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Title: Refining the variational posterior through iterative optimization

Paper content:
Variational inference (VI) is a popular approach for approximate Bayesian inference that is particularly promising for highly parameterized models such as deep neural networks.
A key challenge of variational inference is to approximate the posterior over model parameters with a distribution that is simpler and tractable yet sufficiently expressive.
In this work, we propose a method for training highly flexible variational distributions by starting with a coarse approximation and iteratively refining it.
Each refinement step makes cheap, local adjustments and only requires optimization of simple variational families.
We demonstrate theoretically that our method always improves a bound on the approximation (the Evidence Lower BOund) and observe this empirically across a variety of benchmark tasks.
In experiments, our method consistently outperforms recent variational inference methods for deep learning in terms of log-likelihood and the ELBO.
We see that the gains are further amplified on larger scale models, significantly outperforming standard VI and deep ensembles on residual networks on CIFAR10.
Uncertainty plays a crucial role in a multitude of machine learning applications, ranging from weather prediction to drug discovery.
Poor predictive uncertainty risks potentially poor outcomes, especially in domains such as medical diagnosis or autonomous vehicles where some forms of high confidence errors may be especially costly (Amodei et al., 2016) .
Thus, it is becoming increasingly important that the underlying model provides high quality uncertainty estimates along with its predictions.
Yet, possibly the most widely used models, deep neural networks (LeCun et al., 2015) , are unable to accurately quantify model uncertainty.
They are often overconfident in their predictions, even when their predictions are incorrect (Guo et al., 2017; Ovadia et al., 2019) .
By marginalizing over a posterior distribution over the parameters given the training data, Bayesian inference provides a principled approach to capturing uncertainty.
In contrast, standard training of neural networks employs a point estimate of the parameters, which cannot account for model uncertainty.
Unfortunately, exact Bayesian inference is intractable in general for neural networks.
To model epistemic uncertainty, variational inference (VI) instead approximates the true posterior with a simpler distribution.
The most widely used one for neural networks is the mean-field approximation, where the posterior is represented using an independent Gaussian distribution over all the weights.
Variational inference is appealing since it reduces the problem of inference to an optimization problem, minimizing the discrepancy between the true posterior and the variational posterior.
The key challenge, however, is the task of training expressive posterior approximations that can capture the true posterior without significantly increasing the computational costs.
This paper describes a novel method for training highly flexible posterior approximations.
The idea is to start with a coarse, mean-field approximation q(w) and make iterative, local refinements to it.
The regions of the local refinements are determined by sampling the values of additive auxiliary variables.
The model parameters w are expressed using a number of auxiliary variables a k (Figure 1 left) for k = 1, . . . , K that leave the marginal distribution unchanged.
In each iteration, we sample the value of an auxiliary variable according to the current variational approximation q(a k ) and refine the approximation by conditioning on the newly sampled value q(w) ≈ p(w|x, y, a 1:k ) (Figure 1 right illustrates the process).
Each refinement step makes cheap, local adjustments to the variational posterior in the region of the sampled auxiliary variables.
At the end, we draw one sample from In each iteration the value of an auxiliary variable is fixed and the posterior is locally adjusted.
In the final iteration, a sample is drawn from w.
Through the iterations, the variational distribution is able to approximate well the true posterior in a small region.
the refined q(w).
The refinement iterations have to be repeated for each posterior sample.
The algorithm results in samples from a highly complex distribution, starting from a simple mean-field approximation.
While the distribution of the samples is difficult to quantify, we show that it is not limited to factorized, uni-modal forms, and that the procedure is guaranteed to improve the resulting ELBO without posing a significant computational overhead.
In this paper, we describe a novel algorithm for refining a coarse variational approximation to the Bayesian posterior.
We show, both theoretically and empirically, that the refined posterior is a better approximation to the posterior than the initial variational distribution.
Our method outperforms the baseline variational approximations in both uncertainty estimation as well as computational requirements.
It sets a new state-of-the-art in uncertainty estimation using variational inference at ResNet scale (ResNet-20) on CIFAR10.

assistant: text: The paper proposes an algorithm to increase the flexibility of the variational posterior in Bayesian neural networks through iterative optimization.