Gradient flows

infinitesimal optimization

Hinze et al. (2021) depict a mosquito’s gradient flow in a 3d optimisation problem.

Stochastic models of optimisation, especially stochastic gradience descent.


Gradient flows we can think of a continuous-limit of gradient descent. There is a (deterministic) ODE corresponding to an infinitesimal trainning rate.

Stochastic DE for early stage training

SGD as an SDE (Ljung, Pflug, and Walk 1992; Mandt, Hoffman, and Blei 2017). Worth the price of dusting off the old stochastic calculus. This is used for choosing scaling rules for model training, typically. (Q. Li, Tai, and Weinan 2019; Z. Li, Malladi, and Arora 2021; Malladi et al. 2022)

Stochastic DE around the optimum

The limiting diffusion describes diffusion around an optim, i.e. after we have converged. Interesting for understanding generalisation (Gu et al. 2022; Z. Li, Wang, and Arora 2021; Lyu, Li, and Arora 2023; Wang et al. 2023).

They have an interpretation in terms of sampling from a Bayes posterior: See Bayes by Backprop things.


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Hinze, Annika, Jörgen Lantz, Sharon R. Hill, and Rickard Ignell. 2021. Mosquito Host Seeking in 3D Using a Versatile Climate-Controlled Wind Tunnel System.” Frontiers in Behavioral Neuroscience 15 (March): 643693.
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Lyu, Kaifeng, Zhiyuan Li, and Sanjeev Arora. 2023. Understanding the Generalization Benefit of Normalization Layers: Sharpness Reduction.” In. arXiv.
Malladi, Sadhika, Kaifeng Lyu, Abhishek Panigrahi, and Sanjeev Arora. 2022. On the SDEs and Scaling Rules for Adaptive Gradient Algorithms.” In Advances in Neural Information Processing Systems, 35:7697–7711.
Mandt, Stephan, Matthew D. Hoffman, and David M. Blei. 2017. Stochastic Gradient Descent as Approximate Bayesian Inference.” JMLR, April.
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Wang, Runzhe, Sadhika Malladi, Tianhao Wang, Kaifeng Lyu, and Zhiyuan Li. 2023. The Marginal Value of Momentum for Small Learning Rate SGD.” arXiv.

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