Deep neural networks with remarkably strong generalization performances are usually over-parameterized. Despite explicit regularization strategies are used for practitioners to avoid over-fitting, the impacts are often small. Some theoretical studies have analyzed the implicit regularization effect of stochastic gradient descent (SGD) on simple machine learning models with certain assumptions. However, how it behaves practically in state-of-the-art models and real-world datasets is still unknown. To bridge this gap, we study the role of SGD implicit regularization in deep learning systems. We show pure SGD tends to converge to minimas that have better generalization performances in multiple natural language processing (NLP) tasks. This phenomenon coexists with dropout, an explicit regularizer. In addition, neural network's finite learning capability does not impact the intrinsic nature of SGD's implicit regularization effect. Specifically, under limited training samples or with certain corrupted labels, the implicit regularization effect remains strong. We further analyze the stability by varying the weight initialization range. We corroborate these experimental findings with a decision boundary visualization using a 3-layer neural network for interpretation. Altogether, our work enables a deepened understanding on how implicit regularization affects the deep learning model and sheds light on the future study of the over-parameterized model's generalization ability.
One study shows that the logistic regression model and 2-layer neural networks using monotone decreasing loss functions tend to converge in the direction of the max-margin solution when using GD and SGD @cite_22 . We further enhance the conclusion by doing studies on more practical deep learning systems.
@cite_22
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We examine gradient descent on unregularized logistic regression problems, with homogeneous linear predictors on linearly separable datasets. We show the predictor converges to the direction of the max-margin (hard margin SVM) solution. The result also generalizes to other monotone decreasing loss functions with an infimum at infinity, to multi-class problems, and to training a weight layer in a deep network in a certain restricted setting. Furthermore, we show this convergence is very slow, and only logarithmic in the convergence of the loss itself. This can help explain the benefit of continuing to optimize the logistic or cross-entropy loss even after the training error is zero and the training loss is extremely small, and, as we show, even if the validation loss increases. Our methodology can also aid in understanding implicit regularization n more complex models and with other optimization methods. We study the implicit bias of gradient descent methods in solving a binary classification problem over a linearly separable dataset. The classifier is described by a nonlinear ReLU model and the objective function adopts the exponential loss function. We first characterize the landscape of the loss function and show that there can exist spurious asymptotic local minima besides asymptotic global minima. We then show that gradient descent (GD) can converge to either a global or a local max-margin direction, or may diverge from the desired max-margin direction in a general context. For stochastic gradient descent (SGD), we show that it converges in expectation to either the global or the local max-margin direction if SGD converges. We further explore the implicit bias of these algorithms in learning a multi-neuron network under certain stationary conditions, and show that the learned classifier maximizes the margins of each sample pattern partition under the ReLU activation. We present a formulation of deep learning that aims at producing a large margin classifier. The notion of margin, minimum distance to a decision boundary, has served as the foundation of several theoretically profound and empirically successful results for both classification and regression tasks. However, most large margin algorithms are applicable only to shallow models with a preset feature representation; and conventional margin methods for neural networks only enforce margin at the output layer. Such methods are therefore not well suited for deep networks. In this work, we propose a novel loss function to impose a margin on any chosen set of layers of a deep network (including input and hidden layers). Our formulation allows choosing any norm on the metric measuring the margin. We demonstrate that the decision boundary obtained by our loss has nice properties compared to standard classification loss functions. Specifically, we show improved empirical results on the MNIST, CIFAR-10 and ImageNet datasets on multiple tasks: generalization from small training sets, corrupted labels, and robustness against adversarial perturbations. The resulting loss is general and complementary to existing data augmentation (such as random adversarial input transform) and regularization techniques (such as weight decay, dropout, and batch norm). Multilayer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires optimizing a nonconvex high-dimensional objective (risk function), a problem that is usually attacked using stochastic gradient descent (SGD). Does SGD converge to a global optimum of the risk or only to a local optimum? In the former case, does this happen because local minima are absent or because SGD somehow avoids them? In the latter, why do local minima reached by SGD have good generalization properties? In this paper, we consider a simple case, namely two-layer neural networks, and prove that—in a suitable scaling limit—SGD dynamics is captured by a certain nonlinear partial differential equation (PDE) that we call distributional dynamics (DD). We then consider several specific examples and show how DD can be used to prove convergence of SGD to networks with nearly ideal generalization error. This description allows for “averaging out” some of the complexities of the landscape of neural networks and can be used to prove a general convergence result for noisy SGD.