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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In this paper, we use dynamical system to analyze the nonlinear weight dynamics of two-layered bias-free networks in the form of @math , where @math is ReLU nonlinearity. We assume that the input @math follow Gaussian distribution. The network is trained using gradient descent to mimic the output of a teacher network of the same size with fixed parameters @math using @math loss. We first show that when @math , the nonlinear dynamics can be written in close form, and converges to @math with at least @math probability, if random weight initializations of proper standard derivation ( @math ) is used, verifying empirical practice. For networks with many ReLU nodes ( @math ), we apply our close form dynamics and prove that when the teacher parameters @math forms orthonormal bases, (1) a symmetric weight initialization yields a convergence to a saddle point and (2) a certain symmetry-breaking weight initialization yields global convergence to @math without local minima. To our knowledge, this is the first proof that shows global convergence in nonlinear neural network without unrealistic assumptions on the independence of ReLU activations. In addition, we also give a concise gradient update formulation for a multilayer ReLU network when it follows a teacher of the same size with @math loss. Simulations verify our theoretical analysis.