We present a novel regularization approach to train neural networks that enjoys better generalization and test error than standard stochastic gradient descent. Our approach is based on the principles of cross-validation, where a validation set is used to limit the model overfitting. We formulate such principles as a bilevel optimization problem. This formulation allows us to define the optimization of a cost on the validation set subject to another optimization on the training set. The overfitting is controlled by introducing weights on each mini-batch in the training set and by choosing their values so that they minimize the error on the validation set. In practice, these weights define mini-batch learning rates in a gradient descent update equation that favor gradients with better generalization capabilities. Because of its simplicity, this approach can be integrated with other regularization methods and training schemes. We evaluate extensively our proposed algorithm on several neural network architectures and datasets, and find that it consistently improves the generalization of the model, especially when labels are noisy.
One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization parameter from data by means of optimization. This approach can be interpreted as solving an empirical risk minimization problem, and we analyze its performance in the large data sample size limit for general nonlinear problems. We demonstrate how to implement our framework on linear inverse problems, where we can further show the inverse accuracy does not …
Technical University of Munich（慕尼黑工业大学）；Munich Data Science Institute（慕尼黑数据科学研究所）；Polytechnique Montréal（蒙特利尔理工大学）；Pontifical Catholic University of Rio de Janeiro（里约热内卢天主教大学）
We study the feature-based newsvendor problem, in which a decision-maker has access to historical data consisting of demand observations and exogenous features. In this setting, we investigate feature selection, aiming to derive sparse, explainable models with improved out-of-sample performance. Up to now, state-of-the-art methods utilize regularization, which penalizes the number of selected features or the norm of the solution vector. As an alternative, we introduce a novel bilevel programming formulation. The upper-level problem selects a subset of features that minimizes an estimate of the…
Texas A&M University（得克萨斯农工大学）；University of Iowa（爱荷华大学）
In this paper, we study multi-block min-max bilevel optimization problems, where the upper level is non-convex strongly-concave minimax objective and the lower level is a strongly convex objective, and there are multiple blocks of dual variables and lower level problems. Due to the intertwined multi-block min-max bilevel structure, the computational cost at each iteration could be prohibitively high, especially with a large number of blocks. To tackle this challenge, we present a single-loop randomized stochastic algorithm, which requires updates for only a constant number of blocks at each it…
China University of Petroleum, Beijing Campus (CUP)（中国石油大学（北京）校区）；Department of Automation, College of Information Science and Engineering, China University of Petroleum, Beijing Campus (CUP)（中国石油大学（北京）校区信息科学与工程学院自动化系）
Recent years have witnessed enormous progress of online learning. However, a major challenge on the road to artificial agents is concept drift, that is, the data probability distribution would change where the data instance arrives sequentially in a stream fashion, which would lead to catastrophic forgetting and degrade the performance of the model. In this paper, we proposed a new Bilevel Online Deep Learning (BODL) framework, which combine bilevel optimization strategy and online ensemble classifier. In BODL algorithm, we use an ensemble classifier, which use the output of different hidden l…
IRIT, University of Toulouse（图卢兹大学 IRIT 研究所）；IRT Saint-Exupery（圣埃克苏佩里技术研究所）；ISAE-SUPAERO, University of Toulouse（图卢兹大学 ISAE-SUPAERO 高等航空与航天学院）
Differentiable Architecture Search (DARTS) is a recent neural architecture search (NAS) method based on a differentiable relaxation. Due to its success, numerous variants analyzing and improving parts of the DARTS framework have recently been proposed. By considering the problem as a constrained bilevel optimization, we present and analyze DARTS-PRIME, a variant including improvements to architectural weight update scheduling and regularization towards discretization. We propose a dynamic schedule based on per-minibatch network information to make architecture updates more informed, as well as…
Indian Institute of Management Ahmedabad（印度管理学院艾哈迈达巴德分校）
Hyperparameter tuning is an active area of research in machine learning, where the aim is to identify the optimal hyperparameters that provide the best performance on the validation set. Hyperparameter tuning is often achieved using naive techniques, such as random search and grid search. However, most of these methods seldom lead to an optimal set of hyperparameters and often get very expensive. In this paper, we propose a bilevel solution method for solving the hyperparameter optimization problem that does not suffer from the drawbacks of the earlier studies. The proposed method is general a…
Huawei Research America（华为美国研究院）；Jilin University（吉林大学）；Innopeak Technology（Innopeak科技）
Regularization plays an important role in generalization of deep neural networks, which are often prone to overfitting with their numerous parameters. L1 and L2 regularizers are common regularization tools in machine learning with their simplicity and effectiveness. However, we observe that imposing strong L1 or L2 regularization with stochastic gradient descent on deep neural networks easily fails, which limits the generalization ability of the underlying neural networks. To understand this phenomenon, we first investigate how and why learning fails when strong regularization is imposed on de…
Bilevel Learning via Inexact Stochastic Gradient Descent
EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath（巴斯大学统计应用数学博士培训中心）；Australian Research Council（澳大利亚研究理事会）；Indian Institute of Technology (IIT), Kharagpur（印度理工学院（Kharagpur分校））；University of Melbourne（墨尔本大学）；Department of Electronics & Electrical Communication Engineering（电子与电气通信工程系）；School of Mathematics and Statistics（数学与统计学学院）
Generalization is essential for deep learning. In contrast to previous works claiming that Deep Neural Networks (DNNs) have an implicit regularization implemented by the stochastic gradient descent, we demonstrate explicitly Bayesian regularizations in a specific category of DNNs, i.e., Convolutional Neural Networks (CNNs). First, we introduce a novel probabilistic representation for the hidden layers of CNNs and demonstrate that CNNs correspond to Bayesian networks with the serial connection. Furthermore, we show that the hidden layers close to the input formulate prior distributions, thus CN…
Mila Institute（米拉研究所）；Saint Mary’s University（圣玛丽大学）；Université de Montréal（蒙特利尔大学）
We theoretically analyze the Feedback Alignment (FA) algorithm, an efficient alternative to backpropagation for training neural networks. We provide convergence guarantees with rates for deep linear networks for both continuous and discrete dynamics. Additionally, we study incremental learning phenomena for shallow linear networks. Interestingly, certain specific initializations imply that negligible components are learned before the principal ones, thus potentially negatively affecting the effectiveness of such a learning algorithm; a phenomenon we classify as implicit anti-regularization. We…
This article is an overview of supervised machine learning problems for regression and classification. Topics include: kernel methods, training by stochastic gradient descent, deep learning architecture, losses for classification, statistical learning theory, and dimension independent generalization bounds. Implicit regularization in deep learning examples are presented, including data augmentation, adversarial training, and additive noise. These methods are reframed as explicit gradient regularization.
Unsupervised neural networks, such as restricted Boltzmann machines (RBMs) and deep belief networks (DBNs), are powerful tools for feature selection and pattern recognition tasks. We demonstrate that overfitting occurs in such models just as in deep feedforward neural networks, and discuss possible regularization methods to reduce overfitting. We also propose a "partial" approach to improve the efficiency of Dropout/DropConnect in this scenario, and discuss the theoretical justification of these methods from model convergence and likelihood bounds. Finally, we compare the performance of these …
Recently, a large number of efficient deep learning methods for solving inverse problems have been developed and show outstanding numerical performance. For these deep learning methods, however, a solid theoretical foundation in the form of reconstruction guarantees is missing. In contrast, for classical reconstruction methods, such as convex variational and frame-based regularization, theoretical convergence and convergence rate results are well established. In this paper, we introduce deep synthesis regularization (DESYRE) using neural networks as nonlinear synthesis operator bridging the ga…
University of Bath（巴斯大学）；Australian National University（澳大利亚国立大学）
Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing this strategy leads to a nested optimization problem (known as bilevel optimization) which is computationally very difficult to handle. It is common when solving the upper-level problem to assume access to exact solutions of the lower-level problem, which is practically infe…
Nanyang Technological University（南洋理工大学）；Harvard University（哈佛大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
Deep neural networks (DNNs) have shown superior performances on various multimodal learning problems. However, it often requires huge efforts to adapt DNNs to individual multimodal tasks by manually engineering unimodal features and designing multimodal feature fusion strategies. This paper proposes Bilevel Multimodal Neural Architecture Search (BM-NAS) framework, which makes the architecture of multimodal fusion models fully searchable via a bilevel searching scheme. At the upper level, BM-NAS selects the inter/intra-modal feature pairs from the pretrained unimodal backbones. At the lower lev…
University of Florida（佛罗里达大学）；Georgia State University（佐治亚州立大学）
Purpose: This work aims at developing a generalizable MRI reconstruction model in the meta-learning framework. The standard benchmarks in meta-learning are challenged by learning on diverse task distributions. The proposed network learns the regularization function in a variational model and reconstructs MR images with various under-sampling ratios or patterns that may or may not be seen in the training data by leveraging a heterogeneous dataset. Methods: We propose an unrolling network induced by learnable optimization algorithms (LOA) for solving our nonconvex nonsmooth variational model for…
Facebook AI Research（脸书人工智能研究院）；Google Research（谷歌研究院）
We present SOSELETO (SOurce SELEction for Target Optimization), a new method for exploiting a source dataset to solve a classification problem on a target dataset. SOSELETO is based on the following simple intuition: some source examples are more informative than others for the target problem. To capture this intuition, source samples are each given weights; these weights are solved for jointly with the source and target classification problems via a bilevel optimization scheme. The target therefore gets to choose the source samples which are most informative for its own classification task. F…
Dalian University of Technology（大连理工大学）；Guilin University of Electronic Technology（桂林电子科技大学）；Children’s Hospital of Philadelphia（费城儿童医院）；University of Pennsylvania（宾夕法尼亚大学）
Conventional deformable registration methods aim at solving an optimization model carefully designed on image pairs and their computational costs are exceptionally high. In contrast, recent deep learning based approaches can provide fast deformation estimation. These heuristic network architectures are fully data-driven and thus lack explicit geometric constraints, e.g., topology-preserving, which are indispensable to generate plausible deformations. We design a new deep learning based framework to optimize a diffeomorphic model via multi-scale propagation in order to integrate advantages and …
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…
This paper seeks to answer the question: as the (near-) orthogonality of weights is found to be a favorable property for training deep convolutional neural networks, how can we enforce it in more effective and easy-to-use ways? We develop novel orthogonality regularizations on training deep CNNs, utilizing various advanced analytical tools such as mutual coherence and restricted isometry property. These plug-and-play regularizations can be conveniently incorporated into training almost any CNN without extra hassle. We then benchmark their effects on state-of-the-art models: ResNet, WideResNet,…
In spite of achieving revolutionary successes in machine learning, deep convolutional neural networks have been recently found to be vulnerable to adversarial attacks and difficult to generalize to novel test images with reasonably large geometric transformations. Inspired by a recent neuroscience discovery revealing that primate brain employs disentangled shape and appearance representations for object recognition, we propose a general disentangled deep autoencoding regularization framework that can be easily applied to any deep embedding based classification model for improving the robustnes…
We propose a new algorithm for training deep neural networks (DNNs) with binary weights. In particular, we first cast the problem of training binary neural networks (BiNNs) as a bilevel optimization instance and subsequently construct flexible relaxations of this bilevel program. The resulting training method shares its algorithmic simplicity with several existing approaches to train BiNNs, in particular with the straight-through gradient estimator successfully employed in BinaryConnect and subsequent methods. In fact, our proposed method can be interpreted as an adaptive variant of the origin…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
We present experiments demonstrating that some other form of capacity control, different from network size, plays a central role in learning multilayer feed-forward networks. We argue, partially through analogy to matrix factorization, that this is an inductive bias that can help shed light on deep learning.
We apply concepts from manifold regularization to develop new regularization techniques for training locally stable deep neural networks. Our regularizers are based on a sparsification of the graph Laplacian which holds with high probability when the data is sparse in high dimensions, as is common in deep learning. Empirically, our networks exhibit stability in a diverse set of perturbation models, including $\ell_2$, $\ell_\infty$, and Wasserstein-based perturbations; in particular, we achieve 40% adversarial accuracy on CIFAR-10 against an adaptive PGD attack using $\ell_\infty$ perturbation…
Regularization is one of the crucial ingredients of deep learning, yet the term regularization has various definitions, and regularization methods are often studied separately from each other. In our work we present a systematic, unifying taxonomy to categorize existing methods. We distinguish methods that affect data, network architectures, error terms, regularization terms, and optimization procedures. We do not provide all details about the listed methods; instead, we present an overview of how the methods can be sorted into meaningful categories and sub-categories. This helps revealing lin…
Philips Medical Systems Technologies Ltd.（飞利浦医疗系统科技有限公司）
Deep sparse auto-encoders with mixed structure regularization (MSR) in addition to explicit sparsity regularization term and stochastic corruption of the input data with Gaussian noise have the potential to improve unsupervised abnormality detection. Unsupervised abnormality detection based on identifying outliers using deep sparse auto-encoders is a very appealing approach for medical computer aided detection systems as it requires only healthy data for training rather than expert annotated abnormality. In the task of detecting coronary artery disease from Coronary Computed Tomography Angiogr…
Regularization plays a vital role in the context of deep learning by preventing deep neural networks from the danger of overfitting. This paper proposes a novel deep learning regularization method named as DL-Reg, which carefully reduces the nonlinearity of deep networks to a certain extent by explicitly enforcing the network to behave as much linear as possible. The key idea is to add a linear constraint to the objective function of the deep neural networks, which is simply the error of a linear mapping from the inputs to the outputs of the model. More precisely, the proposed DL-Reg carefully…
The extent to which the arbitrarily selected L2 regularization hyperparameter value affects the outcome of semantic segmentation with deep learning is demonstrated. Demonstrations rely on training U-net on small LGE-MRI datasets using the arbitrarily selected L2 regularization values. The remaining hyperparameters are to be manually adjusted or tuned only when 10 % of all epochs are reached before the training validation accuracy reaches 90%. Semantic segmentation with deep learning outcomes are objectively and subjectively evaluated against the manual ground truth segmentation.
This paper presents a predictive model for estimating regularization parameters of diffeomorphic image registration. We introduce a novel framework that automatically determines the parameters controlling the smoothness of diffeomorphic transformations. Our method significantly reduces the effort of parameter tuning, which is time and labor-consuming. To achieve the goal, we develop a predictive model based on deep convolutional neural networks (CNN) that learns the mapping between pairwise images and the regularization parameter of image registration. In contrast to previous methods that esti…
Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）
In this paper, regularized lightweight deep convolutional neural network models, capable of effectively operating in real-time on devices with restricted computational power for high-resolution video input are proposed. Furthermore, a novel regularization method motivated by the Quadratic Mutual Information, in order to improve the generalization ability of the utilized models is proposed. Extensive experiments on various binary classification problems involved in autonomous systems are performed, indicating the effectiveness of the proposed models as well as of the proposed regularizer.
Overfitting is a crucial problem in deep neural networks, even in the latest network architectures. In this paper, to relieve the overfitting effect of ResNet and its improvements (i.e., Wide ResNet, PyramidNet, and ResNeXt), we propose a new regularization method called ShakeDrop regularization. ShakeDrop is inspired by Shake-Shake, which is an effective regularization method, but can be applied to ResNeXt only. ShakeDrop is more effective than Shake-Shake and can be applied not only to ResNeXt but also ResNet, Wide ResNet, and PyramidNet. An important key is to achieve stability of training.…
L2 regularization for weights in neural networks is widely used as a standard training trick. However, L2 regularization for gamma, a trainable parameter of batch normalization, remains an undiscussed mystery and is applied in different ways depending on the library and practitioner. In this paper, we study whether L2 regularization for gamma is valid. To explore this issue, we consider two approaches: 1) variance control to make the residual network behave like identity mapping and 2) stable optimization through the improvement of effective learning rate. Through two analyses, we specify the …
Recently established equivalences between differential equations and the structure of neural networks enabled some interpretation of training of a neural network as partial-differential-equation (PDE) constrained optimization. We add to the previously established connections, explicit regularization that is particularly beneficial in the case of single large-scale examples with partial annotation. We show that explicit regularization of model parameters in PDE constrained optimization translates to regularization of the network output. Examination of the structure of the corresponding Lagrangi…
IMT Atlantique（IMT大西洋）；LaTIM UMR 1101, Inserm（LaTIM UMR 1101，法国国家健康与医学研究院）；Symbiosis Centre for Medical Image Analysis, Symbiosis International University（共生国际大学共生医学图像分析中心）；Fondation ILDYS（ILDYS基金会）
Automatic segmentation of the musculoskeletal system in pediatric magnetic resonance (MR) images is a challenging but crucial task for morphological evaluation in clinical practice. We propose a deep learning-based regularized segmentation method for multi-structure bone delineation in MR images, designed to overcome the inherent scarcity and heterogeneity of pediatric data. Based on a newly devised shape code discriminator, our adversarial regularization scheme enforces the deep network to follow a learnt shape representation of the anatomy. The novel shape priors based adversarial regulariza…
