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1. Early Stopping in Deep Networks: Double Descent and How to Eliminate it

   Heckel, Reinhard ; Furkan Yilmaz, Fatih ( April 2021 , International Conference on Learning Representations)

   null (Ed.)

   Over-parameterized models, such as large deep networks, often exhibit a double descent phenomenon, whereas a function of model size, error first decreases, increases, and decreases at last. This intriguing double descent behavior also occurs as a function of training epochs and has been conjectured to arise because training epochs control the model complexity. In this paper, we show that such epoch-wise double descent arises for a different reason: It is caused by a superposition of two or more bias-variance tradeoffs that arise because different parts of the network are learned at different epochs, and eliminating this by proper scaling of stepsizes can significantly improve the early stopping performance. We show this analytically for i) linear regression, where differently scaled features give rise to a superposition of bias-variance tradeoffs, and for ii) a two-layer neural network, where the first and second layer each govern a bias-variance tradeoff. Inspired by this theory, we study two standard convolutional networks empirically and show that eliminating epoch-wise double descent through adjusting stepsizes of different layers improves the early stopping performance significantly. 

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2. Reconciling modern machine-learning practice and the classical bias–variance trade-off

   https://doi.org/10.1073/pnas.1903070116  

   Belkin, Mikhail ; Hsu, Daniel ; Ma, Siyuan ; Mandal, Soumik ( August 2019 , Proceedings of the National Academy of Sciences)

   Breakthroughs in machine learning are rapidly changing science and society, yet our fundamental understanding of this technology has lagged far behind. Indeed, one of the central tenets of the field, the bias–variance trade-off, appears to be at odds with the observed behavior of methods used in modern machine-learning practice. The bias–variance trade-off implies that a model should balance underfitting and overfitting: Rich enough to express underlying structure in data and simple enough to avoid fitting spurious patterns. However, in modern practice, very rich models such as neural networks are trained to exactly fit (i.e., interpolate) the data. Classically, such models would be considered overfitted, and yet they often obtain high accuracy on test data. This apparent contradiction has raised questions about the mathematical foundations of machine learning and their relevance to practitioners. In this paper, we reconcile the classical understanding and the modern practice within a unified performance curve. This “double-descent” curve subsumes the textbook U-shaped bias–variance trade-off curve by showing how increasing model capacity beyond the point of interpolation results in improved performance. We provide evidence for the existence and ubiquity of double descent for a wide spectrum of models and datasets, and we posit a mechanism for its emergence. This connection between the performance and the structure of machine-learning models delineates the limits of classical analyses and has implications for both the theory and the practice of machine learning. 

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3. Stochastic Gradient and Langevin Processes

   Cheng, Xiang ; Yin, Dong ; Bartlett, Peter L. ; Jordan, Michael ( January 2020 , Proceedings of the 37th International Conference on Machine Learning)

   Daumé III, Hal ; Singh, Aarti (Ed.)

   We prove quantitative convergence rates at which discrete Langevin-like processes converge to the invariant distribution of a related stochastic differential equation. We study the setup where the additive noise can be non-Gaussian and state-dependent and the potential function can be non-convex. We show that the key properties of these processes depend on the potential function and the second moment of the additive noise. We apply our theoretical findings to studying the convergence of Stochastic Gradient Descent (SGD) for non-convex problems and corroborate them with experiments using SGD to train deep neural networks on the CIFAR-10 dataset. 

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4. Robust Sparse Regularization: Defending Adversarial Attacks Via Regularized Sparse Network

   https://doi.org/10.1145/3386263.3407651  

   Rakin, Adnan Siraj ; He, Zhezhi ; Yang, Li ; Wang, Yanzhi ; Wang, Liqiang ; Fan, Deliang ( September 2020 , Proceedings of the 2020 on Great Lakes Symposium on VLSI)

   null (Ed.)

   Deep Neural Network (DNN) trained by the gradient descent method is known to be vulnerable to maliciously perturbed adversarial input, aka. adversarial attack. As one of the countermeasures against adversarial attacks, increasing the model capacity for DNN robustness enhancement was discussed and reported as an effective approach by many recent works. In this work, we show that shrinking the model size through proper weight pruning can even be helpful to improve the DNN robustness under adversarial attack. For obtaining a simultaneously robust and compact DNN model, we propose a multi-objective training method called Robust Sparse Regularization (RSR), through the fusion of various regularization techniques, including channel-wise noise injection, lasso weight penalty, and adversarial training. We conduct extensive experiments to show the effectiveness of RSR against popular white-box (i.e., PGD and FGSM) and black-box attacks. Thanks to RSR, 85 % weight connections of ResNet-18 can be pruned while still achieving 0.68 % and 8.72 % improvement in clean- and perturbed-data accuracy respectively on CIFAR-10 dataset, in comparison to its PGD adversarial training baseline. 

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5. Implicit regularization for deep neural networks driven by an Orstein-Uhlenbeck like process

   Blanc, Guy ; Gupta, Neha ; Valiant, Gregory ; Valiant, Paul ( January 2020 , 33rd Annual Conference on Learning Theory (COLT))

   We consider networks, trained via stochastic gradient descent to minimize L2 loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization term corresponding to the sum over the datapoints, of the squared L2 of the gradient of the model with respect to the parameter vector, evaluated at each data point. This holds for networks of any connectivity, width,depth, and choice of activation function. We interpret this implicit regularization term for three simple settings: matrix sensing, two layer ReLU networks trained on one-dimensional data, and two layer networks with sigmoid activations trained on a single datapoint. For these settings, we show why this new and general implicit regularization effect drives the networks towards "simple" models. 

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