How does gradient descent work?
centralflows.github.io
How does gradient descent work?
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Re: How does gradient descent work?
#2Related: Gradient descent visualizer: https://uclaacm.github.io/gradient-descent-visualiser/
Re: How does gradient descent work?
#3First 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.
Re: How does gradient descent work?
#4So 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.
Re: How does gradient descent work?
#5Fascinating, 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?
Re: How does gradient descent work?
#6So 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.
Reminds me of Simulated Annealing. Some randomness have always been part of optimization processes that seek a better equilbrium than local. Genetic Algorithms have mutation, Simulated Annealing has temperature, Gradient Descent similarly has random batches.
Re: How does gradient descent work?
#7https://youtu.be/ajGX7odA87k?t=613 the stuff is what the stuff, brother.
Re: How does gradient descent work?
#8Very 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?
Re: How does gradient descent work?
#9Fascinating, 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 expressi…
The authors somewhat address your questions in the accompanying paper https://arxiv.org/abs/2410.24206
> We emphasize that the central flow is a theoretical tool for understanding optimizer behavior, not a practical optimization method. In practice, maintaining an exponential moving average of the iterates (e.g., Morales-Brotons et al., 2024) is likely a computational feasible way to estimate the optimizer’s time-averaged trajectory.
They analyze the behavior of RMSProp (Adam without momentum) using their framework to come up with simplified mathematical models that are able to predict actual training behavior in experiments. It looks like their mathematical models explain why RMSProp works, in a way that is more satisfying than the usual hand waving explanations.
Re: How does gradient descent work?
#10very nice, but it's only useful if it also works with stochastic gradients.