We investigate the capacity control provided by dropout in various machine learning problems. First, we study dropout for matrix completion, where it induces a data-dependent regularizer that, in expectation, equals the weighted trace-norm of the product of the factors. In deep learning, we show that the data-dependent regularizer due to dropout directly controls the Rademacher complexity of the underlying class of deep neural networks. These developments enable us to give concrete generalization error bounds for the dropout algorithm in both matrix completion as well as training deep neural networks. We evaluate our theoretical findings on real-world datasets, including MovieLens, MNIST, and Fashion-MNIST.
Stanford University（斯坦福大学）；Microsoft Research（微软研究院）；University of Washington（华盛顿大学）
Dropout is a widely-used regularization technique, often required to obtain state-of-the-art for a number of architectures. This work demonstrates that dropout introduces two distinct but entangled regularization effects: an explicit effect (also studied in prior work) which occurs since dropout modifies the expected training objective, and, perhaps surprisingly, an additional implicit effect from the stochasticity in the dropout training update. This implicit regularization effect is analogous to the effect of stochasticity in small mini-batch stochastic gradient descent. We disentangle these…
Unlike machines, humans learn through rapid, abstract model-building. The role of a teacher is not simply to hammer home right or wrong answers, but rather to provide intuitive comments, comparisons, and explanations to a pupil. This is what the Learning Under Privileged Information (LUPI) paradigm endeavors to model by utilizing extra knowledge only available during training. We propose a new LUPI algorithm specifically designed for Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs). We propose to use a heteroscedastic dropout (i.e. dropout with a varying variance) and …
Norwegian University of Life Sciences（挪威生命科学大学）；Rosim Company（Rosim公司）
Exploiting capacity of sewer system using decentralized control is a cost effective mean of minimizing the overflow. Given the size of the real sewer system, exploiting all the installed control structures in the sewer pipes can be challenging. This paper presents a divide and conquer solution to implement decentralized control measures based on unsupervised learning algorithms. A sewer system is first divided into a number of subcatchments. A series of natural and built factors that have the impact on sewer system performance is then collected. Clustering algorithms are then applied to groupi…
An important problem in training deep networks with high capacity is to ensure that the trained network works well when presented with new inputs outside the training dataset. Dropout is an effective regularization technique to boost the network generalization in which a random subset of the elements of the given data and the extracted features are set to zero during the training process. In this paper, a new randomized regularization technique in which we withhold a random part of the data without necessarily turning off the neurons/data-elements is proposed. In the proposed method, of which …
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
The cross-entropy loss commonly used in deep learning is closely related to the defining properties of optimal representations, but does not enforce some of the key properties. We show that this can be solved by adding a regularization term, which is in turn related to injecting multiplicative noise in the activations of a Deep Neural Network, a special case of which is the common practice of dropout. We show that our regularized loss function can be efficiently minimized using Information Dropout, a generalization of dropout rooted in information theoretic principles that automatically adapts…
Deep neural networks are learning models with a very high capacity and therefore prone to over-fitting. Many regularization techniques such as Dropout, DropConnect, and weight decay all attempt to solve the problem of over-fitting by reducing the capacity of their respective models (Srivastava et al., 2014), (Wan et al., 2013), (Krogh & Hertz, 1992). In this paper we introduce a new form of regularization that guides the learning problem in a way that reduces over-fitting without sacrificing the capacity of the model. The mistakes that models make in early stages of training carry information …
Istituto Italiano di Tecnologia（意大利技术研究院）；Johns Hopkins University（约翰斯·霍普金斯大学）
Dropout is a simple yet effective algorithm for regularizing neural networks by randomly dropping out units through Bernoulli multiplicative noise, and for some restricted problem classes, such as linear or logistic regression, several theoretical studies have demonstrated the equivalence between dropout and a fully deterministic optimization problem with data-dependent Tikhonov regularization. This work presents a theoretical analysis of dropout for matrix factorization, where Bernoulli random variables are used to drop a factor, thereby attempting to control the size of the factorization. Wh…
Dropout and other feature noising schemes control overfitting by artificially corrupting the training data. For generalized linear models, dropout performs a form of adaptive regularization. Using this viewpoint, we show that the dropout regularizer is first-order equivalent to an L2 regularizer applied after scaling the features by an estimate of the inverse diagonal Fisher information matrix. We also establish a connection to AdaGrad, an online learning algorithm, and find that a close relative of AdaGrad operates by repeatedly solving linear dropout-regularized problems. By casting dropout …
Policy gradient methods have had great success in solving continuous control tasks, yet the stochastic nature of such problems makes deterministic value estimation difficult. We propose an approach which instead estimates a distribution by fitting the value function with a Bayesian Neural Network. We optimize an $α$-divergence objective with Bayesian dropout approximation to learn and estimate this distribution. We show that using the Monte Carlo posterior mean of the Bayesian value function distribution, rather than a deterministic network, improves stability and performance of policy gradien…
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 co…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of Oxford（牛津大学）；Alan Turing Institute（艾伦·图灵研究所）
We propose a novel approach for inferring the individualized causal effects of a treatment (intervention) from observational data. Our approach conceptualizes causal inference as a multitask learning problem; we model a subject's potential outcomes using a deep multitask network with a set of shared layers among the factual and counterfactual outcomes, and a set of outcome-specific layers. The impact of selection bias in the observational data is alleviated via a propensity-dropout regularization scheme, in which the network is thinned for every training example via a dropout probability that …
Monte Carlo (MC) dropout is a simple and efficient ensembling method that can improve the accuracy and confidence calibration of high-capacity deep neural network models. However, MC dropout is not as effective as more compute-intensive methods such as deep ensembles. This performance gap can be attributed to the relatively poor quality of individual models in the MC dropout ensemble and their lack of diversity. These issues can in turn be traced back to the coupled training and substantial parameter sharing of the dropout models. Motivated by this perspective, we propose a strategy to compute…
Google Research（谷歌研究院）；Carnegie Mellon University（卡内基梅隆大学）
Neural networks are often over-parameterized and hence benefit from aggressive regularization. Conventional regularization methods, such as Dropout or weight decay, do not leverage the structures of the network's inputs and hidden states. As a result, these conventional methods are less effective than methods that leverage the structures, such as SpatialDropout and DropBlock, which randomly drop the values at certain contiguous areas in the hidden states and setting them to zero. Although the locations of dropout areas random, the patterns of SpatialDropout and DropBlock are manually designed …
Overfitting is one of the most common problems when training deep neural networks on comparatively small datasets. Here, we demonstrate that neural network activation sparsity is a reliable indicator for overfitting which we utilize to propose novel targeted sparsity visualization and regularization strategies. Based on these strategies we are able to understand and counteract overfitting caused by activation sparsity and filter correlation in a targeted layer-by-layer manner. Our results demonstrate that targeted sparsity regularization can efficiently be used to regularize well-known dataset…
Explaining the unreasonable effectiveness of deep learning has eluded researchers around the globe. Various authors have described multiple metrics to evaluate the capacity of deep architectures. In this paper, we allude to the radius margin bounds described for a support vector machine (SVM) with hinge loss, apply the same to the deep feed-forward architectures and derive the Vapnik-Chervonenkis (VC) bounds which are different from the earlier bounds proposed in terms of number of weights of the network. In doing so, we also relate the effectiveness of techniques like Dropout and Dropconnect …
Lehrstuhl für Robotik und Echtzeitsysteme（机器人与实时系统教席）；Fakultät für Informatik（计算机科学学院）；Technische Universität München（慕尼黑工业大学）
Recurrent Neural Networks (RNNs) are rich models for the processing of sequential data. Recent work on advancing the state of the art has been focused on the optimization or modelling of RNNs, mostly motivated by adressing the problems of the vanishing and exploding gradients. The control of overfitting has seen considerably less attention. This paper contributes to that by analyzing fast dropout, a recent regularization method for generalized linear models and neural networks from a back-propagation inspired perspective. We show that fast dropout implements a quadratic form of an adaptive, pe…
Tencent Inc.（腾讯公司）；University of Manchester（曼彻斯特大学）
Recent success in Deep Reinforcement Learning (DRL) methods has shown that policy optimization with respect to an off-policy distribution via importance sampling is effective for sample reuse. In this paper, we show that the use of importance sampling could introduce high variance in the objective estimate. Specifically, we show in a principled way that the variance of importance sampling estimate grows quadratically with importance ratios and the large ratios could consequently jeopardize the effectiveness of surrogate objective optimization. We then propose a technique called sample dropout …
Dropout and other feature noising schemes have shown promising results in controlling over-fitting by artificially corrupting the training data. Though extensive theoretical and empirical studies have been performed for generalized linear models, little work has been done for support vector machines (SVMs), one of the most successful approaches for supervised learning. This paper presents dropout training for both linear SVMs and the nonlinear extension with latent representation learning. For linear SVMs, to deal with the intractable expectation of the non-smooth hinge loss under corrupting d…
Montréal Institute for Learning Algorithms（蒙特利尔学习算法研究所）；Université de Montréal（蒙特利尔大学）；Jagiellonian University（雅盖隆大学）；McGill University（麦吉尔大学）；University of California, Berkeley（加州大学伯克利分校）；Polytechnique Montréal（蒙特利尔理工学院）；University of Bonn（波恩大学）
We examine the role of memorization in deep learning, drawing connections to capacity, generalization, and adversarial robustness. While deep networks are capable of memorizing noise data, our results suggest that they tend to prioritize learning simple patterns first. In our experiments, we expose qualitative differences in gradient-based optimization of deep neural networks (DNNs) on noise vs. real data. We also demonstrate that for appropriately tuned explicit regularization (e.g., dropout) we can degrade DNN training performance on noise datasets without compromising generalization on real…
Traversing intersections is a challenging problem for autonomous vehicles, especially when the intersections do not have traffic control. Recently deep reinforcement learning has received massive attention due to its success in dealing with autonomous driving tasks. In this work, we address the problem of traversing unsignalized intersections using a novel curriculum for deep reinforcement learning. The proposed curriculum leads to: 1) A faster training process for the reinforcement learning agent, and 2) Better performance compared to an agent trained without curriculum. Our main contribution…
End-to-end trained neural networks (NNs) are a compelling approach to autonomous vehicle control because of their ability to learn complex tasks without manual engineering of rule-based decisions. However, challenging road conditions, ambiguous navigation situations, and safety considerations require reliable uncertainty estimation for the eventual adoption of full-scale autonomous vehicles. Bayesian deep learning approaches provide a way to estimate uncertainty by approximating the posterior distribution of weights given a set of training data. Dropout training in deep NNs approximates Bayesi…
Southern University of Science and Technology（南方科技大学）；Research Institute of Trustworthy Autonomous Systems, SUSTech（南方科技大学可信自主系统研究院）；The Chinese University of Hong Kong（香港中文大学）；University of Tokyo（东京大学）；Huawei Technologies CO.LTD（华为技术有限公司）
Spatial-temporal (ST) graph modeling, such as traffic speed forecasting and taxi demand prediction, is an important task in deep learning area. However, for the nodes in graph, their ST patterns can vary greatly in difficulties for modeling, owning to the heterogeneous nature of ST data. We argue that unveiling the nodes to the model in a meaningful order, from easy to complex, can provide performance improvements over traditional training procedure. The idea has its root in Curriculum Learning which suggests in the early stage of training models can be sensitive to noise and difficult samples…
Ecole Polytechnique（巴黎综合理工学院）；Université Paris Dauphine-PSL（巴黎多芬-PSL大学）
The capacity of a ML model refers to the range of functions this model can approximate. It impacts both the complexity of the patterns a model can learn but also memorization, the ability of a model to fit arbitrary labels. We propose Adaptive Capacity (AdaCap), a training scheme for Feed-Forward Neural Networks (FFNN). AdaCap optimizes the capacity of FFNN so it can capture the high-level abstract representations underlying the problem at hand without memorizing the training dataset. AdaCap is the combination of two novel ingredients, the Muddling labels for Regularization (MLR) loss and the …
University of Chinese Academy of Sciences（中国科学院大学）
Despite its remarkable empirical success as a highly competitive branch of artificial intelligence, deep learning is often blamed for its widely known low interpretation and lack of firm and rigorous mathematical foundation. However, most theoretical endeavor is devoted in discriminative deep learning case, whose complementary part is generative deep learning. To the best of our knowledge, we firstly highlight landscape of empirical error in generative case to complete the full picture through exquisite design of image super resolution under norm based capacity control. Our theoretical advance…
Deep neural networks have amply demonstrated their prowess but estimating the reliability of their predictions remains challenging. Deep Ensembles are widely considered as being one of the best methods for generating uncertainty estimates but are very expensive to train and evaluate. MC-Dropout is another popular alternative, which is less expensive, but also less reliable. Our central intuition is that there is a continuous spectrum of ensemble-like models of which MC-Dropout and Deep Ensembles are extreme examples. The first uses an effectively infinite number of highly correlated models whi…
