We can greatly simplify Hamiltonian and Lagrangian neural nets by working in Cartesian coordinates with explicit constraints, leading to dramatic performance improvements! Our #NeurIPS2020 paper: arxiv.org/abs/2010.13581
with @m_finzi, @KAlexanderWang. 1/5
Complex dynamics can be described more simply with higher levels of abstraction. For example, a trajectory can be found by solving a differential equation. The differential equation can in turn be derived by a simpler Hamiltonian or Lagrangian, which is easier to model. 2/5
We can move further up the hierarchy of abstraction by working in Cartesian coordinates and explicitly representing constraints with Lagrange multipliers, for constrained Hamiltonian and Lagrangian neural networks (CHNNs and CLNNs) that face a much easier learning problem. 3/5
We show how to apply our approach to arbitrary rigid body and extended systems by showing how such systems can be embedded into Cartesian coordinates. We also introduce a series of chaotic and 3D extended body systems that challenge current approaches. 4/5
Much more in the paper! Code is also available at: github.com/mfinzi/constra…

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More from @andrewgwils

26 May
Effective dimension compares favourably to popular path-norm and PAC-Bayes flatness measures, including double descent and width-depth trade-offs! We have just posted this new result in section 7 of our paper on posterior contraction in BDL: arxiv.org/abs/2003.02139. 1/16
The plots are most interpretable for comparing models of similar train loss (e.g. above the green partition). N_eff(Hess) = effective dimension of the Hessian at convergence. 2/16
Both path-norm and PAC-Bayes flatness variants perform well in the recent fantastic generalization measures paper of Jiang et. al (2019): arxiv.org/abs/1912.02178.
Read 16 tweets
21 Feb
Our new paper "Bayesian Deep Learning and a Probabilistic Perspective of Generalization": arxiv.org/abs/2002.08791. Includes (1) benefits of BMA; (2) BMA <-> Deep Ensembles; (3) new methods; (4) BNN priors; (5) generalization in DL; (6) tempering in BDL. With @Pavel_Izmailov. 1/19
Since neural nets can fit images with noisy labels, it has been suggested we should rethink generalization. But this behaviour is understandable from a probabilistic perspective: we want to support any possible solution, but also have good inductive biases. 2/19
The inductive biases determine what solutions are a priori likely. Indeed, we show this seemingly mysterious behaviour is not unique to neural nets: GPs with RBF kernels can perfectly fit noisy CIFAR, but also generalize on the noise free problem. 3/19
Read 20 tweets
27 Dec 19
Bayesian methods are *especially* compelling for deep neural networks. The key distinguishing property of a Bayesian approach is marginalization instead of optimization, not the prior, or Bayes rule. This difference will be greatest for underspecified models like DNNs. 1/18
In particular, the predictive distribution we often want to find is p(y|x,D) = \int p(y|x,w) p(w|D) dw. 'y' is an output, 'x' an input, 'w' the weights, and D the data. This is not a controversial equation, it is simply the sum and product rules of probability. 2/18
Rather than betting everything on a single hypothesis, we want to use every setting of parameters, weighted by posterior probabilities. This procedure is known as a Bayesian model average (BMA). 3/18
Read 18 tweets

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