Regularization and Optimization in Model-Based Clustering
PSR；Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)（里约热内卢天主教大学（PUC-Rio））；LAMPS；CIRRELT；SCALE-AI Chair in Data-Driven Supply Chains（SCALE-AI数据驱动供应链讲席）；Polytechnique Montréal（蒙特利尔理工学院）
Raphael Araujo Sampaio, Joaquim Dias Garcia, Marcus Poggi, Thibaut Vidal
Due to their conceptual simplicity, k-means algorithm variants have been extensively used for unsupervised cluster analysis. However, one main shortcoming of these algorithms is that they essentially fit a mixture of identical spherical Gaussians to data that vastly deviates from such a distribution. In comparison, general Gaussian Mixture Models (GMMs) can fit richer structures but require estimating a quadratic number of parameters per cluster to represent the covariance matrices. This poses two main issues: (i) the underlying optimization problems are challenging due to their larger number of local minima, and (ii) their solutions can overfit the data. In this work, we design search strategies that circumvent both issues. We develop more effective optimization algorithms for general GMMs, and we combine these algorithms with regularization strategies that avoid overfitting. Through extensive computational analyses, we observe that optimization or regularization in isolation does not substantially improve cluster recovery. However, combining these techniques permits a completely new level of performance previously unachieved by k-means algorithm variants, unraveling vastly different cluster structures. These results shed new light on the current status quo between GMM and k-means methods and suggest the more frequent use of general GMMs for data exploration. To facilitate such applications, we provide open-source code as well as Julia packages (UnsupervisedClustering.jl and RegularizedCovarianceMatrices.jl) implementing the proposed techniques.
Clustering is a fundamental machine learning task and can be used in many applications. With the development of deep neural networks (DNNs), combining techniques from DNNs with clustering has become a new research direction and achieved some success. However, few studies have focused on the imbalanced-data problem which commonly occurs in real-world applications. In this paper, we propose a clustering method, regularized deep embedding clustering (RDEC), that integrates virtual adversarial training (VAT), a network regularization technique, with a clustering method called deep embedding cluste…
Universidade Federal de Pernambuco（伯南布哥联邦大学）；Centro de Informática（信息中心）
The clustering methods have been used in a variety of fields such as image processing, data mining, pattern recognition, and statistical analysis. Generally, the clustering algorithms consider all variables equally relevant or not correlated for the clustering task. Nevertheless, in real situations, some variables can be correlated or may be more or less relevant or even irrelevant for this task. This paper proposes partitioning fuzzy clustering algorithms based on Euclidean, City-block and Mahalanobis distances and entropy regularization. These methods are an iterative three steps algorithms …
Rice University（莱斯大学）；Baylor College of Medicine（贝勒医学院）
Convex clustering is a promising new approach to the classical problem of clustering, combining strong performance in empirical studies with rigorous theoretical foundations. Despite these advantages, convex clustering has not been widely adopted, due to its computationally intensive nature and its lack of compelling visualizations. To address these impediments, we introduce Algorithmic Regularization, an innovative technique for obtaining high-quality estimates of regularization paths using an iterative one-step approximation scheme. We justify our approach with a novel theoretical result, gu…
Toulouse Mathematics Institute（图卢兹数学研究所）；University of Toulouse（图卢兹大学）；Metabolic and Cardiovascular Diseases Institute（代谢与心血管疾病研究所）；CNRS（法国国家科学研究中心）
Cluster structure detection is a fundamental task for the analysis of graphs, in order to understand and to visualize their functional characteristics. Among the different cluster structure detection methods, spectral clustering is currently one of the most widely used due to its speed and simplicity. Yet, there are few theoretical guarantee to recover the underlying partitions of the graph for general models. This paper therefore presents a variant of spectral clustering, called 1-spectral clustering, performed on a new random model closely related to stochastic block model. Its goal is to pr…
We propose a simple and easy to implement neural network compression algorithm that achieves results competitive with more complicated state-of-the-art methods. The key idea is to modify the original optimization problem by adding K independent Gaussian priors (corresponding to the k-means objective) over the network parameters to achieve parameter quantization, as well as an L1 penalty to achieve pruning. Unlike many existing quantization-based methods, our method uses hard clustering assignments of network parameters, which adds minimal change or overhead to standard network training. We als…
In this paper, we propose a novel unsupervised clustering approach exploiting the hidden information that is indirectly introduced through a pseudo classification objective. Specifically, we randomly assign a pseudo parent-class label to each observation which is then modified by applying the domain specific transformation associated with the assigned label. Generated pseudo observation-label pairs are subsequently used to train a neural network with Auto-clustering Output Layer (ACOL) that introduces multiple softmax nodes for each pseudo parent-class. Due to the unsupervised objective based …
Huazhong University of Science and Technology（华中科技大学）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）
To effectively optimize Takagi-Sugeno-Kang (TSK) fuzzy systems for regression problems, a mini-batch gradient descent with regularization, DropRule, and AdaBound (MBGD-RDA) algorithm was recently proposed. This paper further proposes FCM-RDpA, which improves MBGD-RDA by replacing the grid partition approach in rule initialization by fuzzy c-means clustering, and AdaBound by Powerball AdaBelief, which integrates recently proposed Powerball gradient and AdaBelief to further expedite and stabilize parameter optimization. Extensive experiments on 22 regression datasets with various sizes and dimen…
Instituto de Telecomunicações（电信研究所）；Instituto Superior Técnico（高等技术学院）
我们提出了一种新的稀疏性和聚类（SPARsity and Clustering，简称SPARMN）正则化器，它是对以前的八角形收缩和聚类回归算法（OSCAR）的改进，其中，所提出的正则化器由$K$稀疏约束和限制在$K$最大分量上的成对$\ell_{\infty}$范数组成。所提出的正则化是能够单独执行$K$ -稀疏性，并鼓励非零是相等的幅度。此外，它可以准确地对特征进行分组，而不会缩小它们的大小。事实上，OSCAR与OSCAR密切相关，因此前者的邻近算子可以基于后者的邻近算子有效地计算，从而允许使用邻近分裂算法来解决OSCAR正则化问题。在合成数据和基准乳腺癌数据上的实验表明，该方法是一种具有竞争力的群体稀疏诱导正则化回归和分类。
The paper has been withdrawn since more effective experiments should be completed. Auto-encoders (AE) has been widely applied in different fields of machine learning. However, as a deep model, there are a large amount of learnable parameters in the AE, which would cause over-fitting and slow learning speed in practice. Many researchers have been study the intrinsic structure of AE and showed different useful methods to regularize those parameters. In this paper, we present a novel regularization method based on a clustering algorithm which is able to classify the parameters into different grou…
Hanoi University of Science and Technology（河内理工大学）；The Graduate University for Advanced Studies, SOKENDAI（综合研究大学院大学（SOKENDAI））；National Institute of Informatics（国立信息学研究所）
Kernel segmentation aims at partitioning a data sequence into several non-overlapping segments that may have nonlinear and complex structures. In general, it is formulated as a discrete optimization problem with combinatorial constraints. A popular algorithm for optimally solving this problem is dynamic programming (DP), which has quadratic computation and memory requirements. Given that sequences in practice are too long, this algorithm is not a practical approach. Although many heuristic algorithms have been proposed to approximate the optimal segmentation, they have no guarantee on the qual…
This paper uses the relationship between graph conductance and spectral clustering to study (i) the failures of spectral clustering and (ii) the benefits of regularization. The explanation is simple. Sparse and stochastic graphs create a lot of small trees that are connected to the core of the graph by only one edge. Graph conductance is sensitive to these noisy `dangling sets'. Spectral clustering inherits this sensitivity. The second part of the paper starts from a previously proposed form of regularized spectral clustering and shows that it is related to the graph conductance on a `regulari…
Sorted $L_1$ penalization estimator (SLOPE) is a regularization technique for sorted absolute coefficients in high-dimensional regression. By arbitrarily setting its regularization weights $λ$ under the monotonicity constraint, SLOPE can have various feature selection and clustering properties. On weight tuning, the selected features and their clusters are very sensitive to the tuning parameters. Moreover, the exhaustive tracking of their changes is difficult using grid search methods. This study presents a solution path algorithm that provides the complete and exact path of solutions for SLOP…
Towards Cohesion-Fairness Harmony: Contrastive Regularization in Individual Fair Graph Clustering
L3S Research Centre, Leibniz University Hannover（莱布尼茨汉诺威大学L3S研究中心）；University of Kurdistan（库尔德斯坦大学）；RI CODE, University of the Bundeswehr Munich（慕尼黑联邦国防军大学RI CODE）
Minimum Description Length based Granular-Ball Tree Regularization for Spectral Clustering
Department of Mathematics and Computer Science, Gannan Normal University（赣南师范大学数学与计算机科学学院）；Key Laboratory of Data Science and Artificial Intelligence of Jiangxi Education Institutes, Gannan Normal University（江西省教育研究院数据科学与人工智能重点实验室）
Chung-Ang University（中央大学）；University of Copenhagen（哥本哈根大学）
Regularization is essential for avoiding over-fitting to training data in network optimization, leading to better generalization of the trained networks. The label noise provides a strong implicit regularization by replacing the target ground truth labels of training examples by uniform random labels. However, it can cause undesirable misleading gradients due to the large loss associated with incorrect labels. We propose a first-order optimization method (Label-Noised Trim-SGD) that uses the label noise with the example trimming in order to remove the outliers based on the loss. The proposed a…
Toyota Technological Institute at Chicago（芝加哥丰田技术学院）；MSR Cambridge（微软研究院剑桥）；Carnegie Mellon University（卡内基梅隆大学）
We argue that the optimization plays a crucial role in generalization of deep learning models through implicit regularization. We do this by demonstrating that generalization ability is not controlled by network size but rather by some other implicit control. We then demonstrate how changing the empirical optimization procedure can improve generalization, even if actual optimization quality is not affected. We do so by studying the geometry of the parameter space of deep networks, and devising an optimization algorithm attuned to this geometry.
Trust Region Policy Optimization (TRPO) is a popular and empirically successful policy search algorithm in reinforcement learning (RL). It iteratively solved the surrogate problem which restricts consecutive policies to be close to each other. TRPO is an on-policy algorithm. On-policy methods bring many benefits, like the ability to gauge each resulting policy. However, they typically discard all the knowledge about the policies which existed before. In this work, we use a replay buffer to borrow from the off-policy learning setting to TRPO. Entropy regularization is usually used to improve po…
Johns Hopkins University（约翰斯·霍普金斯大学）；Beijing University of Posts and Telecommunications（北京邮电大学）
State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with $\ell_1$, $\ell_2$ or nuclear norms. $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. $\ell_2$ and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed $\ell_1$, …
Clustering single-cell RNA sequence (scRNA-seq) data poses statistical and computational challenges due to their high-dimensionality and data-sparsity, also known as `dropout' events. Recently, Regularized Auto-Encoder (RAE) based deep neural network models have achieved remarkable success in learning robust low-dimensional representations. The basic idea in RAEs is to learn a non-linear mapping from the high-dimensional data space to a low-dimensional latent space and vice-versa, simultaneously imposing a distributional prior on the latent space, which brings in a regularization effect. This …
In this paper, we consider a generic probabilistic discriminative learner from the functional viewpoint and argue that, to make it learn well, it is necessary to constrain its hypothesis space to a set of non-trivial piecewise constant functions. To achieve this goal, we present a scalable unsupervised regularization framework. On the theoretical front, we prove that this framework is conducive to a factually confident and smooth discriminative model and connect it to an adversarial Taboo game, spectral clustering and virtual adversarial training. Experimentally, we take deep neural networks a…
Spectral embedding is a popular technique for the representation of graph data. Several regularization techniques have been proposed to improve the quality of the embedding with respect to downstream tasks like clustering. In this paper, we explain on a simple block model the impact of the complete graph regularization, whereby a constant is added to all entries of the adjacency matrix. Specifically, we show that the regularization forces the spectral embedding to focus on the largest blocks, making the representation less sensitive to noise or outliers. We illustrate these results on both on …
Toyota Technological Institute at Chicago（芝加哥丰田技术学院）
In an attempt to better understand generalization in deep learning, we study several possible explanations. We show that implicit regularization induced by the optimization method is playing a key role in generalization and success of deep learning models. Motivated by this view, we study how different complexity measures can ensure generalization and explain how optimization algorithms can implicitly regularize complexity measures. We empirically investigate the ability of these measures to explain different observed phenomena in deep learning. We further study the invariances in neural netwo…
Subspace clustering methods based on $\ell_1$, $\ell_2$ or nuclear norm regularization have become very popular due to their simplicity, theoretical guarantees and empirical success. However, the choice of the regularizer can greatly impact both theory and practice. For instance, $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad conditions (e.g., arbitrary subspaces and corrupted data). However, it requires solving a large scale convex optimization problem. On the other hand, $\ell_2…
Multi-view clustering methods have been a focus in recent years because of their superiority in clustering performance. However, typical traditional multi-view clustering algorithms still have shortcomings in some aspects, such as removal of redundant information, utilization of various views and fusion of multi-view features. In view of these problems, this paper proposes a new multi-view clustering method, low-rank subspace multi-view clustering based on adaptive graph regularization. We construct two new data matrix decomposition models into a unified optimization model. In this framework, …
Ecole Polytechnique F\'ed\'erale de Lausanne (EPFL), LTS4, Lausanne, Switzerland
We are interested in multilayer graph clustering, which aims at dividing the graph nodes into categories or communities. To do so, we propose to learn a clustering-friendly embedding of the graph nodes by solving an optimization problem that involves a fidelity term to the layers of a given multilayer graph, and a regularization on the (single-layer) graph induced by the embedding. The fidelity term uses the contrastive loss to properly aggregate the observed layers into a representative embedding. The regularization pushes for a sparse and community-aware graph, and it is based on a measure o…
Center for Industrial Mathematics（工业数学中心）；University of Bremen（不来梅大学）
Classical approaches in cluster analysis are typically based on a feature space analysis. However, many applications lead to datasets with additional spatial information and a ground truth with spatially coherent classes, which will not necessarily be reconstructed well by standard clustering methods. Motivated by applications in hyperspectral imaging, we introduce in this work clustering models based on orthogonal nonnegative matrix factorization, which include an additional total variation (TV) regularization procedure on the cluster membership matrix to enforce the needed spatial coherence …
Zhejiang University of Science and Technology（浙江科技大学）
Hyper-parameter optimization is a crucial problem in machine learning as it aims to achieve the state-of-the-art performance in any model. Great efforts have been made in this field, such as random search, grid search, Bayesian optimization. In this paper, we model hyper-parameter optimization process as a Markov decision process, and tackle it with reinforcement learning. A novel hyper-parameter optimization method based on soft actor critic and hierarchical mixture regularization has been proposed. Experiments show that the proposed method can obtain better hyper-parameters in a shorter time…
Latent Adversarial Regularization for Offline Preference Optimization
Department of Computer Science, Stanford University（斯坦福大学计算机科学系）；Siebel School of Computing（塞贝尔计算学院）；Data Science, University of Illinois at Urbana-Champaign（数据科学，伊利诺伊大学厄巴纳-香槟分校）
ICTEAM, Université catholique de Louvain（天主教鲁汶大学 ICTEAM 学院）；FNRS（比利时国家科学研究基金会）
Several works have shown that the regularization mechanisms underlying deep neural networks' generalization performances are still poorly understood. In this paper, we hypothesize that deep neural networks are regularized through their ability to extract meaningful clusters among the samples of a class. This constitutes an implicit form of regularization, as no explicit training mechanisms or supervision target such behaviour. To support our hypothesis, we design four different measures of intraclass clustering, based on the neuron- and layer-level representations of the training data. We then…
Convex clustering is an attractive clustering algorithm with favorable properties such as efficiency and optimality owing to its convex formulation. It is thought to generalize both k-means clustering and agglomerative clustering. However, it is not known whether convex clustering preserves desirable properties of these algorithms. A common expectation is that convex clustering may learn difficult cluster types such as non-convex ones. Current understanding of convex clustering is limited to only consistency results on well-separated clusters. We show new understanding of its solutions. We pro…
Entropy regularization is used to get improved optimization performance in reinforcement learning tasks. A common form of regularization is to maximize policy entropy to avoid premature convergence and lead to more stochastic policies for exploration through action space. However, this does not ensure exploration in the state space. In this work, we instead consider the distribution of discounted weighting of states, and propose to maximize the entropy of a lower bound approximation to the weighting of a state, based on latent space state representation. We propose entropy regularization based…
Semi-supervised Laplacian regularization, a standard graph-based approach for learning from both labelled and unlabelled data, was recently demonstrated to have an insignificant high dimensional learning efficiency with respect to unlabelled data (Mai and Couillet 2018), causing it to be outperformed by its unsupervised counterpart, spectral clustering, given sufficient unlabelled data. Following a detailed discussion on the origin of this inconsistency problem, a novel regularization approach involving centering operation is proposed as solution, supported by both theoretical analysis and emp…
Courant Institute of Mathematical Sciences（柯朗数学科学研究所）；New York University（纽约大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；Center of Data Science（数据科学中心）
This research proposes to use the Moreau-Yosida envelope to stabilize the convergence behavior of bi-level Hyperparameter optimization solvers, and introduces the new algorithm called Moreau-Yosida regularized Hyperparameter Optimization (MY-HPO) algorithm. Theoretical analysis on the correctness of the MY-HPO solution and initial convergence analysis is also provided. Our empirical results show significant improvement in loss values for a fixed computation budget, compared to the state-of-art bi-level HPO solvers.
This paper proposes a simple approach to derive efficient error bounds for learning multiple components with sparsity-inducing regularization. We show that for such regularization schemes, known decompositions of the Rademacher complexity over the components can be used in a more efficient manner to result in tighter bounds without too much effort. We give examples of application to switching regression and center-based clustering/vector quantization. Then, the complete workflow is illustrated on the problem of subspace clustering, for which decomposition results were not previously available.…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；New York University（纽约大学）；Georgia Institute of Technology（佐治亚理工学院）
Wasserstein distributionally robust optimization (DRO) has recently achieved empirical success for various applications in operations research and machine learning, owing partly to its regularization effect. Although connection between Wasserstein DRO and regularization has been established in several settings, existing results often require restrictive assumptions, such as smoothness or convexity, that are not satisfied for many problems. In this paper, we develop a general theory on the variation regularization effect of the Wasserstein DRO - a new form of regularization that generalizes tot…
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…
This work is on constrained large-scale non-convex optimization where the constraint set implies a manifold structure. Solving such problems is important in a multitude of fundamental machine learning tasks. Recent advances on Riemannian optimization have enabled the convenient recovery of solutions by adapting unconstrained optimization algorithms over manifolds. However, it remains challenging to scale up and meanwhile maintain stable convergence rates and handle saddle points. We propose a new second-order Riemannian optimization algorithm, aiming at improving convergence rate and reducing …
Computational Research Division, Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室计算研究部）
Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have emerged as a way to provide a more precise and expressive control over the solution, relying on persistent homology to quantify and reduce its roughness. All such existing techniques back-propagate gradients through the persistence diagram, which is a summary of the topological features of a function. Their downside is that they provide information only at the c…
While nowadays most gradient-based optimization methods focus on exploring the high-dimensional geometric features, the random error accumulated in a stochastic version of any algorithm implementation has not been stressed yet. In this work, we propose a universal principle which reduces the random error accumulation by exploiting statistic information hidden in mini-batch gradients. This is achieved by regularizing the learning-rate according to mini-batch variances. Due to the complementarity of our perspective, this regularization could provide a further improvement for stochastic implement…
In this paper, we develop a simple yet effective screening rule strategy to improve the computational efficiency in solving structured optimization involving nonconvex $\ell_{q,p}$ regularization. Based on an iteratively reweighted $\ell_1$ (IRL1) framework, the proposed screening rule works like a preprocessing module that potentially removes the inactive groups before starting the subproblem solver, thereby reducing the computational time in total. This is mainly achieved by heuristically exploiting the dual subproblem information during each iteration.Moreover, we prove that our screening r…
We describe a framework for deriving and analyzing online optimization algorithms that incorporate adaptive, data-dependent regularization, also termed preconditioning. Such algorithms have been proven useful in stochastic optimization by reshaping the gradients according to the geometry of the data. Our framework captures and unifies much of the existing literature on adaptive online methods, including the AdaGrad and Online Newton Step algorithms as well as their diagonal versions. As a result, we obtain new convergence proofs for these algorithms that are substantially simpler than previous…
Nanjing University of Posts and Telecommunications（南京邮电大学）
Bayesian optimization (BO) is a powerful paradigm for derivative-free global optimization of a black-box objective function (BOF) that is expensive to evaluate. However, the overhead of BO can still be prohibitive for problems with highly expensive function evaluations. In this paper, we investigate how to reduce the required number of function evaluations for BO without compromise in solution quality. We explore the idea of posterior regularization to harness low fidelity (LF) data within the Gaussian process upper confidence bound (GP-UCB) framework. The LF data can arise from previous evalu…
University of Pittsburgh（匹兹堡大学）；University of Rochester（罗切斯特大学）
Stochastic composition optimization draws much attention recently and has been successful in many emerging applications of machine learning, statistical analysis, and reinforcement learning. In this paper, we focus on the composition problem with nonsmooth regularization penalty. Previous works either have slow convergence rate or do not provide complete convergence analysis for the general problem. In this paper, we tackle these two issues by proposing a new stochastic composition optimization method for composition problem with nonsmooth regularization penalty. In our method, we apply varian…
