Deep learning as a dynamical system

Image: Donny Darko

A recurring movement within neural network learning research which tries to render the learning of prediction functions tractable by considering them as dynamical systems, and using the theory of stability in the context of Hamiltonians, optimal control and/or ODE solvers, to make it all work.

I’ve been interested by this since seeing the (Haber and Ruthotto 2018) paper, but it’s got a real kick recently since the (Chen et al. 2018) won the prize at NeurIPS for learning the ODEs themselves.

Coming from the ODE side, Chris Rackauckas’ lecture notes christen this development “scientific machine learning”.

Convnets/Resnets as discrete ODE approximations

Arguing that neural networks are in the limit approximants to quadrature solutions of certain ODES, work and gain insights and new tricks into neural nets by using ODE tricks. This is mostly what Haber and Rhutthoto et al do. “Stability of training” is a useful outcome here. Related, but not quite the same, notion of stability, as in input-stability in learning. (Haber and Ruthotto 2018; Haber et al. 2017; Chang, Meng, Haber, Ruthotto, et al. 2018; Ruthotto and Haber 2018)

Neural ODE regression

By which I mean learning an ODE whose solution is the regression problem. This is what e.g. the famous Vector Institute paper (Chen et al. 2018) did, although I’m not sure its as novel as they imply, since it does look like a lot of earlier work. Indeed, author Duvenaud argues that in some ways the hype ran away with this paper, and credits casadi with some of the innovations here. There are various laypersons’ introductions/ tutorials in this area, including the simple and practical magical take in julia. See also the CASADI example

Learning an ODE, a purely deterministic process, feels unsatisfying; As Duvenaud points out, it is not ideal to have time series models which need to encode everything in an initial state.

More generally, learnable SDEs are probably what we want. I’m particularly interested on jump ODE regression.

There are syntheses of these approaches that try to do everything with ODEs, all the time. (Rackauckas et al. 2018; Niu, Horesh, and Chuang 2019), and even some tutorial implementations by the indefatigable Chris Rackauckas, and a whole MIT course.

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Chang, Bo, Lili Meng, Eldad Haber, Lars Ruthotto, David Begert, and Elliot Holtham. 2018. “Reversible Architectures for Arbitrarily Deep Residual Neural Networks.” In.

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