What are Bayesian neural network posteriors really like? With high fidelity HMC, we study approximate inference quality, generalization, cold posteriors, priors, and more.
With @Pavel_Izmailov, @sharadvikram, and Matthew D. Hoffman. 1/10
We show that Bayesian neural networks reassuringly provide good generalization, outperforming deep ensembles, standard training, and many approximate inference procedures, even with a single chain. 2/10
However, we find that BNNs are surprisingly poor at OOD generalization, even worse than SGD, despite the popularity of approximate inference in this setting, and the relatively good performance of BNNs for OOD detection. 3/10
Even though deep ensembles are often talked about as a "non-Bayesian" alternative to standard approximate inference, we find they approximate the HMC predictive distribution better than MFVI, and about as well as standard SGLD. 4/10
There has been much attention lately on "cold posteriors" in BDL, where the posterior raised to a power 1/T with T<1 can lead to better results. We see little evidence for a general cold posterior effect, which we find is largely due to data augmentation. 5/10
We explored Gaussian, mixture of Gaussian, and heavy-tailed logistic priors, which performed similarly, although the heavy-tailed priors did slightly better. We also found performance relatively insensitive to the scale of the Gaussian prior... 6/10
...these results highlight the relative importance of the architecture compared to the distribution over weights in defining the induced prior over functions. Indeed, other work shows that even standard Gaussian priors have many useful properties: arxiv.org/abs/2002.08791. 7/10
We present many other results, including mixing in function space vs. weight space, posterior geometry and mode connecting paths, single chain vs. multi-chain...! 8/10
Many of the results, both positive and negative for BDL, are contrary to conventional wisdom. 9/10
We worked hard to obtain these HMC samples, which we plan to release as a public resource, as a reference for evaluating more practical alternatives to HMC, and for researchers to explore their own questions around approximate inference in BDL. 10/10

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More from @andrewgwils

29 Dec 20
There is a lot of often overlooked evidence that standard p(w) = N(0, a*I) priors combined with a NN f(x,w) induce a distribution over functions p(f(x)) with useful properties!... 1/15
The deep image prior shows this p(f(x)) captures low-level image statistics useful for image denoising, super-resolution, and inpainting. The rethinking generalization paper shows pre-processing data with a randomly initialized CNN can dramatically boost performance. 2/15
We show that the induced p(f(x)) has a reasonable correlation function, such that visually similar images are more correlated a priori. Moreover, the flatness arguments for SGD generalization suggest that good solutions take up a large volume in the corresponding posteriors. 3/15
Read 17 tweets
9 Dec 20
In practice, standard "deep ensembles" of independently trained models provides a relatively compelling Bayesian model average. This point is often overlooked because we are used to viewing Bayesian methods as sampling from some (approximate) posterior... 1/10
...to form a model average, via simple Monte Carlo. But if we instead directly consider what we ultimately want to compute, the integral corresponding to the marginal predictive distribution (the predictive distribution not conditioning on weights)... 2/10
...then deep ensembles are in practice a _better_ approximation to the Bayesian model average than methods that are conventionally accepted as Bayesian (such as Laplace, variational methods with a Gaussian posterior, etc.). 3/10
Read 10 tweets
27 Oct 20
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
Read 5 tweets
26 May 20
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 20
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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