We introduce a novel regularization approach for deep learning that incorporates and respects the underlying graphical structure of the neural network. Existing regularization methods often focus on dropping/penalizing weights in a global manner that ignores the connectivity structure of the neural network. We propose to use the Fiedler value of the neural network's underlying graph as a tool for regularization. We provide theoretical support for this approach via spectral graph theory. We list several useful properties of the Fiedler value that makes it suitable in regularization. We provide an approximate, variational approach for fast computation in practical training of neural networks. We provide bounds on such approximations. We provide an alternative but equivalent formulation of this framework in the form of a structurally weighted L1 penalty, thus linking our approach to sparsity induction. We performed experiments on datasets that compare Fiedler regularization with traditional regularization methods such as dropout and weight decay. Results demonstrate the efficacy of Fiedler regularization.
Texas A&M University（德克萨斯农工大学）；New York University（纽约大学）
We study the implicit regularization of gradient descent towards structured sparsity via a novel neural reparameterization, which we call a diagonally grouped linear neural network. We show the following intriguing property of our reparameterization: gradient descent over the squared regression loss, without any explicit regularization, biases towards solutions with a group sparsity structure. In contrast to many existing works in understanding implicit regularization, we prove that our training trajectory cannot be simulated by mirror descent. We analyze the gradient dynamics of the correspon…
Ecole Centrale de Paris（巴黎中央理工学院）；EPFL（洛桑联邦理工学院）；University at Albany（奥尔巴尼大学）；University College London（伦敦大学学院）
During the past years there has been an explosion of interest in learning methods based on sparsity regularization. In this paper, we discuss a general class of such methods, in which the regularizer can be expressed as the composition of a convex function $ω$ with a linear function. This setting includes several methods such the group Lasso, the Fused Lasso, multi-task learning and many more. We present a general approach for solving regularization problems of this kind, under the assumption that the proximity operator of the function $ω$ is available. Furthermore, we comment on the applicati…
STADIUS, ESAT, KU Leuven（STADIUS、ESAT、鲁汶大学）；Research Center for Intelligent Robotics, Zhejiang Lab（之江实验室智能机器人研究中心）；School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen（哈尔滨工业大学（深圳）机械工程与自动化学院）
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…
Notwithstanding the popularity of conventional clustering algorithms such as K-means and probabilistic clustering, their clustering results are sensitive to the presence of outliers in the data. Even a few outliers can compromise the ability of these algorithms to identify meaningful hidden structures rendering their outcome unreliable. This paper develops robust clustering algorithms that not only aim to cluster the data, but also to identify the outliers. The novel approaches rely on the infrequent presence of outliers in the data which translates to sparsity in a judiciously chosen domain. …
In high-dimensional statistical inference, sparsity regularizations have shown advantages in consistency and convergence rates for coefficient estimation. We consider a generalized version of Sparse-Group Lasso which captures both element-wise sparsity and group-wise sparsity simultaneously. We state one universal theorem which is proved to obtain results on consistency and convergence rates for different forms of double sparsity regularization. The universality of the results lies in an generalization of various convergence rates for single regularization cases such as LASSO and group LASSO a…
Stanford University（斯坦福大学）；Harvard University SEAS（哈佛大学工程与应用科学学院）；University of Basel（巴塞尔大学）；University of Siena（锡耶纳大学）
The lack of interpretability remains a key barrier to the adoption of deep models in many applications. In this work, we explicitly regularize deep models so human users might step through the process behind their predictions in little time. Specifically, we train deep time-series models so their class-probability predictions have high accuracy while being closely modeled by decision trees with few nodes. Using intuitive toy examples as well as medical tasks for treating sepsis and HIV, we demonstrate that this new tree regularization yields models that are easier for humans to simulate than s…
In this paper, we provide a Banach-space formulation of supervised learning with generalized total-variation (gTV) regularization. We identify the class of kernel functions that are admissible in this framework. Then, we propose a variation of supervised learning in a continuous-domain hybrid search space with gTV regularization. We show that the solution admits a multi-kernel expansion with adaptive positions. In this representation, the number of active kernels is upper-bounded by the number of data points while the gTV regularization imposes an $\ell_1$ penalty on the kernel coefficients. F…
Instituto de Telecomunicações（电信研究所）；Instituto Superior Técnico（高等技术学院）
We propose a novel SPARsity and Clustering (SPARC) regularizer, which is a modified version of the previous octagonal shrinkage and clustering algorithm for regression (OSCAR), where, the proposed regularizer consists of a $K$-sparse constraint and a pair-wise $\ell_{\infty}$ norm restricted on the $K$ largest components in magnitude. The proposed regularizer is able to separably enforce $K$-sparsity and encourage the non-zeros to be equal in magnitude. Moreover, it can accurately group the features without shrinking their magnitude. In fact, SPARC is closely related to OSCAR, so that the prox…
We present a data dependent generalization bound for a large class of regularized algorithms which implement structured sparsity constraints. The bound can be applied to standard squared-norm regularization, the Lasso, the group Lasso, some versions of the group Lasso with overlapping groups, multiple kernel learning and other regularization schemes. In all these cases competitive results are obtained. A novel feature of our bound is that it can be applied in an infinite dimensional setting such as the Lasso in a separable Hilbert space or multiple kernel learning with a countable number of ke…
We study the emergence of sparse representations in neural networks. We show that in unsupervised models with regularization, the emergence of sparsity is the result of the input data samples being distributed along highly non-linear or discontinuous manifold. We also derive a similar argument for discriminatively trained networks and present experiments to support this hypothesis. Based on our study of sparsity, we introduce a new loss function which can be used as regularization term for models like autoencoders and MLPs. Further, the same loss function can also be used as a cost function fo…
Mila（米拉人工智能研究所）；DIRO, Université de Montréal（蒙特利尔大学计算机科学与运筹学系）；ServiceNow Research（ServiceNow研究院）；Tübingen AI Center, University of Tübingen（蒂宾根大学蒂宾根人工智能中心）；Google Research（谷歌研究院）
This work introduces a novel principle we call disentanglement via mechanism sparsity regularization, which can be applied when the latent factors of interest depend sparsely on past latent factors and/or observed auxiliary variables. We propose a representation learning method that induces disentanglement by simultaneously learning the latent factors and the sparse causal graphical model that relates them. We develop a rigorous identifiability theory, building on recent nonlinear independent component analysis (ICA) results, that formalizes this principle and shows how the latent variables ca…
Multiple kernel learning (MKL), structured sparsity, and multi-task learning have recently received considerable attention. In this paper, we show how different MKL algorithms can be understood as applications of either regularization on the kernel weights or block-norm-based regularization, which is more common in structured sparsity and multi-task learning. We show that these two regularization strategies can be systematically mapped to each other through a concave conjugate operation. When the kernel-weight-based regularizer is separable into components, we can naturally consider a generati…
Rationalizing which parts of a molecule drive the predictions of a molecular graph convolutional neural network (GCNN) can be difficult. To help, we propose two simple regularization techniques to apply during the training of GCNNs: Batch Representation Orthonormalization (BRO) and Gini regularization. BRO, inspired by molecular orbital theory, encourages graph convolution operations to generate orthonormal node embeddings. Gini regularization is applied to the weights of the output layer and constrains the number of dimensions the model can use to make predictions. We show that Gini and BRO r…
We propose a novel regularization-based continual learning method, dubbed as Adaptive Group Sparsity based Continual Learning (AGS-CL), using two group sparsity-based penalties. Our method selectively employs the two penalties when learning each node based its the importance, which is adaptively updated after learning each new task. By utilizing the proximal gradient descent method for learning, the exact sparsity and freezing of the model is guaranteed, and thus, the learner can explicitly control the model capacity as the learning continues. Furthermore, as a critical detail, we re-initializ…
This paper studies structured sparse training of CNNs with a gradual pruning technique that leads to fixed, sparse weight matrices after a set number of epochs. We simplify the structure of the enforced sparsity so that it reduces overhead caused by regularization. The proposed training methodology Campfire explores pruning at granularities within a convolutional kernel and filter. We study various tradeoffs with respect to pruning duration, level of sparsity, and learning rate configuration. We show that our method creates a sparse version of ResNet-50 and ResNet-50 v1.5 on full ImageNet whil…
University of Turin（都灵大学）；Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院）
Deep neural networks include millions of learnable parameters, making their deployment over resource-constrained devices problematic. SeReNe (Sensitivity-based Regularization of Neurons) is a method for learning sparse topologies with a structure, exploiting neural sensitivity as a regularizer. We define the sensitivity of a neuron as the variation of the network output with respect to the variation of the activity of the neuron. The lower the sensitivity of a neuron, the less the network output is perturbed if the neuron output changes. By including the neuron sensitivity in the cost function…
Istituto Italiano di Tecnologia（意大利技术研究院）；University of Genova（热那亚大学）
In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric model, hence avoiding linear or additive models. The key idea is to measure the importance of each variable in the model by making use of partial derivatives. Based on this intuition we propose a new notion of nonparametric sparsity and a corresponding least squares regularization scheme. Using concepts and results from the theory of reproducing kernel Hilbe…
Polytechnic Institute of New York University（纽约大学理工学院）
Convex optimization with sparsity-promoting convex regularization is a standard approach for estimating sparse signals in noise. In order to promote sparsity more strongly than convex regularization, it is also standard practice to employ non-convex optimization. In this paper, we take a third approach. We utilize a non-convex regularization term chosen such that the total cost function (consisting of data consistency and regularization terms) is convex. Therefore, sparsity is more strongly promoted than in the standard convex formulation, but without sacrificing the attractive aspects of conv…
Factorization machines (FMs) are machine learning predictive models based on second-order feature interactions and FMs with sparse regularization are called sparse FMs. Such regularizations enable feature selection, which selects the most relevant features for accurate prediction, and therefore they can contribute to the improvement of the model accuracy and interpretability. However, because FMs use second-order feature interactions, the selection of features often causes the loss of many relevant feature interactions in the resultant models. In such cases, FMs with regularization specially d…
In this paper, we develop a new sequential regression modeling approach for data streams. Data streams are commonly found around us, e.g in a retail enterprise sales data is continuously collected every day. A demand forecasting model is an important outcome from the data that needs to be continuously updated with the new incoming data. The main challenge in such modeling arises when there is a) high dimensional and sparsity, b) need for an adaptive use of prior knowledge, and/or c) structural changes in the system. The proposed approach addresses these challenges by incorporating an adaptive …
Deep neural networks (DNNs) often require good regularizers to generalize well. Currently, state-of-the-art DNN regularization techniques consist in randomly dropping units and/or connections on each iteration of the training algorithm. Dropout and DropConnect are characteristic examples of such regularizers, that are widely popular among practitioners. However, a drawback of such approaches consists in the fact that their postulated probability of random unit/connection omission is a constant that must be heuristically selected based on the obtained performance in some validation set. To alle…
Non-negative blind source separation (non-negative BSS), which is also referred to as non-negative matrix factorization (NMF), is a very active field in domains as different as astrophysics, audio processing or biomedical signal processing. In this context, the efficient retrieval of the sources requires the use of signal priors such as sparsity. If NMF has now been well studied with sparse constraints in the direct domain, only very few algorithms can encompass non-negativity together with sparsity in a transformed domain since simultaneously dealing with two priors in two different domains i…
Modern neural network architectures typically have many millions of parameters and can be pruned significantly without substantial loss in effectiveness which demonstrates they are over-parameterized. The contribution of this work is two-fold. The first is a method for approximating a multivariate Bernoulli random variable by means of a deterministic and differentiable transformation of any real-valued multivariate random variable. The second is a method for model selection by element-wise multiplication of parameters with approximate binary gates that may be computed deterministically or stoc…
Feature selection is one of the most decisive tools in understanding data and machine learning models. Among other methods, sparsity induced by $L^{1}$ penalty is one of the simplest and best studied approaches to this problem. Although such regularization is frequently used in neural networks to achieve sparsity of weights or unit activations, it is unclear how it can be employed in the feature selection problem. This work aims at extending the neural network with ability to automatically select features by rethinking how the sparsity regularization can be used, namely, by stochastically pena…
University of Notre Dame（圣母大学）；Seoul National University（首尔国立大学）
Recent theoretical studies proved that deep neural network (DNN) estimators obtained by minimizing empirical risk with a certain sparsity constraint can attain optimal convergence rates for regression and classification problems. However, the sparsity constraint requires to know certain properties of the true model, which are not available in practice. Moreover, computation is difficult due to the discrete nature of the sparsity constraint. In this paper, we propose a novel penalized estimation method for sparse DNNs, which resolves the aforementioned problems existing in the sparsity constrai…
Politecnico di Torino（都灵理工大学）；Nuance Communications（Nuance通信公司）；Télécom ParisTech（巴黎高等电信学院）；Telecom Italia（意大利电信公司）
The ever-increasing number of parameters in deep neural networks poses challenges for memory-limited applications. Regularize-and-prune methods aim at meeting these challenges by sparsifying the network weights. In this context we quantify the output sensitivity to the parameters (i.e. their relevance to the network output) and introduce a regularization term that gradually lowers the absolute value of parameters with low sensitivity. Thus, a very large fraction of the parameters approach zero and are eventually set to zero by simple thresholding. Our method surpasses most of the recent techni…
We investigate the learning rate of multiple kernel learning (MKL) with $\ell_1$ and elastic-net regularizations. The elastic-net regularization is a composition of an $\ell_1$-regularizer for inducing the sparsity and an $\ell_2$-regularizer for controlling the smoothness. We focus on a sparse setting where the total number of kernels is large, but the number of nonzero components of the ground truth is relatively small, and show sharper convergence rates than the learning rates have ever shown for both $\ell_1$ and elastic-net regularizations. Our analysis reveals some relations between the …
The Hong Kong University of Science and Technology（香港科技大学）
In this paper, the estimation problem for sparse reduced rank regression (SRRR) model is considered. The SRRR model is widely used for dimension reduction and variable selection with applications in signal processing, econometrics, etc. The problem is formulated to minimize the least squares loss with a sparsity-inducing penalty considering an orthogonality constraint. Convex sparsity-inducing functions have been used for SRRR in literature. In this work, a nonconvex function is proposed for better sparsity inducing. An efficient algorithm is developed based on the alternating minimization (or…
Université de Marne-la-Vallée（马恩河谷大学）；Ruhr-University Bochum（鲁尔大学波鸿分校）
Sparsity has become popular in machine learning, because it can save computational resources, facilitate interpretations, and prevent overfitting. In this paper, we discuss sparsity in the framework of neural networks. In particular, we formulate a new notion of sparsity that concerns the networks' layers and, therefore, aligns particularly well with the current trend toward deep networks. We call this notion layer sparsity. We then introduce corresponding regularization and refitting schemes that can complement standard deep-learning pipelines to generate more compact and accurate networks.
We propose a simple modification to the iterative hard thresholding (IHT) algorithm, which recovers asymptotically sparser solutions as a function of the condition number. When aiming to minimize a convex function $f(x)$ with condition number $κ$ subject to $x$ being an $s$-sparse vector, the standard IHT guarantee is a solution with relaxed sparsity $O(sκ^2)$, while our proposed algorithm, regularized IHT, returns a solution with sparsity $O(sκ)$. Our algorithm significantly improves over ARHT which also finds a solution of sparsity $O(sκ)$, as it does not require re-optimization in each iter…
Bayesian optimization (BO) is a powerful approach to sample-efficient optimization of black-box objective functions. However, the application of BO to areas such as recommendation systems often requires taking the interpretability and simplicity of the configurations into consideration, a setting that has not been previously studied in the BO literature. To make BO useful for this setting, we present several regularization-based approaches that allow us to discover sparse and more interpretable configurations. We propose a novel differentiable relaxation based on homotopy continuation that mak…
High demand for computation resources severely hinders deployment of large-scale Deep Neural Networks (DNN) in resource constrained devices. In this work, we propose a Structured Sparsity Learning (SSL) method to regularize the structures (i.e., filters, channels, filter shapes, and layer depth) of DNNs. SSL can: (1) learn a compact structure from a bigger DNN to reduce computation cost; (2) obtain a hardware-friendly structured sparsity of DNN to efficiently accelerate the DNNs evaluation. Experimental results show that SSL achieves on average 5.1x and 3.1x speedups of convolutional layer com…
Principal component analysis (PCA) is widely used for dimensionality reduction, with well-documented merits in various applications involving high-dimensional data, including computer vision, preference measurement, and bioinformatics. In this context, the fresh look advocated here permeates benefits from variable selection and compressive sampling, to robustify PCA against outliers. A least-trimmed squares estimator of a low-rank bilinear factor analysis model is shown closely related to that obtained from an $\ell_0$-(pseudo)norm-regularized criterion encouraging sparsity in a matrix explici…
Graphs have been commonly used to represent complex data structures. In models dealing with graph-structured data, multivariate parameters may not only exhibit sparse patterns but have structured sparsity and smoothness in the sense that both zero and non-zero parameters tend to cluster together. We propose a new prior for high-dimensional parameters with graphical relations, referred to as the Tree-based Low-rank Horseshoe (T-LoHo) model, that generalizes the popular univariate Bayesian horseshoe shrinkage prior to the multivariate setting to detect structured sparsity and smoothness simultan…
Canonical correlation analysis (CCA) is a multivariate statistical technique for finding the linear relationship between two sets of variables. The kernel generalization of CCA named kernel CCA has been proposed to find nonlinear relations between datasets. Despite their wide usage, they have one common limitation that is the lack of sparsity in their solution. In this paper, we consider sparse kernel CCA and propose a novel sparse kernel CCA algorithm (SKCCA). Our algorithm is based on a relationship between kernel CCA and least squares. Sparsity of the dual transformations is introduced by p…
Inverse inference, or "brain reading", is a recent paradigm for analyzing functional magnetic resonance imaging (fMRI) data, based on pattern recognition and statistical learning. By predicting some cognitive variables related to brain activation maps, this approach aims at decoding brain activity. Inverse inference takes into account the multivariate information between voxels and is currently the only way to assess how precisely some cognitive information is encoded by the activity of neural populations within the whole brain. However, it relies on a prediction function that is plagued by th…
We consider a class of sparse learning problems in high dimensional feature space regularized by a structured sparsity-inducing norm which incorporates prior knowledge of the group structure of the features. Such problems often pose a considerable challenge to optimization algorithms due to the non-smoothness and non-separability of the regularization term. In this paper, we focus on two commonly adopted sparsity-inducing regularization terms, the overlapping Group Lasso penalty $l_1/l_2$-norm and the $l_1/l_\infty$-norm. We propose a unified framework based on the augmented Lagrangian method,…
Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions con…
