Dropout is commonly used to help reduce overfitting in deep neural networks. Sparsity is a potentially important property of neural networks, but is not explicitly controlled by Dropout-based regularization. In this work, we propose Sparseout a simple and efficient variant of Dropout that can be used to control the sparsity of the activations in a neural network. We theoretically prove that Sparseout is equivalent to an (L_q ) penalty on the features of a generalized linear model and that Dropout is a special case of Sparseout for neural networks. We empirically demonstrate that Sparseout is computationally inexpensive and is able to control the desired level of sparsity in the activations. We evaluated Sparseout on image classification and language modelling tasks to see the effect of sparsity on these tasks. We found that sparsity of the activations is favorable for language modelling performance while image classification benefits from denser activations. Sparseout provides a way to investigate sparsity in state-of-the-art deep learning models. Source code for Sparseout could be found at https: github.com najeebkhan sparseout.
Another related technique that normalizes activations in the network so as to have zero mean and unit variance is Batchnorm @cite_35 . Although, the primary purpose of Batchnorm is accelerating training optimization of the neural network rather than regularization, Batchnorm has reduced the need for stochastic regularization in certain domains. The above mentioned sparsity-inducing methods are deterministic and thus may result in correlated activations. In this paper we propose Sparseout that implicitly imposes an @math penalty on the activations thus allowing us to choose the level of sparsity in the activations as well as the stochasticity preventing correlated neural activities. Sparseout is different than Bridgeout @cite_29 in that it is applied to activations rather than the weights. Therefore, Sparseout is orders of magnitude faster and practical than Bridgeout. We believe that Sparseout is the first theoretically-motivated technique that is capable of simultaneously controlling sparsity in activations and reducing correlations between them, besides being equivalent to Dropout for @math .
@cite_35 @cite_29
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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. A major challenge in training deep neural networks is overfitting, i.e. inferior performance on unseen test examples compared to performance on training examples. To reduce overfitting, stochastic regularization methods have shown superior performance compared to deterministic weight penalties on a number of image recognition tasks. Stochastic methods such as Dropout and Shakeout, in expectation, are equivalent to imposing a ridge and elastic-net penalty on the model parameters, respectively. However, the choice of the norm of weight penalty is problem dependent and is not restricted to @math . Therefore, in this paper we propose the Bridgeout stochastic regularization technique and prove that it is equivalent to an @math penalty on the weights, where the norm @math can be learned as a hyperparameter from data. Experimental results show that Bridgeout results in sparse model weights, improved gradients and superior classification performance compared to Dropout and Shakeout on synthetic and real datasets. 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. In this work, we investigate the use of sparsity-inducing regularizers during training of Convolution Neural Networks (CNNs). These regularizers encourage that fewer connections in the convolution and fully connected layers take non-zero values and in effect result in sparse connectivity between hidden units in the deep network. This in turn reduces the memory and runtime cost involved in deploying the learned CNNs. We show that training with such regularization can still be performed using stochastic gradient descent implying that it can be used easily in existing codebases. Experimental evaluation of our approach on MNIST, CIFAR, and ImageNet datasets shows that our regularizers can result in dramatic reductions in memory requirements. For instance, when applied on AlexNet, our method can reduce the memory consumption by a factor of four with minimal loss in accuracy.