We propose a new active learning algorithm for parametric linear regression with random design. We provide finite sample convergence guarantees for general distributions in the misspecified model. This is the first active learner for this setting that provably can improve over passive learning. Unlike other learning settings (such as classification), in regression the passive learning rate of O(1/epsilon) cannot in general be improved upon. Nonetheless, the so-called `constant' in the rate of convergence, which is characterized by a distribution-dependent risk, can be improved in many cases. For a given distribution, achieving the optimal risk requires prior knowledge of the distribution. Following the stratification technique advocated in Monte-Carlo function integration, our active learner approaches a the optimal risk using piecewise constant approximations.

Group LASSO is widely used to enforce the structural sparsity, which achieves the sparsity at the inter-group level. In this paper, we propose a new formulation called "exclusive group LASSO", which brings out sparsity at intragroup level in the context of feature selection. The proposed exclusive group LASSO is applicable on any feature structures, regardless of their overlapping or non-overlapping structures. We provide analysis on the properties of exclusive group LASSO, and propose an effective iteratively re-weighted algorithm to solve the corresponding optimization problem with rigorous convergence analysis. We show applications of exclusive group LASSO for uncorrelated feature selection. Extensive experiments on both synthetic and real-world datasets validate the proposed method.

In this paper, we develop a family of algorithms for optimizing superposition-structured” or “dirty” statistical estimators for high-dimensional problems involving the minimization of the sum of a smooth loss function with a hybrid regularization. Most of the current approaches are first-order methods, including proximal gradient or Alternating Direction Method of Multipliers (ADMM). We propose a new family of second-order methods where we approximate the loss function using quadratic approximation. The superposition structured regularizer then leads to a subproblem that can be efficiently solved by alternating minimization. We propose a general active subspace selection approach to speed up the solver by utilizing the low-dimensional structure given by the regularizers, and provide convergence guarantees for our algorithm. Empirically, we show that our approach is more than 10 times faster than state-of-the-art first-order approaches for the latent variable graphical model selection problems and multi-task learning problems when there is more than one regularizer. For these problems, our approach appears to be the first algorithm that can extend active subspace ideas to multiple regularizers."

We present two related contributions of independent interest: (1) high-probability finite sample rates for $k$-NN density estimation, and (2) practical mode estimators -- based on $k$-NN -- which attain minimax-optimal rates under surprisingly general distributional conditions.

Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of regions and region-specific models. This paper proposes novel convex region-specific linear models, which we refer to as partition-wise linear models. Our key ideas are 1) assigning linear models not to regions but to partitions (region-specifiers) and representing region-specific linear models by linear combinations of partition-specific models, and 2) optimizing regions via partition selection from a large number of given partition candidates by means of convex structured regularizations. In addition to providing initialization-free globally-optimal solutions, our convex formulation makes it possible to derive a generalization bound and to use such advanced optimization techniques as proximal methods and decomposition of the proximal maps for sparsity-inducing regularizations. Experimental results demonstrate that our partition-wise linear models perform better than or are at least competitive with state-of-the-art region-specific or locally linear models.

We investigate a general framework of multiplicative multitask feature learning which decomposes each task's model parameters into a multiplication of two components. One of the components is used across all tasks and the other component is task-specific. Several previous methods have been proposed as special cases of our framework. We study the theoretical properties of this framework when different regularization conditions are applied to the two decomposed components. We prove that this framework is mathematically equivalent to the widely used multitask feature learning methods that are based on a joint regularization of all model parameters, but with a more general form of regularizers. Further, an analytical formula is derived for the across-task component as related to the task-specific component for all these regularizers, leading to a better understanding of the shrinkage effect. Study of this framework motivates new multitask learning algorithms. We propose two new learning formulations by varying the parameters in the proposed framework. Empirical studies have revealed the relative advantages of the two new formulations by comparing with the state of the art, which provides instructive insights into the feature learning problem with multiple tasks.

We describe a framework for defining high-order image models that can be used in a variety of applications. The approach involves modeling local patterns in a multiscale representation of an image. Local properties of a coarsened image reflect non-local properties of the original image. In the case of binary images local properties are defined by the binary patterns observed over small neighborhoods around each pixel. With the multiscale representation we capture the frequency of patterns observed at different scales of resolution. This framework leads to expressive priors that depend on a relatively small number of parameters. For inference and learning we use an MCMC method for block sampling with very large blocks. We evaluate the approach with two example applications. One involves contour detection. The other involves binary segmentation.

A wild bootstrap method for nonparametric hypothesis tests based on kernel distribution embeddings is proposed. This bootstrap method is used to construct provably consistent tests that apply to random processes, for which the naive permutation-based bootstrap fails. It applies to a large group of kernel tests based on V-statistics, which are degenerate under the null hypothesis, and non-degenerate elsewhere. To illustrate this approach, we construct a two-sample test, an instantaneous independence test and a multiple lag independence test for time series. In experiments, the wild bootstrap gives strong performance on synthetic examples, on audio data, and in performance benchmarking for the Gibbs sampler. The code is available at https://github.com/kacperChwialkowski/wildBootstrap.

We propose a structured prediction algorithm for object localization based on Support Vector Machines (SVMs) using privileged information. Privileged information provides useful high-level knowledge for image understanding and facilitates learning a reliable model even with a small number of training examples. In our setting, we assume that such information is available only at training time since it may be difficult to obtain from visual data accurately without human supervision. Our goal is to improve performance by incorporating privileged information into ordinary learning framework and adjusting model parameters for better generalization. We tackle object localization problem based on a novel structural SVM using privileged information, where an alternating loss-augmented inference procedure is employed to handle the term in the objective function corresponding to privileged information. We apply the proposed algorithm to the Caltech-UCSD Birds 200-2011 dataset, and obtain encouraging results suggesting further investigation into the benefit of privileged information in structured prediction.

This paper is concerned with compressive sensing of signals drawn from a Gaussian mixture model (GMM) with sparse precision matrices. Previous work has shown: (i) a signal drawn from a given GMM can be perfectly reconstructed from r noise-free measurements if the (dominant) rank of each covariance matrix is less than r; (ii) a sparse Gaussian graphical model can be efficiently estimated from fully-observed training signals using graphical lasso. This paper addresses a problem more challenging than both (i) and (ii), by assuming that the GMM is unknown and each signal is only partially observed through incomplete linear measurements. Under these challenging assumptions, we develop a hierarchical Bayesian method to simultaneously estimate the GMM and recover the signals using solely the incomplete measurements and a Bayesian shrinkage prior that promotes sparsity of the Gaussian precision matrices. In addition, we provide theoretical performance bounds to relate the reconstruction error to the number of signals for which measurements are available, the sparsity level of precision matrices, and the “incompleteness” of measurements. The proposed method is demonstrated extensively on compressive sensing of imagery and video, and the results with simulated and hardware-acquired real measurements show significant performance improvement over state-of-the-art methods.