This paper is concerned with the problem of top-$K$ ranking from pairwise comparisons. Given a collection of $n$ items and a few pairwise binary comparisons across them, one wishes to identify the set of $K$ items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model---the Bradley-Terry-Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress towards characterizing the performance (e.g. the mean square error for estimating the scores) of several classical methods, including the spectral method and the maximum likelihood estimator (MLE). However, where they stand regarding top-$K$ ranking remains unsettled. We demonstrate that under a random sampling model, the spectral method alone, or the regularized MLE alone, is minimax optimal in terms of the sample complexity---the number of paired comparisons needed to ensure exact top-$K$ identification. This is accomplished via optimal control of the entrywise error of the score estimates. We complement our theoretical studies by numerical experiments, confirming that both methods yield low entrywise errors for estimating the underlying scores. Our theory is established based on a novel leave-one-out trick, which proves effective for analyzing both iterative and non-iterative optimization procedures. Along the way, we derive an elementary eigenvector perturbation bound for probability transition matrices, which parallels the Davis-Kahan $\sin\Theta$ theorem for symmetric matrices. This further allows us to close the gap between the $\ell_2$ error upper bound for the spectral method and the minimax lower limit.

Biclustering techniques have been widely used to identify homogeneous subgroups within large data matrices, such as subsets of genes similarly expressed across subsets of patients. Mining a max-sum sub-matrix is a related but distinct problem for which one looks for a (non-necessarily contiguous) rectangular sub-matrix with a maximal sum of its entries. Le Van et al. (Ranked Tiling, 2014) already illustrated its applicability to gene expression analysis and addressed it with a constraint programming (CP) approach combined with large neighborhood search (CP-LNS). In this work, we exhibit some key properties of this NP-hard problem and define a bounding function such that larger problems can be solved in reasonable time. Two different algorithms are proposed in order to exploit the highlighted characteristics of the problem: a CP approach with a global constraint (CPGC) and mixed integer linear programming (MILP). Practical experiments conducted both on synthetic and real gene expression data exhibit the characteristics of these approaches and their relative benefits over the original CP-LNS method. Overall, the CPGC approach tends to be the fastest to produce a good solution. Yet, the MILP formulation is arguably the easiest to formulate and can also be competitive.

Low-rank modeling has many important applications in computer vision and machine learning. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better empirical performance. However, the resulting optimization problem is much more challenging. Recent state-of-the-art requires an expensive full SVD in each iteration. In this paper, we show that for many commonly-used nonconvex low-rank regularizers, a cutoff can be derived to automatically threshold the singular values obtained from the proximal operator. This allows such operator being efficiently approximated by power method. Based on it, we develop a proximal gradient algorithm (and its accelerated variant) with inexact proximal splitting and prove that a convergence rate of O(1/T) where T is the number of iterations is guaranteed. Furthermore, we show the proposed algorithm can be well parallelized, which achieves nearly linear speedup w.r.t the number of threads. Extensive experiments are performed on matrix completion and robust principal component analysis, which shows a significant speedup over the state-of-the-art. Moreover, the matrix solution obtained is more accurate and has a lower rank than that of the nuclear norm regularizer.

We investigate an existing distributed algorithm for learning sparse signals or data over networks. The algorithm is iterative and exchanges intermediate estimates of a sparse signal over a network. This learning strategy using exchange of intermediate estimates over the network requires a limited communication overhead for information transmission. Our objective in this article is to show that the strategy is good for learning in spite of limited communication. In pursuit of this objective, we first provide a restricted isometry property (RIP)-based theoretical analysis on convergence of the iterative algorithm. Then, using simulations, we show that the algorithm provides competitive performance in learning sparse signals vis-a-vis an existing alternate distributed algorithm. The alternate distributed algorithm exchanges more information including observations and system parameters.

We study the problem of instance segmentation in biological images with crowded and compact cells. We formulate this task as an integer program where variables correspond to cells and constraints enforce that cells do not overlap. To solve this integer program, we propose a column generation formulation where the pricing program is solved via exact optimization of very small scale integer programs. Column generation is tightened using odd set inequalities which fit elegantly into pricing problem optimization. Our column generation approach achieves fast stable anytime inference for our instance segmentation problems. We demonstrate on three distinct light microscopy datasets, with several hundred cells each, that our proposed algorithm rapidly achieves or exceeds state of the art accuracy. Keywords: Combinatorial optimization, Column generation, Integer programming, Large scale optimization, Linear programming 1. Introduction Cell detection and instance segmentation are fundamental tasks for the study of bioimages (Meijering, 2012) in the era of big data.

A core problem of learning control is to determine optimal feedback controllers for (partially) unknown nonlinear systems from experimental data. Reinforcement learning (RL) [1], [2] is a promising framework for this, yet often requires performing many experiments on the physical system to even find suitable controllers, which limits the applicability of such techniques. Therefore, a lot of research effort has been invested into data efficiency of RL aiming at learning controllers from as few experiments as possible. Recently, Bayesian optimization (BO) has been proposed for RL as a promising approach in this direction. BO employs a probabilistic description of the latent objective function (typically a Gaussian process (GP)), which allows for selecting next control experiments in a principled manner, e.g., to maximize information gain [3] or perform safe exploration [4]. While BO provides a promising framework for learning controllers in fairly general settings, the full power of Bayesian learning is often not exploited. A key advantage of Bayesian methods is that they allow for combining prior problem knowledge with learning from data in a principled manner. In case of GP models, this concerns specifically the choice of the kernel, which captures the covariance between function values at different inputs and is thus the core component to model prior knowledge about the function shape. By choosing standard kernels, however, naive BO approaches do often not exploit this opportunity to improve learning performance.

Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the Sherali-Adams hierarchy. Despite the popularity of these algorithms, it is well understood that the Sum-of-Squares (SOS) hierarchy based on semidefinite programming (SDP) can provide superior guarantees. Unfortunately, SOS relaxations for a graph with $n$ vertices require solving an SDP with $n^{\Theta(d)}$ variables where $d$ is the degree in the hierarchy. In practice, for $d\ge 4$, this approach does not scale beyond a few tens of variables. In this paper, we propose binary SDP relaxations for MAP inference using the SOS hierarchy with two innovations focused on computational efficiency. Firstly, in analogy to BP and its variants, we only introduce decision variables corresponding to contiguous regions in the graphical model. Secondly, we solve the resulting SDP using a non-convex Burer-Monteiro style method, and develop a sequential rounding procedure. We demonstrate that the resulting algorithm can solve problems with tens of thousands of variables within minutes, and outperforms BP and GBP on practical problems such as image denoising and Ising spin glasses. Finally, for specific graph types, we establish a sufficient condition for the tightness of the proposed partial SOS relaxation.

Some real-world problems revolve to solve the optimization problem \max_{x\in\mathcal{X}}f\left(x\right) where f\left(.\right) is a black-box function and X might be the set of non-vectorial objects (e.g., distributions) where we can only define a symmetric and non-negative similarity score on it. This setting requires a novel view for the standard framework of Bayesian Optimization that generalizes the core insightful spirit of this framework. With this spirit, in this paper, we propose Analogical-based Bayesian Optimization that can maximize black-box function over a domain where only a similarity score can be defined. Our pathway is as follows: we first base on the geometric view of Gaussian Processes (GP) to define the concept of influence level that allows us to analytically represent predictive means and variances of GP posteriors and base on that view to enable replacing kernel similarity by a more genetic similarity score. Furthermore, we also propose two strategies to find a batch of query points that can efficiently handle high dimensional data.

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using ${\rm T{\small ENSOR}F{\small LOW}}$ in ${\rm P{\small YTHON}}$ illustrate the efficiency and the accuracy of the method in the cases of a $100$-dimensional Black-Scholes-Barenblatt equation, a $100$-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a $ 100 $-dimensional $ G $-Brownian motion.

ECENTLY, Vapnik and Vashist [1] provided a new learning paradigm termed learning using privileged information (LUPI), which is aimed at enhancing the generalization performance of learning algorithms. Generally speaking, in classical supervised learning paradigm, the training data and test data must come from the same distribution. Although in this new learning paradigm the training data is also considered an unbiased representation for the test data, the LUPI provides a set of additional information for the training data during the training stage, which is called privileged information. In the LUPI paradigm, we use the new training set containing privileged information to train a learning algorithm, while the privileged information is not available in the test stage. We note that the new learning paradigm is analogous to human learning process. In class, a teacher can provide some important and helpful information about this course for students, and these information provided by the teacher can help students acquire knowledge better. Therefore, a teacher plays an essential role in human leaning process. The LUPI paradigm resembling the classroom teaching model can achieve better generalization performance than the traditional learning paradigm. The author is with Department of Industrial Engineering and Logistics Management, School of Engineering, Hong Kong University of Science and Technology, Hong Kong 999077, China.(Email: