Comment by rsp1984 | Hacker News Reader

Comment by rsp1984 | Hacker News Reader

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Leveraging thermodynamics to more efficiently compute second-order updates is certainly cool and worth exploring, however specifically in the context of deep learning I remain skeptical of its usefulness.

We already have very efficient second-order methods running on classical hardware [1] but they are basically not being used at all in practice, as they are outperformed by ADAM and other 1st-order methods. This is because optimizing highly nonlinear loss functions, such as the ones in deep learning models, only really works with very low learning rates, regardless of whether a 1st or a 2nd order method is used. So, comparatively speaking, a 2nd order method might give you a slightly better parameter update per step but at a more-than-slightly-higher cost, so most of the time it's simply not worth doing.

[1] https://andrew.gibiansky.com/blog/machine-learning/hessian-f...

Agreed that it's very cool, and also about how hard it is to make second order methods worthwhile. We're just using such huge datasets that sometimes it's hard to even get a decent estimate of the gradient for a minibatch. Getting a useful estimate of second order information over the dataset is even harder, especially when the whole point of using minibatches is computational feasibility.

Those are valid points! Hessian-free (HF) optimization is a really nice method, but as you say remains costly so people don't use it. The key idea in this paper is that if you are able to solve linear systems faster by using an analog device, the cost of a HF-like method is brought down, so the method can become competitive.

About the noise, it is true that the second-order information will be noisier than the gradient for a given batch size (and a lot of results out there for HF optimization are with impractically large batch sizes). In the paper we use relatively small batch sizes (eg 32 for the fine-tuning example) and show that you can still get an advantage from second-order information. Of course it would be interesting to study in more detail how noisy 2nd order information can be, and on more datasets.

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