In this work, we consider the problem of learning a positive semidefinite matrix. The critical issue is how to preserve positive semidefiniteness during the course of learning. Our algorithm is mainly inspired by LPBoost [1] and the general greedy convex optimization framework of Zhang [2]. We demonstrate the essence of the algorithm, termed PSDBoost (positive semidefinite Boosting), by focusing on a few different applications in machine learning. The proposed PSDBoost algorithm extends traditional Boosting algorithms in that its parameter is a positive semidefinite matrix with trace being one instead of a classifier.

Many popular optimization algorithms, like the Levenberg-Marquardt algorithm (LMA), use heuristic-based controllers'' that modulate the behavior of the optimizer during the optimization process. For example, in the LMA a damping parameter is dynamically modified based on a set rules that were developed using various heuristic arguments. Reinforcement learning (RL) is a machine learning approach to learn optimal controllers by examples and thus is an obvious candidate to improve the heuristic-based controllers implicit in the most popular and heavily used optimization algorithms. Improving the performance of off-the-shelf optimizers is particularly important for time-constrained optimization problems. For example the LMA algorithm has become popular for many real-time computer vision problems, including object tracking from video, where only a small amount of time can be allocated to the optimizer on each incoming video frame.

Object matching is a fundamental operation in data analysis. It typically requires the definition of a similarity measure between the classes of objects to be matched. Instead, we develop an approach which is able to perform matching by requiring a similarity measure only within each of the classes. This is achieved by maximizing the dependency between matched pairs of observations by means of the Hilbert Schmidt Independence Criterion. This problem can be cast as one of maximizing a quadratic assignment problem with special structure and we present a simple algorithm for finding a locally optimal solution. Papers published at the Neural Information Processing Systems Conference.

Most algorithms for solving Markov decision processes rely on a discount factor, which ensures their convergence. In fact, it is often used in problems with is no intrinsic motivation. In this paper, we show that when used in approximate dynamic programming, an artificially low discount factor may significantly improve the performance on some problems, such as Tetris. We propose two explanations for this phenomenon. Our first justification follows directly from the standard approximation error bounds: using a lower discount factor may decrease the approximation error bounds.

Bayesian methods of matrix factorization (MF) have been actively explored recently as promising alternatives to classical singular value decomposition. In this paper, we show that, despite the fact that the optimization problem is non-convex, the global optimal solution of variational Bayesian (VB) MF can be computed analytically by solving a quartic equation. This is highly advantageous over a popular VBMF algorithm based on iterated conditional modes since it can only find a local optimal solution after iterations. We further show that the global optimal solution of empirical VBMF (hyperparameters are also learned from data) can also be analytically computed. We illustrate the usefulness of our results through experiments.

We consider a class of learning problems that involve a structured sparsity-inducing norm defined as the sum of $\ell_\infty$-norms over groups of variables. Whereas a lot of effort has been put in developing fast optimization methods when the groups are disjoint or embedded in a specific hierarchical structure, we address here the case of general overlapping groups. To this end, we show that the corresponding optimization problem is related to network flow optimization. More precisely, the proximal problem associated with the norm we consider is dual to a quadratic min-cost flow problem. We propose an efficient procedure which computes its solution exactly in polynomial time.

A novel center-based clustering algorithm is proposed in this paper. We first formulate clustering as an NP-hard linear integer program and we then use linear programming and the duality theory to derive the solution of this optimization problem. This leads to an efficient and very general algorithm, which works in the dual domain, and can cluster data based on an arbitrary set of distances. Despite its generality, it is independent of initialization (unlike EM-like methods such as K-means), has guaranteed convergence, and can also provide online optimality bounds about the quality of the estimated clustering solutions. To deal with the most critical issue in a center-based clustering algorithm (selection of cluster centers), we also introduce the notion of stability of a cluster center, which is a well defined LP-based quantity that plays a key role to our algorithm's success.

Several key problems in machine learning, such as feature selection and active learning, can be formulated as submodular set function maximization. We present herein a novel algorithm for maximizing a submodular set function under a cardinality constraint --- the algorithm is based on a cutting-plane method and is implemented as an iterative small-scale binary-integer linear programming procedure. It is well known that this problem is NP-hard, and the approximation factor achieved by the greedy algorithm is the theoretical limit for polynomial time. As for (non-polynomial time) exact algorithms that perform reasonably in practice, there has been very little in the literature although the problem is quite important for many applications. Our algorithm is guaranteed to find the exact solution in finite iterations, and it converges fast in practice due to the efficiency of the cutting-plane mechanism.

We present a multi-label multiple kernel learning (MKL) formulation, in which the data are embedded into a low-dimensional space directed by the instance-label correlations encoded into a hypergraph. We formulate the problem in the kernel-induced feature space and propose to learn the kernel matrix as a linear combination of a given collection of kernel matrices in the MKL framework. The proposed learning formulation leads to a non-smooth min-max problem, and it can be cast into a semi-infinite linear program (SILP). We further propose an approximate formulation with a guaranteed error bound which involves an unconstrained and convex optimization problem. In addition, we show that the objective function of the approximate formulation is continuously differentiable with Lipschitz gradient, and hence existing methods can be employed to compute the optimal solution efficiently.

Many problems in machine learning and statistics can be formulated as (generalized) eigenproblems. In terms of the associated optimization problem, computing linear eigenvectors amounts to finding critical points of a quadratic function subject to quadratic constraints. In this paper we show that a certain class of constrained optimization problems with nonquadratic objective and constraints can be understood as nonlinear eigenproblems. We derive a generalization of the inverse power method which is guaranteed to converge to a nonlinear eigenvector. We apply the inverse power method to 1-spectral clustering and sparse PCA which can naturally be formulated as nonlinear eigenproblems.