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Comment by stephanheijl | Hacker News Reader

To be more exact, LoRA adds two matrices `A` and `B` to any layers that contain trainable weights. The original weights (`W_0`) have the shape `d × k`. These are frozen. Matrix `A` has dimensions `d × <rank>` (`rank` is configurable) and matrix `B` has the shape `<rank> × k`. A and B are then multiplied and added to `W_0` to get altered weights. The benefit here is that the extra matrices are small compared to `W_0`, which means less parameters need to be optimized, so less activations need to be stored in memory.

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Ah, so the resulting model contains both the large matrix of original weights, and also the two small matrices of alterations? But this is smaller than the alternative of a model which contains the large matrix of original weights, and an equally large matrix of alterations.

Why is fine-tuning done with separate alterations, rather than by mutating the original weights?

> But this is smaller than the alternative of a model which contains the large matrix of original weights, and an equally large matrix of alterations.

It's actually larger. If you just have two equally large matrices of the same dimension, one original, and one of "altercations"... then you can just add them together.

> Why is fine-tuning done with separate alterations, rather than by mutating the original weights?

Then you'd have to compute the gradients for the whole network, which is very expensive when the model has 7b, 65b, 165b parameters. The intent is to make that cheaper by only computing gradients for a low rank representation of the change in the weight matrix from training.

Correct me if I'm wrong, but I think you still need to compute gradients of non-trained weights in order to compute the gradients of the LoRA weights. What you don't have to do is store and update the optimizer state for all those non-trained weights.

I mean the derivative of a constant is 0. So if all of the original weights are considered constants, then computing their gradients is trivial, since they’re just zero.

Computing gradients is easy/cheap. What this technique solves is that you no longer need to store the computed values of the gradient until the backpropagation phase, which saves on expensive GPU RAM, allowing you to use commodity hardware.

>Then you'd have to compute the gradients for the whole network

You have to do that with LoRA regardless, to compute the gradients for the lowest-level LoRA weights.

It's larger, but there are less parameters to train for your specific use case since you are training the small matrix only, while the original ones remain unaltered.

> Why is fine-tuning done with separate alterations, rather than by mutating the original weights?

The goal of most parameter-efficient methods is to store one gold copy of the original model, and learn minor modifications/additions to the model. The easiest way to think about this is in some kind of deployment setting, where you have 1 capable model and you learn different sets of LoRA weights for different tasks and applications.

The original intent of parameter-efficient methods is to reduce the amount of storage space needed for models (do you really want to keep a whole additional copy of LLaMA for each different task?). A secondary benefit is that because you are fine-tuning a smaller number of parameters, the optimizer states (can take up to 2x the size of your model) are also heavily shrunk, which makes it more economical (memory-wise) to (parameter-efficient) fine-tune your model.

That’s probably what OpenAI does with their custom fine tuned models, no?

Can rank decomposition be used to reduce the original weight matrices as well? Or are they assumed to be compressed already?

Those fully trained networks are usually considered full-rank. At least that is what they say in the LoRA paper.

Your explanation is crystal clear. I suppose it works well in practice, but is there any reason it works that well?

Per the original paper, empirically it’s been found that neural network weights often have low intrinsic rank. It follows, then, that the change in the weights as you train also have low intrinsic rank, which means that you should be able represent them with a lower rank matrix.

Since we are in ELI5, it seems that the concept of low rank approximation is required to understand this method.

(1) https://en.wikipedia.org/wiki/Low-rank_approximation

Edited: By the way, it seems to me that there is an error in the wikipedia page because if the Low-rank approximation takes a larger rank then the bound of the error should decrease, and in this page the error increases.

>> that the change in the weights as you train also have low intrinsic rank

It seems that the initial matrix of weights has a low rank approximation A and this implies that the difference E = W - A is small, also it seems that PCA fails when E is sparse because PCA is designed to be optimum when the error is gaussian.

In terms of PCA, PCA is also quite expensive computationally. Additionally, you'd probably have to do SVD instead.

Since the weights are derived from gradient descent, yeah we don't really know what the distributions would be.

A random projection empirically works quite well for very high dimensions, and is of course very cheap computationally.

Does this mean the matrices are highly compressible?

kinda/yes. To translate to more intuitive concepts: the matrices don't contain much variance in as many degrees of freedom as they could.

Think of a point cloud of a piece of paper floating in the wind. It would be a 3xn list of points, but "really" it's a 2d piece of paper.

Just like I can rewrite the number 27 as 333 or 8+19 or (2^3)+(2^4)+3.. Given a single matrix one can find myriad ways to rewrite it as a sequence of matrices that have the same (or similar) numeric value, but with interesting or desirable properties. :D

My favorite example (which is used in signal processing) is to take your ugly matrix and rewrite it as a set of smaller matrices where most of the elements are zero, or a power of 2.

It turns out, computers can multiply by zeros and powers of two very fast

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