Regularization is typically understood as improving generalization by altering the landscape of local extrema to which the model eventually converges. Deep neural networks (DNNs), however, challenge this view: We show that removing regularization after an initial transient period has little effect on generalization, even if the final loss landscape is the same as if there had been no regularization. In some cases, generalization even improves after interrupting regularization. Conversely, if regularization is applied only after the initial transient, it has no effect on the final solution, whose generalization gap is as bad as if regularization never happened. This suggests that what matters for training deep networks is not just whether or how, but when to regularize. The phenomena we observe are manifest in different datasets (CIFAR-10, CIFAR-100), different architectures (ResNet-18, All-CNN), different regularization methods (weight decay, data augmentation), different learning rate schedules (exponential, piece-wise constant). They collectively suggest that there is a critical period'' for regularizing deep networks that is decisive of the final performance. More analysis should, therefore, focus on the transient rather than asymptotic behavior of learning.
@cite_13: 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.
@cite_8: Over the past few years, Batch-Normalization has been commonly used in deep networks, allowing faster training and high performance for a wide variety of applications. However, the reasons behind its merits remained unanswered, with several shortcomings that hindered its use for certain tasks. In this work, we present a novel view on the purpose and function of normalization methods and weight-decay, as tools to decouple weights' norm from the underlying optimized objective. This property highlights the connection between practices such as normalization, weight decay and learning-rate adjustments. We suggest several alternatives to the widely used L2 batch-norm, using normalization in L1 and L∞ spaces that can substantially improve numerical stability in low-precision implementations as well as provide computational and memory benefits. We demonstrate that such methods enable the first batch-norm alternative to work for half-precision implementations. Finally, we suggest a modification to weight-normalization, which improves its performance on large-scale tasks.
@cite_9: In this chapter we introduce a new principle for minimizing the expected risk called the vicinal risk minimization (VRM) principle.’ We use this principle for solving our main problems: pattern recognition, regression estimation, and density estimation.
@cite_1: Weight decay is one of the standard tricks in the neural network toolbox, but the reasons for its regularization effect are poorly understood, and recent results have cast doubt on the traditional interpretation in terms of @math regularization. Literal weight decay has been shown to outperform @math regularization for optimizers for which they differ. We empirically investigate weight decay for three optimization algorithms (SGD, Adam, and K-FAC) and a variety of network architectures. We identify three distinct mechanisms by which weight decay exerts a regularization effect, depending on the particular optimization algorithm and architecture: (1) increasing the effective learning rate, (2) approximately regularizing the input-output Jacobian norm, and (3) reducing the effective damping coefficient for second-order optimization. Our results provide insight into how to improve the regularization of neural networks.
@cite_2: 
@cite_5: Batch Normalization is a commonly used trick to improve the training of deep neural networks. These neural networks use L2 regularization, also called weight decay, ostensibly to prevent overfitting. However, we show that L2 regularization has no regularizing effect when combined with normalization. Instead, regularization has an influence on the scale of weights, and thereby on the effective learning rate. We investigate this dependence, both in theory, and experimentally. We show that popular optimization methods such as ADAM only partially eliminate the influence of normalization on the learning rate. This leads to a discussion on other ways to mitigate this issue.
@cite_10: It has been observed in numerical simulations that a weight decay can improve generalization in a feed-forward neural network. This paper explains why. It is proven that a weight decay has two effects in a linear network. First, it suppresses any irrelevant components of the weight vector by choosing the smallest vector that solves the learning problem. Second, if the size is chosen right, a weight decay can suppress some of the effects of static noise on the targets, which improves generalization quite a lot. It is then shown how to extend these results to networks with hidden layers and non-linear units. Finally the theory is confirmed by some numerical simulations using the data from NetTalk.
@cite_11: Weight decay was proposed to reduce overfitting which often appears in the generalization tasks of artificial neural nets. Here weight decay is applied to a well defined model system based on a single layer perceptron, which exhibits strong overfitting. Since we know for this system the optimal nonoverfitting solution we can compare the effect of the weight decay with this solution. A strategy to find the optimal weight decay strength, which leads to the optimal solution for any number of examples, is proposed.
Also related to our work, there have been attempts to interpret the mechanisms of action of certain regularization methods, such as weight decay @cite_1 @cite_5 @cite_2 @cite_8 @cite_10 @cite_11 , data augmentation @cite_9 , dropout @cite_13 . It has been pointed out in that the Gauss-Newton norm correlates with generalization, and with the Fisher Information Matrix , a measure of the flatness of the minimum, to conclude that the Fisher Information at convergence correlates with generalization. However, there is no causal link proven. In fact, we suggest this correlation may be an epi-phenomenon: Weight decay causes an increase in Fisher information during the transient, which is responsible for generalization ( fig:fisher ), whereas the asymptotic value of the Fisher norm ( , sharpness of the minimum) is not causative. In particular, we show that increasing Fisher Information can actually improve generalization.