Homotopy Continuation Approaches for Robust SV Classification and Regression

In support vector machine (SVM) applications with unreliable data that contains a portion of outliers, non-robustness of SVMs often causes considerable performance deterioration. Although many approaches for improving the robustness of SVMs have been studied, two major challenges remain in robust SVM learning. First, robust learning algorithms are essentially formulated as non-convex optimization problems. It is thus important to develop a non-convex optimization method for robust SVM that can find a good local optimal solution. The second practical issue is how one can tune the hyperparameter that controls the balance between robustness and efficiency. Unfortunately, due to the non-convexity, robust SVM solutions with slightly different hyper-parameter values can be significantly different, which makes model selection highly unstable. In this paper, we address these two issues simultaneously by introducing a novel homotopy approach to non-convex robust SVM learning. Our basic idea is to introduce parametrized formulations of robust SVM which bridge the standard SVM and fully robust SVM via the parameter that represents the influence of outliers. We characterize the necessary and sufficient conditions of the local optimal solutions of robust SVM, and develop an algorithm that can trace a path of local optimal solutions when the influence of outliers is gradually decreased. An advantage of our homotopy approach is that it can be interpreted as simulated annealing, a common approach for finding a good local optimal solution in non-convex optimization problems. In addition, our homotopy method allows stable and efficient model selection based on the path of local optimal solutions. Empirical performances of the proposed approach are demonstrated through intensive numerical experiments both on robust classification and regression problems.

Direct Density-Derivative Estimation and Its Application in KL-Divergence Approximation

Estimation of density derivatives is a versatile tool in statistical data analysis. A naive approach is to first estimate the density and then compute its derivative. However, such a two-step approach does not work well because a good density estimator does not necessarily mean a good density-derivative estimator. In this paper, we give a direct method to approximate the density derivative without estimating the density itself. Our proposed estimator allows analytic and computationally efficient approximation of multi-dimensional high-order density derivatives, with the ability that all hyper-parameters can be chosen objectively by cross-validation. We further show that the proposed density-derivative estimator is useful in improving the accuracy of non-parametric KL-divergence estimation via metric learning. The practical superiority of the proposed method is experimentally demonstrated in change detection and feature selection.

Fast learning rate of multiple kernel learning: Trade-off between sparsity and smoothness

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 choice of a regularization function and the performance. If the ground truth is smooth, we show a faster convergence rate for the elastic-net regularization with less conditions than $ell_1$-regularization; otherwise, a faster convergence rate for the $ell_1$-regularization is shown.

Super-Linear Convergence of Dual Augmented-Lagrangian Algorithm for Sparsity Regularized Estimation

We analyze the convergence behaviour of a recently proposed algorithm for regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is based on a new interpretation of DAL as a proximal minimization algorithm. We theoretically show under some conditions that DAL converges super-linearly in a non-asymptotic and global sense. Due to a special modelling of sparse estimation problems in the context of machine learning, the assumptions we make are milder and more natural than those made in conventional analysis of augmented Lagrangian algorithms. In addition, the new interpretation enables us to generalize DAL to wide varieties of sparse estimation problems. We experimentally confirm our analysis in a large scale $ell_1$-regularized logistic regression problem and extensively compare the efficiency of DAL algorithm to previously proposed algorithms on both synthetic and benchmark datasets.

Dual Augmented Lagrangian Method for Efficient Sparse Reconstruction

We propose an efficient algorithm for sparse signal reconstruction problems. The proposed algorithm is an augmented Lagrangian method based on the dual sparse reconstruction problem. It is efficient when the number of unknown variables is much larger than the number of observations because of the dual formulation. Moreover, the primal variable is explicitly updated and the sparsity in the solution is exploited. Numerical comparison with the state-of-the-art algorithms shows that the proposed algorithm is favorable when the design matrix is poorly conditioned or dense and very large.