>If the Hessian-vector product is Hv for some fixed vector v, we're interested in solving Hx=v for x. The hope is to soon use this as a preconditioner to speed up stochastic gradient descent.
Silly question, but if you have some clever way to compute the inverse Hessian, why not go all the way and use it for Newton's method, rather than as a preconditioner for SGD?
Good q. The method computes Hessian-inverse on a batch. When people say "Newton's method" they're often thinking H^{-1} g, where both the Hessian and the gradient g are on the full dataset. I thought saying "preconditioner" instead of "Newton's method" would make it clear this is solving H^{-1} g on a batch, not on the full dataset.
Just a heads up in case you didn't know, taking the Hessian over batches is indeed referred to as Stochastic Newton, and methods of this kind have been studied for quite some time. Inverting the Hessian is often done with CG, which tends to work pretty well. The only problem is that the Hessian is often not invertible so you need a regularizer (same as here I believe). Newton methods work at scale, but no-one with the resources to try them at scale seems to be aware of them.
It's an interesting trick though, so I'd be curious to see how it compares to CG.
Absolutely plausible (BFGS is awesome), but this is situation dependent (no free lunch and all that). In the context of training neural networks, it gets even more complicated when one takes implicit regularisation coming from the optimizer into account. It's often worthwhile to try a SGD-type optimizer, BFGS, and a Newton variant to see which type works best for a particular problem.
I am not a mathematician, but I do enough weird stuff that I encounter things referring to Hessians, yet I don't really know what they are, because everyone who writes about them does so in terms that assumes the reader knows what they are.
In the context of optimizing parameters of a model, the Gradient consists of all the derivatives of the output being optimized (i.e. the total error measure) with respect to each of the models parameters.
This creates a simplified version of the model, linearized around its current parameter values, making it easy to see which direction to take a small step to move the ultimate output in the direction that is desired.
And easy to see which parameters adjust the desired output more vs. less.
The Hessian consists of all 2nd order derivatives, i.e. not just slope, but the curvature of the model, around the current parameter values.
Calculating all the first and 2nd degree derivatives takes more calculations and memory, but allows for more information as to which direction to take a learning step. As not only do we know how the output will respond linearly to a small parameter change, but whether larger changes will produce higher or lower than linear responses.
This can allow for the calculation of much larger changes to parameters, with high output improvements, speeding up training considerably, per training step.
But the trade off is each learning step requires more derivative calculations and memory. So a conducive model architecture, and clever tricks, are often needed to make the Hessian worth using, on larger models.
Another derivative type is the Jacobian, which is the derivate of every individual output (i.e. all those numbers we normally think of as the outputs, not just the final error measure), with respect to every parameter.
Jacobians can become enormous matrices. For billions of parameters, on billions of examples, with 100's of output elements, we would get a billions x 100's of billions derivative matrix. So the Jacobians calculation can take enormous amounts of extra computation and memory. But there are still occasions (much fewer) when using it can radically speed up training.
[EDIT] NxQxM 1st derivative matrix, N = #parameters, Q = #samples, M = #output elements
At this point, we have enough computer power and memory available, that all small enough problems should be trained with Jacobians in my view. Levenberg-Marquardt is an optimization algorithm that uses Jacobians. It can be orders of magnitude faster than gradient descent.
If I understand it in a nutshell. If Gradient is the angle Hessian is the curvature.
and Jacobians let you know how much weights contributed to the blue component of something identified as a big blue cat.
I think.
Jacobians look like they could be used to train concept splitters. For instance if an LLM has a grab bag of possible conversation paths, the final embedding would have information for each path, but once the selection is made it could filter the embedding to that path, which would be beneficial for chain of thought using the filtered embedding instead of the predicted token. I always wondered how much the thinking in embedding space carried around remnants of conversation paths not taken.
You explain well so what I never understood is how the Jacobians aren't the first derivatives themselves?
Also if you have happen to have any suggestions for linear algebra for someone who uses it without really understanding it (I can write a measurement function for an EKF from scratch OK, but I don't really understand why the maths does what it does) I would really appreciate it.
The Jacobian is first derivatives, but for a function mapping N to M dimensions. It's the first derivative of every output wrt every input, so it will be an N x M matrix.
The gradient is a special case of the Jacobian for functions mapping N to 1 dimension, such as loss functions. The gradient is an N x 1 vector.
[EDIT] Updated original comment to include matrix dimensions.
If you want a serious text that goes through the relevant linear algebra and optimization mathematics in depth up front, Neural Network Design, 2nd edition is a good one. [Disclaimer, co-author]. We took great pains to walk through every conceptual and mathematical topic before we apply those concepts to machine learning. We use MATLAB a lot, which may or may not be helpful.
Another potential option is "Linear Algebra and Optimization for Machine Learning", which looks good and also starts out with linear algebra before machine learning. I haven't read it, but the first 2020 edition gets good reviews, and a second 2026 edition just came out, apparently with a fair amount of positive revision. Given the speed of change, that's nice to see.
Great work. Making the Hessian calculation linear in depth is a solid intermediate step. Thanks for sharing this; I look forward to seeing the final results as this research matures.
I haven't looked into it in years, but would the inverse of a block bi-diagonal matrix have some semiseperable structure? Maybe that would be good to look into?
just to be clear, semiseparate in this context means H = D + CC', where D is block diagonal and C is tall & skinny?
If so, it would be nice if this were the case, because you could then just use the Woodbury formula to invert H. But I don't think such a decomposition exists. I tried to exhaustively search through all the decompositions of H that involved one dummy variable (of which the above is a special case) and I couldn't find one. I ended up having to introduce two dummy variables instead.
> just to be clear, semiseparate in this context means H = D + CC', where D is block diagonal and C is tall & skinny?
Not quite, it means any submatrix taken from the upper(lower) part of the matrix has some low rank. Like a matrix is {3,4}-semiseperable if any sub matrix taken from the lower triangular part has at most rank 3 and any submatrix taken from the upper triangular part has at most rank 4.
The inverse of an upper bidiagonal matrix is {0,1}-semiseperable.
There are a lot of fast algorithms if you know a matrix is semiseperable.
Agreed. But these things have a way of not working out, and one the sadness, one forgets to celebrate the intermediate victories. I wanted to share an intermediate victory before reality crushes the joy.
>If the Hessian-vector product is Hv for some fixed vector v, we're interested in solving Hx=v for x. The hope is to soon use this as a preconditioner to speed up stochastic gradient descent.
Silly question, but if you have some clever way to compute the inverse Hessian, why not go all the way and use it for Newton's method, rather than as a preconditioner for SGD?
Good q. The method computes Hessian-inverse on a batch. When people say "Newton's method" they're often thinking H^{-1} g, where both the Hessian and the gradient g are on the full dataset. I thought saying "preconditioner" instead of "Newton's method" would make it clear this is solving H^{-1} g on a batch, not on the full dataset.
Just a heads up in case you didn't know, taking the Hessian over batches is indeed referred to as Stochastic Newton, and methods of this kind have been studied for quite some time. Inverting the Hessian is often done with CG, which tends to work pretty well. The only problem is that the Hessian is often not invertible so you need a regularizer (same as here I believe). Newton methods work at scale, but no-one with the resources to try them at scale seems to be aware of them.
It's an interesting trick though, so I'd be curious to see how it compares to CG.
[1] https://arxiv.org/abs/2204.09266 [2] https://arxiv.org/abs/1601.04737 [3] https://pytorch-minimize.readthedocs.io/en/latest/api/minimi...
For solving physics equations there is also Jacobian-free Newton-Krylov methods.
Yes the combination of Krylov and quasi-Newton methods are very successful for physics problems (https://en.wikipedia.org/wiki/Quasi-Newton_method).
Iirc eg GMRES is a popular Krylov subspace method.
I lately used these methods and BFGS worked better than CG for me.
Absolutely plausible (BFGS is awesome), but this is situation dependent (no free lunch and all that). In the context of training neural networks, it gets even more complicated when one takes implicit regularisation coming from the optimizer into account. It's often worthwhile to try a SGD-type optimizer, BFGS, and a Newton variant to see which type works best for a particular problem.
I'd call it "Stochastic Newton's Method" then. :-)
fair. thanks. i'll sleep on it and update the paper if it still sounds right tomorrow.
probably my nomenclature bias is that i started this project as a way to find new preconditioners on deep nets.
I am not a mathematician, but I do enough weird stuff that I encounter things referring to Hessians, yet I don't really know what they are, because everyone who writes about them does so in terms that assumes the reader knows what they are.
Any hints? The Battenburg graphics of matrices?
This helped me, coming from an ml background: https://randomrealizations.com/posts/xgboost-explained/
GRADIENT
In the context of optimizing parameters of a model, the Gradient consists of all the derivatives of the output being optimized (i.e. the total error measure) with respect to each of the models parameters.
This creates a simplified version of the model, linearized around its current parameter values, making it easy to see which direction to take a small step to move the ultimate output in the direction that is desired.
And easy to see which parameters adjust the desired output more vs. less.
[EDIT] Nx1 1st derivative vector, N = #parameters, 1 = scalar output.
HESSIAN
The Hessian consists of all 2nd order derivatives, i.e. not just slope, but the curvature of the model, around the current parameter values.
Calculating all the first and 2nd degree derivatives takes more calculations and memory, but allows for more information as to which direction to take a learning step. As not only do we know how the output will respond linearly to a small parameter change, but whether larger changes will produce higher or lower than linear responses.
This can allow for the calculation of much larger changes to parameters, with high output improvements, speeding up training considerably, per training step.
But the trade off is each learning step requires more derivative calculations and memory. So a conducive model architecture, and clever tricks, are often needed to make the Hessian worth using, on larger models.
[EDIT] NxNx1 = NxN 2nd derivative matrix, N = #parameters, 1 = scalar output.
JACOBIAN
Another derivative type is the Jacobian, which is the derivate of every individual output (i.e. all those numbers we normally think of as the outputs, not just the final error measure), with respect to every parameter.
Jacobians can become enormous matrices. For billions of parameters, on billions of examples, with 100's of output elements, we would get a billions x 100's of billions derivative matrix. So the Jacobians calculation can take enormous amounts of extra computation and memory. But there are still occasions (much fewer) when using it can radically speed up training.
[EDIT] NxQxM 1st derivative matrix, N = #parameters, Q = #samples, M = #output elements
At this point, we have enough computer power and memory available, that all small enough problems should be trained with Jacobians in my view. Levenberg-Marquardt is an optimization algorithm that uses Jacobians. It can be orders of magnitude faster than gradient descent.
Thank you very much for this description.
If I understand it in a nutshell. If Gradient is the angle Hessian is the curvature.
and Jacobians let you know how much weights contributed to the blue component of something identified as a big blue cat.
I think.
Jacobians look like they could be used to train concept splitters. For instance if an LLM has a grab bag of possible conversation paths, the final embedding would have information for each path, but once the selection is made it could filter the embedding to that path, which would be beneficial for chain of thought using the filtered embedding instead of the predicted token. I always wondered how much the thinking in embedding space carried around remnants of conversation paths not taken.
You explain well so what I never understood is how the Jacobians aren't the first derivatives themselves?
Also if you have happen to have any suggestions for linear algebra for someone who uses it without really understanding it (I can write a measurement function for an EKF from scratch OK, but I don't really understand why the maths does what it does) I would really appreciate it.
The Jacobian is first derivatives, but for a function mapping N to M dimensions. It's the first derivative of every output wrt every input, so it will be an N x M matrix.
The gradient is a special case of the Jacobian for functions mapping N to 1 dimension, such as loss functions. The gradient is an N x 1 vector.
[EDIT] Updated original comment to include matrix dimensions.
If you want a serious text that goes through the relevant linear algebra and optimization mathematics in depth up front, Neural Network Design, 2nd edition is a good one. [Disclaimer, co-author]. We took great pains to walk through every conceptual and mathematical topic before we apply those concepts to machine learning. We use MATLAB a lot, which may or may not be helpful.
Another potential option is "Linear Algebra and Optimization for Machine Learning", which looks good and also starts out with linear algebra before machine learning. I haven't read it, but the first 2020 edition gets good reviews, and a second 2026 edition just came out, apparently with a fair amount of positive revision. Given the speed of change, that's nice to see.
Great work. Making the Hessian calculation linear in depth is a solid intermediate step. Thanks for sharing this; I look forward to seeing the final results as this research matures.
I haven't looked into it in years, but would the inverse of a block bi-diagonal matrix have some semiseperable structure? Maybe that would be good to look into?
just to be clear, semiseparate in this context means H = D + CC', where D is block diagonal and C is tall & skinny?
If so, it would be nice if this were the case, because you could then just use the Woodbury formula to invert H. But I don't think such a decomposition exists. I tried to exhaustively search through all the decompositions of H that involved one dummy variable (of which the above is a special case) and I couldn't find one. I ended up having to introduce two dummy variables instead.
> just to be clear, semiseparate in this context means H = D + CC', where D is block diagonal and C is tall & skinny?
Not quite, it means any submatrix taken from the upper(lower) part of the matrix has some low rank. Like a matrix is {3,4}-semiseperable if any sub matrix taken from the lower triangular part has at most rank 3 and any submatrix taken from the upper triangular part has at most rank 4.
The inverse of an upper bidiagonal matrix is {0,1}-semiseperable.
There are a lot of fast algorithms if you know a matrix is semiseperable.
edit: link https://people.cs.kuleuven.be/~raf.vandebril/homepage/public...
thanks for the explanation! sorry i had misread the AI summary on "semiseparable".
i need to firm my intuition on this first before i can say anything clever, but i agree it's worth thinking about!
Would be great to see this work continued with some training runs
Agreed. But these things have a way of not working out, and one the sadness, one forgets to celebrate the intermediate victories. I wanted to share an intermediate victory before reality crushes the joy.
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