A recurring problem when building probabilistic latent variable models is regularization and model selection, for instance, the choice of the dimensionality of the latent space. In the context of belief networks with latent variables, this problem has been adressed with Automatic Relevance Determination (ARD) employing Monte Carlo inference. We present a variational inference approach to ARD for Deep Generative Models using doubly stochastic variational inference to provide fast and scalable learning. We show empirical results on a standard dataset illustrating the effects of contracting the latent space automatically. We show that the resulting latent representations are significantly more compact without loss of expressive power of the learned models.
An initial Bayesian treatment for regularization of Neural Networks was performed in seminal work by @cite_3 and @cite_5 . They introduced a notion called Automatic Relevance Determination (ARD), which consists of the idea of using a prior distribution on generative weights attached to the latent variable which encourages the weights to be zero. Effectively, by integrating over such priors using Monte Carlo, settings for the variances of the prior can be inferred from data leading to pruning of unnecessary latent dimensions. ARD was also notably used in a variational treatment of the relevance vector machine @cite_8 to infer a mask over the data features needed for a preditive model. The idea of relevance determination for belief networks in combination with variational inference has also been explored before in @cite_1 , but is different from our model.
@cite_5 @cite_1 @cite_3 @cite_8
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The Bayesian analysis of neural networks is difficult because a simple prior over weights implies a complex prior distribution over functions. In this paper we investigate the use of Gaussian process priors over functions, which permit the predictive Bayesian analysis for fixed values of hyperparameters to be carried out exactly using matrix operations. Two methods, using optimization and averaging (via Hybrid Monte Carlo) over hyperparameters have been tested on a number of challenging problems and have produced excellent results. Bayesian neural networks (BNNs) hold great promise as a flexible and principled solution to deal with uncertainty when learning from finite data. Among approaches to realize probabilistic inference in deep neural networks, variational Bayes (VB) is theoretically grounded, generally applicable, and computationally efficient. With wide recognition of potential advantages, why is it that variational Bayes has seen very limited practical use for BNNs in real applications? We argue that variational inference in neural networks is fragile: successful implementations require careful initialization and tuning of prior variances, as well as controlling the variance of Monte Carlo gradient estimates. We fix VB and turn it into a robust inference tool for Bayesian neural networks. We achieve this with two innovations: first, we introduce a novel deterministic method to approximate moments in neural networks, eliminating gradient variance; second, we introduce a hierarchical prior for parameters and a novel empirical Bayes procedure for automatically selecting prior variances. Combining these two innovations, the resulting method is highly efficient and robust. On the application of heteroscedastic regression we demonstrate strong predictive performance over alternative approaches. The Bayesian framework for model comparison and regularisation is demonstrated by studying interpolation and classification problems modelled with both linear and non-linear models. This framework quantitatively embodies 'Occam's razor'. Over-complex and under-regularised models are automatically inferred to be less probable, even though their flexibility allows them to fit the data better. When applied to 'neural networks', the Bayesian framework makes possible (1) objective comparison of solutions using alternative network architectures; (2) objective stopping rules for network pruning or growing procedures; (3) objective choice of type of weight decay terms (or regularisers); (4) on-line techniques for optimising weight decay (or regularisation constant) magnitude; (5) a measure of the effective number of well-determined parameters in a model; (6) quantified estimates of the error bars on network parameters and on network output. In the case of classification models, it is shown that the careful incorporation of error bar information into a classifier's predictions yields improved performance. Comparisons of the inferences of the Bayesian framework with more traditional cross-validation methods help detect poor underlying assumptions in learning models. The relationship of the Bayesian learning framework to 'active learning' is examined. Objective functions are discussed which measure the expected informativeness of candidate data measurements, in the context of both interpolation and classification problems. The concepts and methods described in this thesis are quite general and will be applicable to other data modelling problems whether they involve regression, classification or density estimation. 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.