Variance reduction has been commonly used in stochastic optimization. It relies crucially on the assumption that the data set is finite. However, when the data are imputed with random noise as in data augmentation, the perturbed data set be- comes essentially infinite. Recently, the stochastic MISO (S-MISO) algorithm is introduced to address this expected risk minimization problem. Though it converges faster than SGD, a significant amount of memory is required. In this pa- per, we propose two SGD-like algorithms for expected risk minimization with random perturbation, namely, stochastic sample average gradient (SSAG) and stochastic SAGA (S-SAGA). The memory cost of SSAG does not depend on the sample size, while that of S-SAGA is the same as those of variance reduction methods on un- perturbed data. Theoretical analysis and experimental results on logistic regression and AUC maximization show that SSAG has faster convergence rate than SGD with comparable space requirement, while S-SAGA outperforms S-MISO in terms of both iteration complexity and storage.
Deep neural nets with a large number of parameters are very powerful machine learning systems. However, overfitting is a serious problem in such networks. Large networks are also slow to use, making it difficult to deal with overfitting by combining the predictions of many different large neural nets at test time. Dropout is a technique for addressing this problem. The key idea is to randomly drop units (along with their connections) from the neural network during training. This prevents units from co-adapting too much. During training, dropout samples from an exponential number of different "thinned" networks. At test time, it is easy to approximate the effect of averaging the predictions of all these thinned networks by simply using a single unthinned network that has smaller weights. This significantly reduces overfitting and gives major improvements over other regularization methods. We show that dropout improves the performance of neural networks on supervised learning tasks in vision, speech recognition, document classification and computational biology, obtaining state-of-the-art results on many benchmark data sets.  The goal of machine learning is to develop predictors that generalize well to test data. Ideally, this is achieved by training on very large (infinite) training data sets that capture all variations in the data distribution. In the case of finite training data, an effective solution is to extend the training set with artificially created examples--which, however, is also computationally costly. We propose to corrupt training examples with noise from known distributions within the exponential family and present a novel learning algorithm, called marginalized corrupted features (MCF), that trains robust predictors by minimizing the expected value of the loss function under the corrupting distribution-- essentially learning with infinitely many (corrupted) training examples. We show empirically on a variety of data sets that MCF classifiers can be trained efficiently, may generalize substantially better to test data, and are more robust to feature deletion at test time. It is well known that the addition of noise to the input data of a neural network during training can, in some circumstances, lead to significant improvements in generalization performance. Previous work has shown that such training with noise is equivalent to a form of regularization in which an extra term is added to the error function. However, the regularization term, which involves second derivatives of the error function, is not bounded below, and so can lead to difficulties if used directly in a learning algorithm based on error minimization. In this paper we show that for the purposes of network training, the regularization term can be reduced to a positive semi-definite form that involves only first derivatives of the network mapping. For a sum-of-squares error function, the regularization term belongs to the class of generalized Tikhonov regularizers. Direct minimization of the regularized error function provides a practical alternative to training with noise.