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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

with @m_finzi, @KAlexanderWang. 1/5

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.

3/16

3/16

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

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