Estimating graphical model structure from high-dimensional and undersampled data is a fundamental problem in many scientific fields. Existing approaches, such as GLASSO, latent variable GLASSO, and latent tree models, suffer from high computational complexity and may impose unrealistic sparsity priors in some cases. We introduce a novel method that leverages a newly discovered connection between information-theoretic measures and structured latent factor models to derive an optimization objective which encourages modular structures where each observed variable has a single latent parent. The proposed method has linear stepwise computational complexity w.r.t. the number of observed variables. Our experiments on synthetic data demonstrate that our approach is the only method that recovers modular structure better as the dimensionality increases. We also use our approach for estimating covariance structure for a number of real-world datasets and show that it consistently outperforms state-of-the-art estimators at a fraction of the computational cost. Finally, we apply the proposed method to high-resolution fMRI data (with more than 10^5 voxels) and show that it is capable of extracting meaningful patterns.
We study \emph{TV regularization}, a widely used technique for eliciting structured sparsity. In particular, we propose efficient algorithms for computing prox-operators for $\ell_p$-norm TV. The most important among these is $\ell_1$-norm TV, for whose prox-operator we present a new geometric analysis which unveils a hitherto unknown connection to taut-string methods. This connection turns out to be remarkably useful as it shows how our geometry guided implementation results in efficient weighted and unweighted 1D-TV solvers, surpassing state-of-the-art methods. Our 1D-TV solvers provide the …
In this paper, we give a new generalization error bound of Multiple Kernel Learning (MKL) for a general class of regularizations, and discuss what kind of regularization gives a favorable predictive accuracy. Our main target in this paper is dense type regularizations including \ellp-MKL. According to the recent numerical experiments, the sparse regularization does not necessarily show a good performance compared with dense type regularizations. Motivated by this fact, this paper gives a general theoretical tool to derive fast learning rates of MKL that is applicable to arbitrary mixed-norm-ty…
Natural policy gradient (NPG) methods are among the most widely used policy optimization algorithms in contemporary reinforcement learning. This class of methods is often applied in conjunction with entropy regularization -- an algorithmic scheme that encourages exploration -- and is closely related to soft policy iteration and trust region policy optimization. Despite the empirical success, the theoretical underpinnings for NPG methods remain limited even for the tabular setting. This paper develops $\textit{non-asymptotic}$ convergence guarantees for entropy-regularized NPG methods under sof…
We investigate the learning rate of multiple kernel leaning (MKL) with elastic-net regularization, which consists 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 non-zero components of the ground truth is relatively small, and prove that elastic-net MKL achieves the minimax learning rate on the $\ell_2$-mixed-norm ball. Our bound is sharper than the convergence rates ever shown, and has a property that the smoother the truth is, the faster t…
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 …
This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(ε^{-3.5})$ stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the $\mathcal{\tilde{O}}(ε^{-4})$ rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima withou…
Big Data Lab, Baidu Research（百度研究院大数据实验室）；School of Artificial Intelligence and Automation Huazhong University of Science and Technology（华中科技大学人工智能与自动化学院）
Principal component analysis (PCA) has been widely used as an effective technique for feature extraction and dimension reduction. In the High Dimension Low Sample Size (HDLSS) setting, one may prefer modified principal components, with penalized loadings, and automated penalty selection by implementing model selection among these different models with varying penalties. The earlier work [1, 2] has proposed penalized PCA, indicating the feasibility of model selection in $L_2$- penalized PCA through the solution path of Ridge regression, however, it is extremely time-consuming because of the int…
Regression is an important task in machine learning and data mining. It has several applications in various domains, including finance, biomedical, and computer vision. Recently, network Lasso, which estimates local models by making clusters using the network information, was proposed and its superior performance was demonstrated. In this study, we propose a simple yet effective local model training algorithm called the fast anchor regularized local linear method (FALL). More specifically, we train a local model for each sample by regularizing it with precomputed anchor models. The key advanta…
Paradigm. Inc.（4Paradigm公司）；Department of Computer Science and Engineering, Hong Kong University of Science and Technology（香港科技大学计算机科学与工程系）；RIKEN（理化学研究所（RIKEN））；Hong Kong University of Science and Technology（香港科技大学）
Low-rank tensor completion problem aims to recover a tensor from limited observations, which has many real-world applications. Due to the easy optimization, the convex overlapping nuclear norm has been popularly used for tensor completion. However, it over-penalizes top singular values and lead to biased estimations. In this paper, we propose to use the nonconvex regularizer, which can less penalize large singular values, instead of the convex one for tensor completion. However, as the new regularizer is nonconvex and overlapped with each other, existing algorithms are either too slow or suffe…
Fast Gradient Sign Method (FGSM) is a popular method to generate adversarial examples that make neural network models robust against perturbations. Despite its empirical success, its theoretical property is not well understood. This paper develops theory to explain the regularization effect of Generalized FGSM, a class of methods to generate adversarial examples. Motivated from the relationship between FGSM and LASSO penalty, the asymptotic properties of Generalized FGSM are derived in the Generalized Linear Model setting, which is essentially the 1-layer neural network setting with certain ac…
This paper addresses image classification through learning a compact and discriminative dictionary efficiently. Given a structured dictionary with each atom (columns in the dictionary matrix) related to some label, we propose cross-label suppression constraint to enlarge the difference among representations for different classes. Meanwhile, we introduce group regularization to enforce representations to preserve label properties of original samples, meaning the representations for the same class are encouraged to be similar. Upon the cross-label suppression, we don't resort to frequently-used …
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…
Overcoming Saturation in Density Ratio Estimation by Iterated Regularization
ELLIS Unit Linz（ELLIS林茨单元）；LIT AI Lab（LIT人工智能实验室）；Institute for Machine Learning, Johannes Kepler University Linz（约翰内斯·开普勒大学林茨分校机器学习研究所）；NXAI GmbH（NXAI有限公司）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉东计算与应用数学研究所）
Georgia Institute of Technology（佐治亚理工学院）；East China Normal University（华东师范大学）
Wasserstein distance plays increasingly important roles in machine learning, stochastic programming and image processing. Major efforts have been under way to address its high computational complexity, some leading to approximate or regularized variations such as Sinkhorn distance. However, as we will demonstrate, regularized variations with large regularization parameter will degradate the performance in several important machine learning applications, and small regularization parameter will fail due to numerical stability issues with existing algorithms. We address this challenge by developi…
A standard technique for understanding underlying dependency structures among a set of variables posits a shared conditional probability distribution for the variables measured on individuals within a group. This approach is often referred to as module networks, where individuals are represented by nodes in a network, groups are termed modules, and the focus is on estimating the network structure among modules. However, estimation solely from node-specific variables can lead to spurious dependencies, and unverifiable structural assumptions are often used for regularization. Here, we propose an…
