First I thought this would be just another gradient descent tutorial for beginners. But the article goes quite deep into gradient descent dynamics, looking into third order approximations of the loss function and eventually motivating a concept called "central flows." Their central flow model was able to predict loss graphs for various training runs across different neural network architectures.
So all the classic optimization theory about staying in the stable region is basically what deep learning doesn't do. The model literally learns by becoming unstable, oscillating, and then using that energy to self-correct.
The chaos is the point. What a crazy, beautiful mess.
Fascinating, do the gained insights allow to directly compute the central flow in order to speed up convergence? Or is this preliminary exploration to understand how it had been working?
They explicitly ignore momentum and exponentially weighted moving average, but that should result in the time-averaged gradient descent (along the valley, not across it). But that requires multiple evaluations, do any of the expressions for the central flow admit fast / computationally efficient central flow calculation?
Very neat stuff! So one question is, if we had an analog computer that could run these flows exactly, would we get better results if we ran the gradient flow or this central flow?
It's a little easier to see what's happening if you fully write out the central flow:
-1/η * dw/dt = ∇L - ∇S * ⟨∇L, ∇S⟩/‖∇S‖²
We're projecting the loss gradient onto the sharpness gradient, and subtracting it off. If you didn't read the article, the sharpness S is the sum of the eigenvalues of the Hessian of the loss that are larger than 2/η, a measure of how unstable the learning dynamics are.
This is almost Sobolev preconditioning:
-1/η * dw/dt = ∇L - ∇S = ∇(I - Δ)L
where this time S is the sum of all the eigenvalues (so, the Laplacian of L).
I apparently didn't get the memo and used stochastic gradient descent with momentum outside of deep learning without running into any problems given a sufficiently low learning rate.
I'm not really convinced that their explanation truly captures why this success should be exclusive to deep learning.
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[ 2.7 ms ] story [ 30.8 ms ] threadThe chaos is the point. What a crazy, beautiful mess.
They explicitly ignore momentum and exponentially weighted moving average, but that should result in the time-averaged gradient descent (along the valley, not across it). But that requires multiple evaluations, do any of the expressions for the central flow admit fast / computationally efficient central flow calculation?
This is almost Sobolev preconditioning:
where this time S is the sum of all the eigenvalues (so, the Laplacian of L).I'm not really convinced that their explanation truly captures why this success should be exclusive to deep learning.