This is less about numerical instability and more that iterative algorithms with error control their error, but when you run AD on them you are ADing the approximation and a derivative of an approximation can be arbitrarily different from an approximation of a derivative.
Yeah, perhaps the actual title would be better: "The Numerical Analysis of Differentiable Simulation". (Rather than the subtitle, which is itself a poor rewording of the actual subtitle in the video.)
It can be both. A mistake in AD primitives can lead to theoretically incorrect derivatives. With the system I use I have run into a few scenarios where edge cases aren't totally covered leading to the wrong result.
> It's also not necessarily immediately obvious that the derivatives ARE wrong if the implementation is wrong.
It's neither full proof or fool proof but an absolute must is a check that the loss function is reducing. It quickly detects a common error that the sign came out wrong in my gradient call. Part of good practice one learns in grad school.
I have been using sympy while learning electron physics to automatically integrate linear charge densities. It works great symbolically, but often fails silently when the symbols are substituted with floats before integration.
With floats getting smaller and smaller in ML, It's hard to imagine anyone failing to learn this as one of their early experiences in the field.
The focus should not be on the possibility of error, but managing the error to be within acceptable limits. There's a hour long video there, and it's 3am, so I'm not sure how much of this covers management. Anyone familiar with it care to say?
Floats are getting smaller and smaller only in neural networks - which works ok because NNs are designed for nice easy backprop. Normalization and skip connections really help avoid numerical instability.
The author is talking about ODEs and PDEs, which come from physical systems and don't have such nice properties.
I'm not sure your claim is correct, nor particularly precise. Skip connections do propagate feature maps from earlier to later/deeper in the network, and one can model this mathematically as using an identity function to do this, but I wouldn't say that this is "about preserving identity". Also, the reason the identity is useful here is because it prevents vanishing gradients (due to the identity function having an identity tensor as the gradient). Arguably, preventing vanishing gradients is exactly about numerical stability, so I would in fact argue that skip connections are there to make gradient descent of certain architectures more numerically stable.
Interesting to see this here! This example is one of the ones mentioned in the appendix of https://arxiv.org/abs/2406.09699, specifically "4.1.2.4 When AD is algorithmically correct but numerically wrong".
If people want a tl;dr, the main idea is you can construct an ODE where the forward pass is trivial, i.e. the ODE solution going forwards is exact, but its derivative is "hard". An easy way to do this is to make it so you have for example `x' = x - y, y' = y - x`, with initial conditions x(0)=y(0). If you start with things being the same value, then the solution to the ODE is constant since `x' = y' = 0`. But the derivative of the ODE solution with respect to its initial condition is very non-zero: a small change away from equality and boom the solution explodes. You write out the expression for dy(t)/dy(0) and what you get is a non-trivial ODE that has to be solved.
What happens in this case though is that automatic differentiation "runs the same code as the primal case". I.e., automatic differentiation has the property that it walks through your code and differentiates the steps of your code, and so the control flow always matches the control flow of the non-differentiated code, slapping the derivative parts on each step. But here the primal case is trivial, so the ODE solver goes "this is easy, let's make dt as big as possible and step through this easily". But this means that `dt` is not error controlled in the derivative (the derivative, being a non-trivial ODE, needs to have a smaller dt and take small steps in order to get an accurate answer), so the derivative then gets error due to this large dt (or small number of steps). Via this construction you can make automatic differentiation give as much error as you want just by tweaking the parameters around.
Thus by this construction, automatic differentiation has no bound to the error it can give, and no this isn't floating point errors this is strictly building a function that does not converge to the correct derivative. It just goes to show that automatic differentiation of a function that computes a nice error controlled answer does not necessarily give a derivative that also has any sense of error control, that is a property that has to be proved and is not true in general. This of course is a bit of a contrived example to show the point that the error is unbounded, but then it points to real issues that can show up in user code (in fact, this example was found because a user opened an issue with a related model).
Then one thing that's noted in here too is that the Julia differential equation solvers hook into the AD system to explicitly "not do forward-mode AD correctly", incorporating the derivative terms into the time stepping adaptivity calculation, so that it is error controlled. The property that you get is that for these solvers you get more steps to the ODE when running it in AD mode than outside of AD mode, and that is a requirement if you want to ensure that the user's tolerance is respected. But that's explicitly "not correct" in terms of what forward-mode AD is, so calling forward-mode AD on the solver doesn't quite do forward mode AD of the solver's code "correctly" in order to give a more precise solution. That of course is a choice, you could instead choose to follow standard AD rules in such a code. The trade-off is between accuracy and complexity.
I don't think this is a failure of Automatic Differentiation. It's a failure of assuming that the derivative of an approximation is equal to an approximation of the derivative. That's obviously nonsense, as the example f(x) = atan(10000 x)/10000, which is nearly 0, shows you. This was first pointed out here in this comment: https://news.ycombinator.com/item?id=45293066
The way I explained it to myself in the past why so much of the CUDA algorithms don't care much about numerical stability is that the error is a form of regularization (i.e. less overfitting over the data) in deep learning.
But reasons why deep learning training is very robust to moderate inaccuracy in gradients:
1. Locally, sigmoid and similar functions are the simplest smoothest possible non-linearity to propagate gradients through.
2. Globally, outside of deep recurrent networks, there is no recursion which makes the total function smooth and well behaved.
3. While the perfect gradient indicates the ideal direction to adjust parameters, for fastest improvement, all that is really needed to reduce error is to move parameters in the direction of the gradient signs, with a small enough step. That is a very low bar.
It's like telling an archer they just need to shoot an arrow so it lands closer to the target than where the archer is standing, but not worry about hitting it!
4. Finally, the perfect first order gradient is only meaningful at one point of the optimization surface. Moving away from that point, i.e. updating the parameters at all, and the gradient changes quickly.
So we are in gradient heuristic land even with "perfect" first order gradients. The most perfectly calculated gradient isn't actually "accurate" already.
To actually get an accurate gradient over a parameter step, would take fitting the local gradient with a second or third order polynomial. I.e. not just first, but second and third order derivatives. At vastly greater computational and working memory cost.
--
The only critical issue for calculating gradients, is that there is enough precision that at least directional gradient information makes it from errors back to the parameters to update. If precision is too low, then the variable magnitude rounding inherent to floating point arithmetic can completely drop directional information for smaller gradients. Without accurate gradient signs, learning stalls.
Typically for matrix multiplications there is a wide range of algorithms you could use to compute it, on one extreme end you could use numerically stable summation and the other extreme you could have tiled matmul with FP8, the industry trend seems to go further away from numerical stable algorithms without much quality drop it seems. My claim is probably unfair since it ignores the scale you gain from the speed/precision tradeoff, so I assumed numerical stability is not that beneficial compared to something precision heavy like physics simulation in HPC.
This article is saying that it can be numerically unstable in certain situations, not that it's theoretically incorrect.
This is less about numerical instability and more that iterative algorithms with error control their error, but when you run AD on them you are ADing the approximation and a derivative of an approximation can be arbitrarily different from an approximation of a derivative.
That makes more sense. The title is flat out wrong IMO.
Yeah, perhaps the actual title would be better: "The Numerical Analysis of Differentiable Simulation". (Rather than the subtitle, which is itself a poor rewording of the actual subtitle in the video.)
It can be both. A mistake in AD primitives can lead to theoretically incorrect derivatives. With the system I use I have run into a few scenarios where edge cases aren't totally covered leading to the wrong result.
I have also run into numerical instability too.
> A mistake in AD primitives can lead to theoretically incorrect derivatives
Ok but that's true of any program. A mistake in the implementation of the program can lead to mistakes in the result of the program...
That's true! But it's also true that any program dealing with floats can run into numerical instability if care isn't taken to avoid it, no?
It's also not necessarily immediately obvious that the derivatives ARE wrong if the implementation is wrong.
You can pretty concretely and easily check that the AD primatives are correct by comparing them to numerical differentiation.
> It's also not necessarily immediately obvious that the derivatives ARE wrong if the implementation is wrong.
It's neither full proof or fool proof but an absolute must is a check that the loss function is reducing. It quickly detects a common error that the sign came out wrong in my gradient call. Part of good practice one learns in grad school.
I have been using sympy while learning electron physics to automatically integrate linear charge densities. It works great symbolically, but often fails silently when the symbols are substituted with floats before integration.
https://github.com/sympy/sympy/issues/27675
With floats getting smaller and smaller in ML, It's hard to imagine anyone failing to learn this as one of their early experiences in the field.
The focus should not be on the possibility of error, but managing the error to be within acceptable limits. There's a hour long video there, and it's 3am, so I'm not sure how much of this covers management. Anyone familiar with it care to say?
Floats are getting smaller and smaller only in neural networks - which works ok because NNs are designed for nice easy backprop. Normalization and skip connections really help avoid numerical instability.
The author is talking about ODEs and PDEs, which come from physical systems and don't have such nice properties.
Skip connections are about preserving identity, not to make it more numerically stable.
I'm not sure your claim is correct, nor particularly precise. Skip connections do propagate feature maps from earlier to later/deeper in the network, and one can model this mathematically as using an identity function to do this, but I wouldn't say that this is "about preserving identity". Also, the reason the identity is useful here is because it prevents vanishing gradients (due to the identity function having an identity tensor as the gradient). Arguably, preventing vanishing gradients is exactly about numerical stability, so I would in fact argue that skip connections are there to make gradient descent of certain architectures more numerically stable.
[video]?
Interesting to see this here! This example is one of the ones mentioned in the appendix of https://arxiv.org/abs/2406.09699, specifically "4.1.2.4 When AD is algorithmically correct but numerically wrong".
If people want a tl;dr, the main idea is you can construct an ODE where the forward pass is trivial, i.e. the ODE solution going forwards is exact, but its derivative is "hard". An easy way to do this is to make it so you have for example `x' = x - y, y' = y - x`, with initial conditions x(0)=y(0). If you start with things being the same value, then the solution to the ODE is constant since `x' = y' = 0`. But the derivative of the ODE solution with respect to its initial condition is very non-zero: a small change away from equality and boom the solution explodes. You write out the expression for dy(t)/dy(0) and what you get is a non-trivial ODE that has to be solved.
What happens in this case though is that automatic differentiation "runs the same code as the primal case". I.e., automatic differentiation has the property that it walks through your code and differentiates the steps of your code, and so the control flow always matches the control flow of the non-differentiated code, slapping the derivative parts on each step. But here the primal case is trivial, so the ODE solver goes "this is easy, let's make dt as big as possible and step through this easily". But this means that `dt` is not error controlled in the derivative (the derivative, being a non-trivial ODE, needs to have a smaller dt and take small steps in order to get an accurate answer), so the derivative then gets error due to this large dt (or small number of steps). Via this construction you can make automatic differentiation give as much error as you want just by tweaking the parameters around.
Thus by this construction, automatic differentiation has no bound to the error it can give, and no this isn't floating point errors this is strictly building a function that does not converge to the correct derivative. It just goes to show that automatic differentiation of a function that computes a nice error controlled answer does not necessarily give a derivative that also has any sense of error control, that is a property that has to be proved and is not true in general. This of course is a bit of a contrived example to show the point that the error is unbounded, but then it points to real issues that can show up in user code (in fact, this example was found because a user opened an issue with a related model).
Then one thing that's noted in here too is that the Julia differential equation solvers hook into the AD system to explicitly "not do forward-mode AD correctly", incorporating the derivative terms into the time stepping adaptivity calculation, so that it is error controlled. The property that you get is that for these solvers you get more steps to the ODE when running it in AD mode than outside of AD mode, and that is a requirement if you want to ensure that the user's tolerance is respected. But that's explicitly "not correct" in terms of what forward-mode AD is, so calling forward-mode AD on the solver doesn't quite do forward mode AD of the solver's code "correctly" in order to give a more precise solution. That of course is a choice, you could instead choose to follow standard AD rules in such a code. The trade-off is between accuracy and complexity.
I don't think this is a failure of Automatic Differentiation. It's a failure of assuming that the derivative of an approximation is equal to an approximation of the derivative. That's obviously nonsense, as the example f(x) = atan(10000 x)/10000, which is nearly 0, shows you. This was first pointed out here in this comment: https://news.ycombinator.com/item?id=45293066
Thank you for this good description.
The way I explained it to myself in the past why so much of the CUDA algorithms don't care much about numerical stability is that the error is a form of regularization (i.e. less overfitting over the data) in deep learning.
I am not quite sure what that means! :)
But reasons why deep learning training is very robust to moderate inaccuracy in gradients:
1. Locally, sigmoid and similar functions are the simplest smoothest possible non-linearity to propagate gradients through.
2. Globally, outside of deep recurrent networks, there is no recursion which makes the total function smooth and well behaved.
3. While the perfect gradient indicates the ideal direction to adjust parameters, for fastest improvement, all that is really needed to reduce error is to move parameters in the direction of the gradient signs, with a small enough step. That is a very low bar.
It's like telling an archer they just need to shoot an arrow so it lands closer to the target than where the archer is standing, but not worry about hitting it!
4. Finally, the perfect first order gradient is only meaningful at one point of the optimization surface. Moving away from that point, i.e. updating the parameters at all, and the gradient changes quickly.
So we are in gradient heuristic land even with "perfect" first order gradients. The most perfectly calculated gradient isn't actually "accurate" already.
To actually get an accurate gradient over a parameter step, would take fitting the local gradient with a second or third order polynomial. I.e. not just first, but second and third order derivatives. At vastly greater computational and working memory cost.
--
The only critical issue for calculating gradients, is that there is enough precision that at least directional gradient information makes it from errors back to the parameters to update. If precision is too low, then the variable magnitude rounding inherent to floating point arithmetic can completely drop directional information for smaller gradients. Without accurate gradient signs, learning stalls.
Typically for matrix multiplications there is a wide range of algorithms you could use to compute it, on one extreme end you could use numerically stable summation and the other extreme you could have tiled matmul with FP8, the industry trend seems to go further away from numerical stable algorithms without much quality drop it seems. My claim is probably unfair since it ignores the scale you gain from the speed/precision tradeoff, so I assumed numerical stability is not that beneficial compared to something precision heavy like physics simulation in HPC.