Search

We consider the problem of computing an equilibrium in a class of nonlinear generalized Nash equilibrium problems (NGNEPs) in which the strategy sets for each player are defined by equality and inequality constraints that may depend on the choices of rival players. While the asymptotic global convergence and local convergence rates of algorithms to solve this problem have been extensively investigated, the analysis of nonasymptotic iteration complexity is still in its infancy. This paper presents two first-order algorithms--based on the quadratic penalty method (QPM) and augmented Lagrangian method (ALM), respectively--with an accelerated mirror-prox algorithm as the solver in each inner loop. We establish a global convergence guarantee for solving monotone and strongly monotone NGNEPs and provide nonasymptotic complexity bounds expressed in terms of the number of gradient evaluations. Experimental results demonstrate the efficiency of our algorithms in practice. [ABSTRACT FROM AUTHOR]

The objective of ordinal embedding is to find a Euclidean representation of a set of abstract items, using only answers to triplet comparisons of the form "Is item i closer to item j or item k?". In recent years, numerous algorithms have been proposed to solve this problem. However, there does not exist a fair and thorough assessment of these embedding methods and therefore several key questions remain unanswered: Which algorithms perform better when the embedding dimension is constrained or few triplet comparisons are available? Which ones scale better with increasing sample size or dimension? In our paper, we address these questions and provide an extensive and systematic empirical evaluation of existing algorithms as well as a new neural network approach. We find that simple, relatively unknown, non-convex methods consistently outperform all other algorithms across a broad range of tasks including more recent and elaborate methods based on neural networks or landmark approaches. This finding can be explained by the insight that many of the non-convex optimization approaches do not suffer from local optima. Our comprehensive assessment is enabled by our unified library of popular embedding algorithms that leverages GPU resources and allows for fast and accurate embeddings of millions of data points. [ABSTRACT FROM AUTHOR]

In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in machine learning including training Generative Adversarial Nets (GANs). We propose an algorithmic framework motivated by the inexact proximal point method, where the weakly monotone variational inequality (VI) corresponding to the original min-max problem is solved through approximately solving a sequence of strongly monotone VIs constructed by adding a strongly monotone mapping to the original gradient mapping. We prove first-order convergence to a nearly stationary solution of the original min-max problem of the generic algorithmic framework and establish different rates by employing different algorithms for solving each strongly monotone VI. Experiments verify the convergence theory and also demonstrate the effectiveness of the proposed methods on training GANs. [ABSTRACT FROM AUTHOR]

Generalized Linear Models (GLM) form a wide class of regression and classification models, where prediction is a function of a linear combination of the input variables. For statistical inference in high dimension, sparsity inducing regularizations have proven to be useful while offering statistical guarantees. However, solving the resulting optimization problems can be challenging: even for popular iterative algorithms such as coordinate descent, one needs to loop over a large number of variables. To mitigate this, techniques known as screening rules and working sets diminish the size of the optimization problem at hand, either by progressively removing variables, or by solving a growing sequence of smaller problems. For both techniques, significant variables are identified thanks to convex duality arguments. In this paper, we show that the dual iterates of a GLM exhibit a Vector AutoRegressive (VAR) behavior after sign identification, when the primal problem is solved with proximal gradient descent or cyclic coordinate descent. Exploiting this regularity, one can construct dual points that offer tighter certificates of optimality, enhancing the performance of screening rules and working set algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we consider high-dimensional nonconvex square-root-loss regression problems and introduce a proximal majorization-minimization (PMM) algorithm for solving these problems. Our key idea for making the proposed PMM to be efficient is to develop a sparse semismooth Newton method to solve the corresponding subproblems. By using the Kurdyka- Lojasiewicz property exhibited in the underlining problems, we prove that the PMM algorithm converges to a d-stationary point. We also analyze the oracle property of the initial subproblem used in our algorithm. Extensive numerical experiments are presented to demonstrate the high efficiency of the proposed PMM algorithm. [ABSTRACT FROM AUTHOR]

In this work we address the Ev-SVM model proposed by Pérez-Cruz et al. as an extension of the traditional v support vector classification model (v-SVM). Through an enhancement of the range of admissible values for the regularization parameter v, the Ev-SVM has been shown to be able to produce a wider variety of decision functions, giving rise to a better adaptability to the data. However, while a clear and intuitive geometric interpretation can be given for the v-SVM model as a nearest-point problem in reduced convex hulls (RCH-NPP), no previous work has been made in developing such intuition for the Ev-SVM model. In this paper we show how Ev-SVM can be reformulated as a geometrical problem that generalizes RCH-NPP, providing new insights into this model. Under this novel point of view, we propose the RapMinos algorithm, able to solve Ev-SVM more efficiently than the current methods. Furthermore, we show how RapMinos is able to address the Ev-SVM model for any choice of regularization norm lp ≥ 1 seamlessly, which further extends the SVM model exibility beyond the usual Ev-SVM models. [ABSTRACT FROM AUTHOR]

This paper considers the problem of clustering a partially observed unweighted graph--i.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within clusters, and sparse across clusters. We take a novel yet natural approach to this problem, by focusing on finding the clustering that minimizes the number of "disagreements"--i.e., the sum of the number of (observed) missing edges within clusters, and (observed) present edges across clusters. Our algorithm uses convex optimization; its basis is a reduction of disagreement minimization to the problem of recovering an (unknown) low-rank matrix and an (unknown) sparse matrix from their partially observed sum. We evaluate the performance of our algorithm on the classical Planted Partition/Stochastic Block Model. Our main theorem provides sufficient conditions for the success of our algorithm as a function of the minimum cluster size, edge density and observation probability; in particular, the results characterize the tradeoff between the observation probability and the edge density gap. When there are a constant number of clusters of equal size, our results are optimal up to logarithmic factors. [ABSTRACT FROM AUTHOR]

Many algorithms and applications involve repeatedly solving variations of the same inference problem, for example to introduce new evidence to the model or to change conditional dependencies. As the model is updated, the goal of adaptive inference is to take advantage of previously computed quantities to perform inference more rapidly than from scratch. In this paper, we present algorithms for adaptive exact inference on general graphs that can be used to efficiently compute marginals and update MAP configurations under arbitrary changes to the input factor graph and its associated elimination tree. After a linear time preprocessing step, our approach enables updates to the model and the computation of any marginal in time that is logarithmic in the size of the input model. Moreover, in contrast to max-product our approach can also be used to update MAP configurations in time that is roughly proportional to the number of updated entries, rather than the size of the input model. To evaluate the practical effectiveness of our algorithms, we implement and test them using synthetic data as well as for two real-world computational biology applications. Our experiments show that adaptive inference can achieve substantial speedups over performing complete inference as the model undergoes small changes over time. [ABSTRACT FROM AUTHOR]

Previous studies of Non-Parametric Kernel Learning (NPKL) usually formulate the learning task as a Semi-Definite Programming (SDP) problem that is often solved by some general purpose SDP solvers. However, for N data examples, the time complexity of NPKL using a standard interiorpoint SDP solver could be as high as O(N6.5), which prohibits NPKL methods applicable to real applications, even for data sets of moderate size. In this paper, we present a family of efficient NPKL algorithms, termed "SimpleNPKL", which can learn non-parametric kernels from a large set of pairwise constraints efficiently. In particular, we propose two efficient SimpleNPKL algorithms. One is SimpleNPKL algorithm with linear loss, which enjoys a closed-form solution that can be efficiently computed by the Lanczos sparse eigen decomposition technique. Another one is SimpleNPKL algorithm with other loss functions (including square hinge loss, hinge loss, square loss) that can be re-formulated as a saddle-point optimization problem, which can be further resolved by a fast iterative algorithm. In contrast to the previous NPKL approaches, our empirical results show that the proposed new technique, maintaining the same accuracy, is significantly more efficient and scalable. Finally, we also demonstrate that the proposed new technique is also applicable to speed up many kernel learning tasks, including colored maximum variance unfolding, minimum volume embedding, and structure preserving embedding. [ABSTRACT FROM AUTHOR]

Repair-based search algorithms start with an initial solution and attempt to improve it by iteratively applying repair operators. Such algorithms can often handle large-scale problems that may be difficult for systematic search algorithms. Nevertheless, the computational cost of solving such problems is still very high. We observed that many of the repair steps applied by such algorithms are redundant in the sense that they do not eventually contribute to finding a solution. Such redundant steps are particularly harmful in repair-based search, where each step carries high cost due to the very high branching factor typically associated with it. Accurately identifying and avoiding such redundant steps would result in faster local search without harming the algorithm's problem-solving ability. In this paper we propose a speedup learning methodology for attaining this goal. It consists of the following steps: defining the concept of a redundant step; acquiring this concept during off-line learning by analyzing solution paths for training problems, tagging all the steps along the paths according to the redundancy definition and using an induction algorithm to infer a classifier based on the tagged examples; and using the acquired classifier to filter out redundant steps while solving unseen problems. Our algorithm was empirically tested on instances of real-world employee timetabling problems (ETP). The problem solver to be improved is based on one of the best methods for solving some large ETP instances. Our results show a significant improvement in speed for test problems that are similar to the given example problems. [ABSTRACT FROM AUTHOR]