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

A D.C. Programming Approach to the Sparse Generalized Eigenvalue Problem

In this paper, we consider the sparse eigenvalue problem wherein the goal is to obtain a sparse solution to the generalized eigenvalue problem. We achieve this by constraining the cardinality of the solution to the generalized eigenvalue problem and obtain sparse principal component analysis (PCA), sparse canonical correlation analysis (CCA) and sparse Fisher discriminant analysis (FDA) as special cases. Unlike the $\ell_1$-norm approximation to the cardinality constraint, which previous methods have used in the context of sparse PCA, we propose a tighter approximation that is related to the negative log-likelihood of a Student's t-distribution. The problem is then framed as a d.c. (difference of convex functions) program and is solved as a sequence of convex programs by invoking the majorization-minimization method. The resulting algorithm is proved to exhibit \emph{global convergence} behavior, i.e., for any random initialization, the sequence (subsequence) of iterates generated by the algorithm converges to a stationary point of the d.c. program. The performance of the algorithm is empirically demonstrated on both sparse PCA (finding few relevant genes that explain as much variance as possible in a high-dimensional gene dataset) and sparse CCA (cross-language document retrieval and vocabulary selection for music retrieval) applications.Comment: 40 page