We address the problem of generating multiple hypotheses for structured prediction tasks that involve interaction with users or successive components in a cascaded architecture. Given a set of multiple hypotheses, such components/users typically have the ability to retrieve the best (or approximately the best) solution in this set. The standard approach for handling such a scenario is to first learn a single-output model and then produce M-Best Maximum a Posteriori (MAP) hypotheses from this model. In contrast, we learn to produce multiple outputs by formulating this task as a multiple-output structured-output prediction problem with a loss-function that effectively captures the setup of the problem. We present a max-margin formulation that minimizes an upper-bound on this lossfunction. Experimental results on image segmentation and protein side-chain prediction show that our method outperforms conventional approaches used for this type of scenario and leads to substantial improvements in prediction accuracy.

Probabilistic graphical models are powerful tools for analyzing constrained, continuous domains. However, finding most-probable explanations (MPEs) in these models can be computationally expensive. In this paper, we improve the scalability of MPE inference in a class of graphical models with piecewise-linear and piecewise-quadratic dependencies and linear constraints over continuous domains. We derive algorithms based on a consensus-optimization framework and demonstrate their superior performance over state of the art. We show empirically that in a large-scale voter-preference modeling problem our algorithms scale linearly in the number of dependencies and constraints.

Although many variants of stochastic gradient descent have been proposed for large-scale convex optimization, most of them require projecting the solution at each iteration to ensure that the obtained solution stays within the feasible domain. For complex domains (e.g., positive semidefinite cone), the projection step can be computationally expensive, making stochastic gradient descent unattractive for large-scale optimization problems. We address this limitation by developing novel stochastic optimization algorithms that do not need intermediate projections. Instead, only one projection at the last iteration is needed to obtain a feasible solution in the given domain. Our theoretical analysis shows that with a high probability, the proposed algorithms achieve an O(1/ T) convergence rate for general convex optimization, and an O(ln T/T) rate for strongly convex optimization under mild conditions about the domain and the objective function.

We consider an abstract class of optimization problems that are parameterized concavely in a single parameter, and show that the solution path along the parameter can always be approximated with accuracy ε > 0 by a set of size O(1/ ε). A lower bound of size Ω(1/ ε) shows that the upper bound is tight up to a constant factor. We also devise an algorithm that calls a step-size oracle and computes an approximate path of size O(1/ ε). Finally, we provide an implementation of the oracle for soft-margin support vector machines, and a parameterized semi-definite program for matrix completion.

Finding maximum a posteriori (MAP) assignments in graphical models is an important task in many applications. Since the problem is generally hard, linear programming (LP) relaxations are often used. Solving these relaxations efficiently is thus an important practical problem. In recent years, several authors have proposed message passing updates corresponding to coordinate descent in the dual LP. However, these are generally not guaranteed to converge to a global optimum.

Nonnegative Matrix Factorization (NMF) is a promising relaxation technique for clustering analysis. However, conventional NMF methods that directly approximate the pairwise similarities using the least square error often yield mediocre performance for data in curved manifolds because they can capture only the immediate similarities between data samples. Here we propose a new NMF clustering method which replaces the approximated matrix with its smoothed version using random walk. Our method can thus accommodate farther relationships between data samples. Furthermore, we introduce a novel regularization in the proposed objective function in order to improve over spectral clustering. The new learning objective is optimized by a multiplicative Majorization-Minimization algorithm with a scalable implementation for learning the factorizing matrix. Extensive experimental results on real-world datasets show that our method has strong performance in terms of cluster purity.

We propose two classes of second-order optimization methods for solving the sparse inverse covariance estimation problem.

We propose a novel Bayesian approach to solve stochastic optimization problems that involve finding extrema of noisy, nonlinear functions. Previous work has focused on representing possible functions explicitly, which leads to a two-step procedure of first, doing inference over the function space and second, finding the extrema of these functions. Here we skip the representation step and directly model the distribution over extrema. To this end, we devise a non-parametric conjugate prior based on a kernel regressor.

In many graph-based machine learning and data mining approaches, the quality of the graph is critical. However, in real-world applications, especially in semisupervised learning and unsupervised learning, the evaluation of the quality of a graph is often expensive and sometimes even impossible, due the cost or the unavailability of ground truth. In this paper, we proposed a robust approach with convex optimization to "forge" a graph: with an input of a graph, to learn a graph with higher quality. Our major concern is that an ideal graph shall satisfy all the following constraints: non-negative, symmetric, low rank, and positive semidefinite. We develop a graph learning algorithm by solving a convex optimization problem and further develop an efficient optimization to obtain global optimal solutions with theoretical guarantees. With only one non-sensitive parameter, our method is shown by experimental results to be robust and achieve higher accuracy in semi-supervised learning and clustering under various settings. As a preprocessing of graphs, our method has a wide range of potential applications machine learning and data mining.

We use a graphical model for representing policies in Markov Decision Processes. This new representation can easily incorporate domain knowledge in the form of a state similarity graph that loosely indicates which states are supposed to have similar optimal actions. A bias is then introduced into the policy search process by sampling policies from a distribution that assigns high probabilities to policies that agree with the provided state similarity graph, i.e. smoother policies.