We investigate the use of alternative divergences to Kullback-Leibler (KL) in variational inference(VI), based on the Variational Dropout kingma2015 . Stochastic gradient variational Bayes (SGVB) aevb is a general framework for estimating the evidence lower bound (ELBO) in Variational Bayes. In this work, we extend the SGVB estimator with using Alpha-Divergences, which are alternative to divergences to VI' KL objective. The Gaussian dropout can be seen as a local reparametrization trick of the SGVB objective. We extend the Variational Dropout to use alpha divergences for variational inference. Our results compare @math -divergence variational dropout with standard variational dropout with correlated and uncorrelated weight noise. We show that the @math -divergence with @math (or KL divergence) is still a good measure for use in variational inference, in spite of the efficient use of Alpha-divergences for Dropout VI Li17 . @math can yield the lowest training error, and optimizes a good lower bound for the evidence lower bound (ELBO) among all values of the parameter @math .
Our work extends the variational dropout technique originally proposed in @cite_0 . It was suggested that applying a multiplicative random noise to inputs of a fully-connected neural network layer is equivalent to computing the posterior distribution of the pre-activation outputs ( @cite_0 ). Using the reparametrization of the weights @math as @math , where @math is a deterministic weight value and @math is a random variable determining the scale of the @math weight. As described in @cite_0 , it is possible to split the posterior distribution over the weights @math into an additive (i.e., location) parameter @math and a multiplicative (i.e., scale) parameter @math . Formally, @math , using set notation.
@cite_0
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Multi-layer neural networks have lead to remarkable performance on many kinds of benchmark tasks in text, speech and image processing. Nonlinear parameter estimation in hierarchical models is known to be subject to overfitting and misspecification. One approach to these estimation and related problems (local minima, colinearity, feature discovery etc.) is called Dropout (Hinton, et al 2012, 2016). The Dropout algorithm removes hidden units according to a Bernoulli random variable with probability @math prior to each update, creating random "shocks" to the network that are averaged over updates. In this paper we will show that Dropout is a special case of a more general model published originally in 1990 called the Stochastic Delta Rule, or SDR (Hanson, 1990). SDR redefines each weight in the network as a random variable with mean @math and standard deviation @math . Each weight random variable is sampled on each forward activation, consequently creating an exponential number of potential networks with shared weights. Both parameters are updated according to prediction error, thus resulting in weight noise injections that reflect a local history of prediction error and local model averaging. SDR therefore implements a more sensitive local gradient-dependent simulated annealing per weight converging in the limit to a Bayes optimal network. Tests on standard benchmarks (CIFAR) using a modified version of DenseNet shows the SDR outperforms standard Dropout in test error by approx. @math with DenseNet-BC 250 on CIFAR-100 and approx. @math in smaller networks. We also show that SDR reaches the same accuracy that Dropout attains in 100 epochs in as few as 35 epochs.