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Title: Bayesian Hypernetworks

Paper content:
We propose Bayesian hypernetworks: a framework for approximate Bayesian inference in neural networks.
A Bayesian hypernetwork, h, is a neural network which learns to transform a simple noise distribution, p(e) = N(0,I), to a distribution q(t) := q(h(e)) over the parameters t of another neural network (the ``primary network).
We train q with variational inference, using an invertible h to enable efficient estimation of the variational lower bound on the posterior p(t | D) via sampling.
In contrast to most methods for Bayesian deep learning, Bayesian hypernets can represent a complex multimodal approximate posterior with correlations between parameters, while enabling cheap iid sampling of q(t).
In practice, Bayesian hypernets provide a better defense against adversarial examples than dropout, and also exhibit competitive performance on a suite of tasks which evaluate model uncertainty, including regularization, active learning, and anomaly detection.
Simple and powerful techniques for Bayesian inference of deep neural networks' (DNNs) parameters have the potential to dramatically increase the scope of applications for deep learning techniques.
In real-world applications, unanticipated mistakes may be costly and dangerous, whereas anticipating mistakes allows an agent to seek human guidance (as in active learning), engage safe default behavior (such as shutting down), or use a "reject option" in a classification context.
DNNs are typically trained to find the single most likely value of the parameters (the "MAP estimate"), but this approach neglects uncertainty about which parameters are the best ("parameter uncertainty"), which may translate into higher predictive uncertainty when likely parameter values yield highly confident but contradictory predictions.
Conversely, Bayesian DNNs model the full posterior distribution of a model's parameters given the data, and thus provides better calibrated confidence estimates, with corresponding safety benefits BID9 BID0 .
1 Maintaining a distribution over parameters is also one of the most effective defenses against adversarial attacks BID4 .Techniques
for Bayesian DNNs are an active research topic. The most popular
approach is variational inference BID2 BID8 , which typically restricts the variational posterior to a simple family of distributions, for instance a factorial Gaussian BID2 BID16 . Unfortunately, from
a safety perspective, variational approximations tend to underestimate uncertainty, by heavily penalizing approximate distributions which place mass in regions where the true posterior has low density. This problem can be
exacerbated by using a restricted family of posterior distribution; for instance a unimodal approximate posterior will generally only capture a single mode of the true posterior. With this in mind,
we propose learning an extremely flexible and powerful posterior, parametrized by a DNN h, which we refer to as a Bayesian hypernetwork in reference to BID17 .A Bayesian hypernetwork
(BHN) takes random noise ∼ N (0, I) as input and outputs a sample from the approximate posterior q(θ) for another DNN of interest (the "primary network"). The key insight for building
such a model is the use of an invertible hypernet, which enables Monte Carlo estimation of the entropy term − logq(θ) in the variational inference training objective.We begin the paper by reviewing previous work on Bayesian DNNs, and explaining the necessary components of our approach (Section 2). Then we explain how to compose
these techniques to yield Bayesian hypernets, as well as design choices which make training BHNs efficient, stable and robust (Section 3). Finally, we present experiments
which validate the expressivity of BHNs, and demonstrate their competitive performance across several tasks (Section 4).
We introduce Bayesian hypernets (BHNs), a new method for variational Bayesian deep learning which uses an invertible hypernetwork as a generative model of parameters.
BHNs feature efficient we found the BALD values our implementation computes provide a better-than-random acquisition function (compare the blue line in the top and bottom plots).
10 Li & Gal (2017) and BID28 used 10 and 1 model samples, respectively, to estimate gradient.
We report the result with 1 sample; results with more samples are given in the appendix.
when more perturbation is added to the data (left), uncertainty measures also increase (first row).
In particular, the BALD and Mean STD scores, which measure epistemic uncertainty, are strongly increasing for BHNs, but not for dropout.
The second row and third row plots show results for adversary detection and error detection (respectively) in terms of the AUC of ROC (y-axis) with increasing perturbation along the x-axis.
Gradient direction is estimated with one Monte Carlo sample of the weights/dropout mask.training and sampling, and can express complicated multimodal distributions, thereby addressing issues of overconfidence present in simpler variational approximations.
We present a method of parametrizing BHNs which allows them to scale successfully to real world tasks, and show that BHNs can offer significant benefits over simpler methods for Bayesian deep learning.
Future work could explore other methods of parametrizing BHNs, for instance using the same hypernet to output different subsets of the primary net parameters.A

assistant: text: We propose Bayesian hypernetworks: a framework for approximate Bayesian inference in neural networks.