The Default Risk of Firms Examined with Smooth Support Vector Machines

In the era of Basel II a powerful tool for bankruptcy prognosis is vital for banks. The tool must be precise but also easily adaptable to the bank's objections regarding the relation of false acceptances (Type I error) and false rejections (Type II error). We explore the suitability of Smooth Support Vector Machines (SSVM), and investigate how important factors such as selection of appropriate accounting ratios (predictors), length of training period and structure of the training sample influence the precision of prediction. Furthermore we showthat oversampling can be employed to gear the tradeoff between error types. Finally, we illustrate graphically how different variants of SSVM can be used jointly to support the decision task of loan officers.Insolvency Prognosis, SVMs, Statistical Learning Theory, Non-parametric Classification

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

Optimistic Robust Optimization With Applications To Machine Learning

Robust Optimization has traditionally taken a pessimistic, or worst-case viewpoint of uncertainty which is motivated by a desire to find sets of optimal policies that maintain feasibility under a variety of operating conditions. In this paper, we explore an optimistic, or best-case view of uncertainty and show that it can be a fruitful approach. We show that these techniques can be used to address a wide variety of problems. First, we apply our methods in the context of robust linear programming, providing a method for reducing conservatism in intuitive ways that encode economically realistic modeling assumptions. Second, we look at problems in machine learning and find that this approach is strongly connected to the existing literature. Specifically, we provide a new interpretation for popular sparsity inducing non-convex regularization schemes. Additionally, we show that successful approaches for dealing with outliers and noise can be interpreted as optimistic robust optimization problems. Although many of the problems resulting from our approach are non-convex, we find that DCA or DCA-like optimization approaches can be intuitive and efficient

The Default Risk of Firms Examined with Smooth Support Vector Machines

In the era of Basel II a powerful tool for bankruptcy prognosis is vital for banks. The tool must be precise but also easily adaptable to the bank's objections regarding the relation of false acceptances (Type I error) and false rejections (Type II error). We explore the suitabil- ity of Smooth Support Vector Machines (SSVM), and investigate how important factors such as selection of appropriate accounting ratios (predictors), length of training period and structure of the training sample in°uence the precision of prediction. Furthermore we show that oversampling can be employed to gear the tradeo® between error types. Finally, we illustrate graphically how di®erent variants of SSVM can be used jointly to support the decision task of loan o±cers.Insolvency Prognosis, SVMs, Statistical Learning Theory, Non-parametric Classification models, local time-homogeneity

Differentially Private Empirical Risk Minimization

Privacy-preserving machine learning algorithms are crucial for the increasingly common setting in which personal data, such as medical or financial records, are analyzed. We provide general techniques to produce privacy-preserving approximations of classifiers learned via (regularized) empirical risk minimization (ERM). These algorithms are private under the $\epsilon$-differential privacy definition due to Dwork et al. (2006). First we apply the output perturbation ideas of Dwork et al. (2006), to ERM classification. Then we propose a new method, objective perturbation, for privacy-preserving machine learning algorithm design. This method entails perturbing the objective function before optimizing over classifiers. If the loss and regularizer satisfy certain convexity and differentiability criteria, we prove theoretical results showing that our algorithms preserve privacy, and provide generalization bounds for linear and nonlinear kernels. We further present a privacy-preserving technique for tuning the parameters in general machine learning algorithms, thereby providing end-to-end privacy guarantees for the training process. We apply these results to produce privacy-preserving analogues of regularized logistic regression and support vector machines. We obtain encouraging results from evaluating their performance on real demographic and benchmark data sets. Our results show that both theoretically and empirically, objective perturbation is superior to the previous state-of-the-art, output perturbation, in managing the inherent tradeoff between privacy and learning performance.Comment: 40 pages, 7 figures, accepted to the Journal of Machine Learning Researc