Simulated annealing is a popular method for approaching the solution of a global optimization problem. Existing results on its performance apply to discrete com- binatorial optimization where the optimization variables can assume only a finite set of possible values. We introduce a new general formulation of simulated an- nealing which allows one to guarantee finite-time performance in the optimiza- tion of functions of continuous variables. The results hold universally for any optimization problem on a bounded domain and establish a connection between simulated annealing and up-to-date theory of convergence of Markov chain Monte Carlo methods on continuous domains. This work is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory.

The problem of obtaining the maximum a posteriori estimate of a general discrete random field (i.e. a random field defined using a finite and discrete set of labels) is known to be N P-hard. However, due to its central importance in many applications, several approximate algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on convex relaxations: (i) L P - S: the linear programming (L P) relaxation proposed by Schlesinger [20] for a special case and independently in [4, 12, 23] for the general case; (ii) Q P - R L: the quadratic programming (Q P) relaxation by Ravikumar and Lafferty [18]; and (iii) S O C P - M S: the second order cone programming (S O C P) relaxation first proposed by Muramatsu and Suzuki [16] for two label problems and later extended in [14] for a general label set. We show that the S O C P - M S and the Q P - R L relaxations are equivalent. Furthermore, we prove that despite the flexibility in the form of the constraints/objective function offered by Q P and S O C P, the L P - S relaxation strictly dominates (i.e. Based on these results we propose some novel S O C P relaxations which strictly dominate the previous approaches.

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 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.

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.

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.

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.

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.

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.