Variational dropout (VD) is a generalization of Gaussian dropout, which aims at inferring the posterior of network weights based on a log-uniform prior on them to learn these weights as well as dropout rate simultaneously. The log-uniform prior not only interprets the regularization capacity of Gaussian dropout in network training, but also underpins the inference of such posterior. However, the log-uniform prior is an improper prior (i.e., its integral is infinite) which causes the inference of posterior to be ill-posed, thus restricting the regularization performance of VD. To address this problem, we present a new generalization of Gaussian dropout, termed variational Bayesian dropout (VBD), which turns to exploit a hierarchical prior on the network weights and infer a new joint posterior. Specifically, we implement the hierarchical prior as a zero-mean Gaussian distribution with variance sampled from a uniform hyper-prior. Then, we incorporate such a prior into inferring the joint posterior over network weights and the variance in the hierarchical prior, with which both the network training and the dropout rate estimation can be cast into a joint optimization problem. More importantly, the hierarchical prior is a proper prior which enables the inference of posterior to be well-posed. In addition, we further show that the proposed VBD can be seamlessly applied to network compression. Experiments on both classification and network compression tasks demonstrate the superior performance of the proposed VBD in terms of regularizing network training.
We explore a recently proposed Variational Dropout technique that provided an elegant Bayesian interpretation to Gaussian Dropout. We extend Variational Dropout to the case when dropout rates are unbounded, propose a way to reduce the variance of the gradient estimator and report first experimental results with individual dropout rates per weight. Interestingly, it leads to extremely sparse solutions both in fully-connected and convolutional layers. This effect is similar to automatic relevance determination effect in empirical Bayes but has a number of advantages. We reduce the number of parameters up to 280 times on LeNet architectures and up to 68 times on VGG-like networks with a negligible decrease of accuracy. We investigate a local reparameterizaton technique for greatly reducing the variance of stochastic gradients for variational Bayesian inference (SGVB) of a posterior over model parameters, while retaining parallelizability. This local reparameterization translates uncertainty about global parameters into local noise that is independent across datapoints in the minibatch. Such parameterizations can be trivially parallelized and have variance that is inversely proportional to the mini-batch size, generally leading to much faster convergence. Additionally, we explore a connection with dropout: Gaussian dropout objectives correspond to SGVB with local reparameterization, a scale-invariant prior and proportionally fixed posterior variance. Our method allows inference of more flexibly parameterized posteriors; specifically, we propose variational dropout, a generalization of Gaussian dropout where the dropout rates are learned, often leading to better models. The method is demonstrated through several experiments. Gaussian multiplicative noise is commonly used as a stochastic regularisation technique in training of deterministic neural networks. A recent paper reinterpreted the technique as a specific algorithm for approximate inference in Bayesian neural networks; several extensions ensued. We show that the log-uniform prior used in all the above publications does not generally induce a proper posterior, and thus Bayesian inference in such models is ill-posed. Independent of the log-uniform prior, the correlated weight noise approximation has further issues leading to either infinite objective or high risk of overfitting. The above implies that the reported sparsity of obtained solutions cannot be explained by Bayesian or the related minimum description length arguments. We thus study the objective from a non-Bayesian perspective, provide its previously unknown analytical form which allows exact gradient evaluation, and show that the later proposed additive reparametrisation introduces minima not present in the original multiplicative parametrisation. Implications and future research directions are discussed. Dropout-based regularization methods can be regarded as injecting random noise with pre-defined magnitude to different parts of the neural network during training. It was recently shown that Bayesian dropout procedure not only improves gener- alization but also leads to extremely sparse neural architectures by automatically setting the individual noise magnitude per weight. However, this sparsity can hardly be used for acceleration since it is unstructured. In the paper, we propose a new Bayesian model that takes into account the computational structure of neural net- works and provides structured sparsity, e.g. removes neurons and or convolutional channels in CNNs. To do this we inject noise to the neurons outputs while keeping the weights unregularized. We establish the probabilistic model with a proper truncated log-uniform prior over the noise and truncated log-normal variational approximation that ensures that the KL-term in the evidence lower bound is com- puted in closed-form. The model leads to structured sparsity by removing elements with a low SNR from the computation graph and provides significant acceleration on a number of deep neural architectures. The model is easy to implement as it can be formulated as a separate dropout-like layer.