Continuation of Nesterov's Smoothing for Regression with Structured Sparsity in High-Dimensional Neuroimaging

Predictive models can be used on high-dimensional brain images for diagnosis of a clinical condition. Spatial regularization through structured sparsity offers new perspectives in this context and reduces the risk of overfitting the model while providing interpretable neuroimaging signatures by forcing the solution to adhere to domain-specific constraints. Total Variation (TV) enforces spatial smoothness of the solution while segmenting predictive regions from the background. We consider the problem of minimizing the sum of a smooth convex loss, a non-smooth convex penalty (whose proximal operator is known) and a wide range of possible complex, non-smooth convex structured penalties such as TV or overlapping group Lasso. Existing solvers are either limited in the functions they can minimize or in their practical capacity to scale to high-dimensional imaging data. Nesterov's smoothing technique can be used to minimize a large number of non-smooth convex structured penalties but reasonable precision requires a small smoothing parameter, which slows down the convergence speed. To benefit from the versatility of Nesterov's smoothing technique, we propose a first order continuation algorithm, CONESTA, which automatically generates a sequence of decreasing smoothing parameters. The generated sequence maintains the optimal convergence speed towards any globally desired precision. Our main contributions are: To propose an expression of the duality gap to probe the current distance to the global optimum in order to adapt the smoothing parameter and the convergence speed. We provide a convergence rate, which is an improvement over classical proximal gradient smoothing methods. We demonstrate on both simulated and high-dimensional structural neuroimaging data that CONESTA significantly outperforms many state-of-the-art solvers in regard to convergence speed and precision.Comment: 11 pages, 6 figures, accepted in IEEE TMI, IEEE Transactions on Medical Imaging 201

Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure

Residual Weighted Learning for Estimating Individualized Treatment Rules

Personalized medicine has received increasing attention among statisticians, computer scientists, and clinical practitioners. A major component of personalized medicine is the estimation of individualized treatment rules (ITRs). Recently, Zhao et al. (2012) proposed outcome weighted learning (OWL) to construct ITRs that directly optimize the clinical outcome. Although OWL opens the door to introducing machine learning techniques to optimal treatment regimes, it still has some problems in performance. In this article, we propose a general framework, called Residual Weighted Learning (RWL), to improve finite sample performance. Unlike OWL which weights misclassification errors by clinical outcomes, RWL weights these errors by residuals of the outcome from a regression fit on clinical covariates excluding treatment assignment. We utilize the smoothed ramp loss function in RWL, and provide a difference of convex (d.c.) algorithm to solve the corresponding non-convex optimization problem. By estimating residuals with linear models or generalized linear models, RWL can effectively deal with different types of outcomes, such as continuous, binary and count outcomes. We also propose variable selection methods for linear and nonlinear rules, respectively, to further improve the performance. We show that the resulting estimator of the treatment rule is consistent. We further obtain a rate of convergence for the difference between the expected outcome using the estimated ITR and that of the optimal treatment rule. The performance of the proposed RWL methods is illustrated in simulation studies and in an analysis of cystic fibrosis clinical trial data.Comment: 48 pages, 3 figure

A PAUC-based Estimation Technique for Disease Classification and Biomarker Selection.

The partial area under the receiver operating characteristic curve (PAUC) is a well-established performance measure to evaluate biomarker combinations for disease classification. Because the PAUC is defined as the area under the ROC curve within a restricted interval of false positive rates, it enables practitioners to quantify sensitivity rates within pre-specified specificity ranges. This issue is of considerable importance for the development of medical screening tests. Although many authors have highlighted the importance of PAUC, there exist only few methods that use the PAUC as an objective function for finding optimal combinations of biomarkers. In this paper, we introduce a boosting method for deriving marker combinations that is explicitly based on the PAUC criterion. The proposed method can be applied in high-dimensional settings where the number of biomarkers exceeds the number of observations. Additionally, the proposed method incorporates a recently proposed variable selection technique (stability selection) that results in sparse prediction rules incorporating only those biomarkers that make relevant contributions to predicting the outcome of interest. Using both simulated data and real data, we demonstrate that our method performs well with respect to both variable selection and prediction accuracy. Specifically, if the focus is on a limited range of specificity values, the new method results in better predictions than other established techniques for disease classification