Journal of Machine Learning Research最新文献

Cluster analysis is a fundamental tool for pattern discovery of complex heterogeneous data. Prevalent clustering methods mainly focus on vector or matrix-variate data and are not applicable to general-order tensors, which arise frequently in modern scientific and business applications. Moreover, there is a gap between statistical guarantees and computational efficiency for existing tensor clustering solutions due to the nature of their non-convex formulations. In this work, we bridge this gap by developing a provable convex formulation of tensor co-clustering. Our convex co-clustering (CoCo) estimator enjoys stability guarantees and its computational and storage costs are polynomial in the size of the data. We further establish a non-asymptotic error bound for the CoCo estimator, which reveals a surprising "blessing of dimensionality" phenomenon that does not exist in vector or matrix-variate cluster analysis. Our theoretical findings are supported by extensive simulated studies. Finally, we apply the CoCo estimator to the cluster analysis of advertisement click tensor data from a major online company. Our clustering results provide meaningful business insights to improve advertising effectiveness.

Proximal distance algorithms combine the classical penalty method of constrained minimization with distance majorization. If f(x) is the loss function, and C is the constraint set in a constrained minimization problem, then the proximal distance principle mandates minimizing the penalized loss $f\left(x\right)+\frac{\rho }{2}\text{dist}{\left(x,C\right)}^2$ and following the solution x _ρ to its limit as ρ tends to ∞. At each iteration the squared Euclidean distance dist(x,C)² is majorized by the spherical quadratic ‖x- P _C (x _k )‖², where P _C (x _k ) denotes the projection of the current iterate x _k onto C. The minimum of the surrogate function $f\left(x\right)+\frac{\rho }{2}\Vert x-P_C\left(x_k\right){\Vert }^2$ is given by the proximal map prox _ρ -_1f [P _C (x _k )]. The next iterate x _k+1 automatically decreases the original penalized loss for fixed ρ. Since many explicit projections and proximal maps are known, it is straightforward to derive and implement novel optimization algorithms in this setting. These algorithms can take hundreds if not thousands of iterations to converge, but the simple nature of each iteration makes proximal distance algorithms competitive with traditional algorithms. For convex problems, proximal distance algorithms reduce to proximal gradient algorithms and therefore enjoy well understood convergence properties. For nonconvex problems, one can attack convergence by invoking Zangwill's theorem. Our numerical examples demonstrate the utility of proximal distance algorithms in various high-dimensional settings, including a) linear programming, b) constrained least squares, c) projection to the closest kinship matrix, d) projection onto a second-order cone constraint, e) calculation of Horn's copositive matrix index, f) linear complementarity programming, and g) sparse principal components analysis. The proximal distance algorithm in each case is competitive or superior in speed to traditional methods such as the interior point method and the alternating direction method of multipliers (ADMM). Source code for the numerical examples can be found at https://github.com/klkeys/proxdist.

We consider the task of estimating a high-dimensional directed acyclic graph, given observations from a linear structural equation model with arbitrary noise distribution. By exploiting properties of common random graphs, we develop a new algorithm that requires conditioning only on small sets of variables. The proposed algorithm, which is essentially a modified version of the PC-Algorithm, offers significant gains in both computational complexity and estimation accuracy. In particular, it results in more efficient and accurate estimation in large networks containing hub nodes, which are common in biological systems. We prove the consistency of the proposed algorithm, and show that it also requires a less stringent faithfulness assumption than the PC-Algorithm. Simulations in low and high-dimensional settings are used to illustrate these findings. An application to gene expression data suggests that the proposed algorithm can identify a greater number of clinically relevant genes than current methods.

This paper frames causal structure estimation as a machine learning task. The idea is to treat indicators of causal relationships between variables as 'labels' and to exploit available data on the variables of interest to provide features for the labelling task. Background scientific knowledge or any available interventional data provide labels on some causal relationships and the remainder are treated as unlabelled. To illustrate the key ideas, we develop a distance-based approach (based on bivariate histograms) within a manifold regularization framework. We present empirical results on three different biological data sets (including examples where causal effects can be verified by experimental intervention), that together demonstrate the efficacy and general nature of the approach as well as its simplicity from a user's point of view.

Individualized treatment rules aim to identify if, when, which, and to whom treatment should be applied. A globally aging population, rising healthcare costs, and increased access to patient-level data have created an urgent need for high-quality estimators of individualized treatment rules that can be applied to observational data. A recent and promising line of research for estimating individualized treatment rules recasts the problem of estimating an optimal treatment rule as a weighted classification problem. We consider a class of estimators for optimal treatment rules that are analogous to convex large-margin classifiers. The proposed class applies to observational data and is doubly-robust in the sense that correct specification of either a propensity or outcome model leads to consistent estimation of the optimal individualized treatment rule. Using techniques from semiparametric efficiency theory, we derive rates of convergence for the proposed estimators and use these rates to characterize the bias-variance trade-off for estimating individualized treatment rules with classification-based methods. Simulation experiments informed by these results demonstrate that it is possible to construct new estimators within the proposed framework that significantly outperform existing ones. We illustrate the proposed methods using data from a labor training program and a study of inflammatory bowel syndrome.

To find optimal decision rule, Fan et al. (2016) proposed an innovative concordance-assisted learning algorithm which is based on maximum rank correlation estimator. It makes better use of the available information through pairwise comparison. However the objective function is discontinuous and computationally hard to optimize. In this paper, we consider a convex surrogate loss function to solve this problem. In addition, our algorithm ensures sparsity of decision rule and renders easy interpretation. We derive the L ₂ error bound of the estimated coefficients under ultra-high dimension. Simulation results of various settings and application to STAR*D both illustrate that the proposed method can still estimate optimal treatment regime successfully when the number of covariates is large.

We extend the adaptive regression spline model by incorporating saturation, the natural requirement that a function extend as a constant outside a certain range. We fit saturating splines to data via a convex optimization problem over a space of measures, which we solve using an efficient algorithm based on the conditional gradient method. Unlike many existing approaches, our algorithm solves the original infinite-dimensional (for splines of degree at least two) optimization problem without pre-specified knot locations. We then adapt our algorithm to fit generalized additive models with saturating splines as coordinate functions and show that the saturation requirement allows our model to simultaneously perform feature selection and nonlinear function fitting. Finally, we briefly sketch how the method can be extended to higher order splines and to different requirements on the extension outside the data range.

Community detection is the process of grouping strongly connected nodes in a network. Many community detection methods for un-weighted networks have a theoretical basis in a null model. Communities discovered by these methods therefore have interpretations in terms of statistical significance. In this paper, we introduce a null for weighted networks called the continuous configuration model. First, we propose a community extraction algorithm for weighted networks which incorporates iterative hypothesis testing under the null. We prove a central limit theorem for edge-weight sums and asymptotic consistency of the algorithm under a weighted stochastic block model. We then incorporate the algorithm in a community detection method called CCME. To benchmark the method, we provide a simulation framework involving the null to plant "background" nodes in weighted networks with communities. We show that the empirical performance of CCME on these simulations is competitive with existing methods, particularly when overlapping communities and background nodes are present. To further validate the method, we present two real-world networks with potential background nodes and analyze them with CCME, yielding results that reveal macro-features of the corresponding systems.

In statistics and machine learning, we are interested in the eigenvectors (or singular vectors) of certain matrices (e.g. covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. The Davis-Kahan sin θ theorem is often used to bound the difference between the eigenvectors of a matrix A and those of a perturbed matrix $\tilde{A}=A+E$ , in terms of $l_2$ norm. In this paper, we prove that when A is a low-rank and incoherent matrix, the $l_{\infty }$ norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of $\sqrt{d_1}$ or $\sqrt{d_2}$ for left and right vectors, where d ₁ and d ₂ are the matrix dimensions. The power of this new perturbation result is shown in robust covariance estimation, particularly when random variables have heavy tails. There, we propose new robust covariance estimators and establish their asymptotic properties using the newly developed perturbation bound. Our theoretical results are verified through extensive numerical experiments.

We consider joint estimation of multiple graphical models arising from heterogeneous and high-dimensional observations. Unlike most previous approaches which assume that the cluster structure is given in advance, an appealing feature of our method is to learn cluster structure while estimating heterogeneous graphical models. This is achieved via a high dimensional version of Expectation Conditional Maximization (ECM) algorithm (Meng and Rubin, 1993). A joint graphical lasso penalty is imposed on the conditional maximization step to extract both homogeneity and heterogeneity components across all clusters. Our algorithm is computationally efficient due to fast sparse learning routines and can be implemented without unsupervised learning knowledge. The superior performance of our method is demonstrated by extensive experiments and its application to a Glioblastoma cancer dataset reveals some new insights in understanding the Glioblastoma cancer. In theory, a non-asymptotic error bound is established for the output directly from our high dimensional ECM algorithm, and it consists of two quantities: statistical error (statistical accuracy) and optimization error (computational complexity). Such a result gives a theoretical guideline in terminating our ECM iterations.