly I would expect that transformers trained on the same loss function for predicting the next word, if the data is at all similar (like human language), would converge to approx the same space, up to some, likely linear, transformations. And to represent that same space probably weights are similar, too. Weights in general seem to occupy low-dimensional spaces.

All in all, I don’t think this is that surprising, and I think the theoretical angle should be (have been?) to find mathematical proofs like this paper https://openreview.net/forum?id=ONfWFluZBI

They also have a previous paper (”CEBRA”) published in Nature with similar results.

[westoncb]

> So, they found an underlying commonality among the post-training structures in 50 LLaMA3-8B models, 177 GPT-2 models, and 8 Flan-T5 models; and, they demonstrated that the commonality could in every case be substituted for those in the original models with no loss of function; and noted that they seem to be the first to discover this.

Could someone clarify what this means in practice? If there is a 'commonality' why would substituting it do anything? Like if there's some subset of weights X found in all these models, how would substituting X with X be useful?

I see how this could be useful in principle (and obviously it's very interesting), but not clear on how it works in practice. Could you e.g. train new models with that weight subset initialized to this universal set? And how 'universal' is it? Just for like like models of certain sizes and architectures, or in some way more durable than that?

[farhanhubble]

It might we worth it to use that subset to initialize the weights of future models but more importantly you could save a huge number of computational cycles by using the lower dimensional weights at the time of inference.

[westoncb]

Ah interesting, I missed that possibility. Digging a little more though my understanding is that what's universal is a shared basis in weight space, and particular models of same architecture can express their specific weights via coefficients in a lower-dimensional subspace using that universal basis (so we get weight compression, simplified param search). But it also sounds like to what extent there will be gains during inference is in the air?

Key point being: the parameters might be picked off a lower dimensional manifold (in weight space), but this doesn't imply that lower-rank activation space operators will be found. So translation to inference-time isn't clear.

[farhanhubble]

My understanding differs and I might be wrong. Here's what I inferred:

Let's say you finetune a Mistral-7B. Now, there are hundreds of other fine-tuned Mistral-7B's, which means it's easy to find the universal subspace U of the weights of all these models combined. You can then decompose the weights of your specific model using U and a coefficient matrix C specific to your model. Then you can convert any operation of the type `out=Wh` to `out=U(C*x)` Both U and C are much smaller dimension that W and so the number of matrix operations as well as the memory required is drastically lower.

[altairprime]

Prior to this paper, no one knew that X existed. If this paper proves sound, then now we know that X exists at all.

No matter how large X is, one copy of X baked into the OS / into the silicon / into the GPU / into CUDA, is less than 50+177+8 copies of X baked into every single model. Would that permit future models to be shipped with #include <X.model> as line 1? How much space would that save us? Could X.model be baked into chip silicon so that we can just take it for granted as we would the mathlib constant "PI"? Can we hardware-accelerate the X.model component of these models more than we can a generic model, if X proves to be a 'mathematical' constant?

Given a common X, theoretically, training for models could now start from X rather than from 0. The cost of developing X could be brutal; we've never known to measure it before. Thousands of dollars of GPU per complete training at minimum? Between Google, Meta, Apple, and ChatGPT, the world has probably spent a billion dollars recalculating X a million times. In theory, they probably would have spent another billion dollars over the next year calculating X from scratch. Perhaps now they won't have to?

We don't have a lot of "in practice" experience here yet, because this was first published 4 days ago, and so that's why I'm suggesting possible, plausible, ways this could help us in the future. Perhaps the authors are mistaken, or perhaps I'm mistaken, or perhaps we'll find that the human brain has X in it too. As someone who truly loathes today's "AI", and in an alternate timeline would have completed a dual-major CompSci/NeuralNet degree in ~2004, I'm extremely excited to have read this paper, and to consider what future discoveries and optimizations could result from it.

EDIT:

Imagine if you had to calculate 3.14159 from basic principles every single time you wanted to use pi in your program. Draw a circle to the buffer, measure it, divide it, increase the memory usage of your buffer and resolution of your circle if necessary to get a higher precision pi. Eventually you want pi to a billion digits, so every time your program starts, you calculate pi from scratch to a billion digits. Then, someday, someone realizes that we've all been independently calculating the exact same mathematical constant! Someone publishes Pi: An Encyclopedia (Volume 1 of ∞). It becomes inconceivably easier to render cones and spheres in computer graphics, suddenly! And then someone invents radians, because now that we can map 0..360° onto 0..τ, and no one predicted radians at all but it's incredibly obvious in hindsight.

We take for granted knowledge of things like Pi, but there was a time when we did not know it existed at all. And then for a long time it was 3. And then someone realized the underlying commonality of every circle and defined it plainly, and now we have Pi Day, and Tau Day, because not only do we know it exists, but we can argue about it. How cool is that! So if someone has discovered a new 'constant', then that's always a day of celebration in my book, because it means that we're about to see not only things we consider "possible, but difficult" to instead be "so easy that we celebrate their existence with a holiday", but also things that we could never have remotely dreamed of before we knew that X existed at all.

(In less tangible analogies, see also: postfix notation which was repeatedly invented for decades (by e.g. Dijkstra) as a programming advance, or the movie "Arrival" (2019) as a linguistic advance, or the BLIT Parrot (don't look!) as a biological advance. :)

[AIchemist]

If even remotely fact what you suggest here, I see two antipodal trajectories the authors secretly huddled and voted on:

1. As John Napier, who freely, generously, gifted his `Mirifici' for the benefit of all.

2. Here we go, patent trolls, have at it. OpenAI, et al burning midnight oil to grab as much real estate on this to erase any (even future?) debt stress, deprecating the AGI Philospher's Stone to first owning everything conceivable from a new miraculous `my precious' ring, not `open', closed.

[scotty79]

16 seems like a suspiciously round number ... why not 17 or 13? ... is this just result of some bug in the code they used to do their science?

or is it just that 16 was arbitrarily chosen by them as close enough to the actual minimal number of dimensions necessary?

[woopsn]

It's a little arbitrary. Look at the graph on page 6, there's no steep gap in the spectrum there. 16 just about the balance point

[moi2388]

But there is a steep gap in the spectrum at 16 on page 7

[yorwba]

That's the spectrum of LoRAs, which are LoW RAnk by design.

[moi2388]

Yes. But from their paper: “In our analysis, we present compelling empirical evidence for the existence of universal subspaces within LoRA adapters across different modalities and tasks.”

I also don’t understand what they write under figure 2, since resnet50 has 50 layers, not 31.

[altairprime]

There’s lots of hockey stick charts in the paper that might answer this visually, if that’s of interest.

[N_Lens]

If models naturally occupy shared spectral subspaces, this could dramatically reduce

- Training costs: We might discover these universal subspaces without training thousands of models

- Storage requirements: Models could share common subspace representations

[scotty79]

"16 dimensions is all you need" ... to do human achievable stuff at least