Confidence bands for a log-concave density

We present a new approach for inference about a log-concave distribution: Instead of using the method of maximum likelihood, we propose to incorporate the log-concavity constraint in an appropriate nonparametric confidence set for the cdf $F$. This approach has the advantage that it automatically provides a measure of statistical uncertainty and it thus overcomes a marked limitation of the maximum likelihood estimate. In particular, we show how to construct confidence bands for the density that have a finite sample guaranteed confidence level. The nonparametric confidence set for $F$ which we introduce here has attractive computational and statistical properties: It allows to bring modern tools from optimization to bear on this problem via difference of convex programming, and it results in optimal statistical inference. We show that the width of the resulting confidence bands converges at nearly the parametric $n^{-\frac{1}{2}}$ rate when the log density is $k$-affine.Comment: Added more experiments, other minor change

Energy Efficiency Maximization for C-RANs: Discrete Monotonic Optimization, Penalty, and l0-Approximation Methods

We study downlink of multiantenna cloud radio access networks (C-RANs) with finite-capacity fronthaul links. The aim is to propose joint designs of beamforming and remote radio head (RRH)-user association, subject to constraints on users' quality-of-service, limited capacity of fronthaul links and transmit power, to maximize the system energy efficiency. To cope with the limited-capacity fronthaul we consider the problem of RRH-user association to select a subset of users that can be served by each RRH. Moreover, different to the conventional power consumption models, we take into account the dependence of baseband signal processing power on the data rate, as well as the dynamics of the efficiency of power amplifiers. The considered problem leads to a mixed binary integer program (MBIP) which is difficult to solve. Our first contribution is to derive a globally optimal solution for the considered problem by customizing a discrete branch-reduce-and-bound (DBRB) approach. Since the global optimization method requires a high computational effort, we further propose two suboptimal solutions able to achieve the near optimal performance but with much reduced complexity. To this end, we transform the design problem into continuous (but inherently nonconvex) programs by two approaches: penalty and \ell_{0}-approximation methods. These resulting continuous nonconvex problems are then solved by the successive convex approximation framework. Numerical results are provided to evaluate the effectiveness of the proposed approaches.Comment: IEEE Transaction on Signal Processing, September 2018 (15 pages, 12 figures

Partial Identification with Proxy of Latent Confoundings via Sum-of-ratios Fractional Programming

Due to the unobservability of confoundings, there has been widespread concern about how to compute causality quantitatively. To address this challenge, proxy-based negative control approaches have been commonly adopted, where auxiliary outcome variables $\bm{W}$ are introduced as the proxy of confoundings $\bm{U}$. However, these approaches rely on strong assumptions such as reversibility, completeness, or bridge functions. These assumptions lack intuitive empirical interpretation and solid verification techniques, hence their applications in the real world are limited. For instance, these approaches are inapplicable when the transition matrix $P(\bm{W} \mid \bm{U})$ is irreversible. In this paper, we focus on a weaker assumption called the partial observability of $P(\bm{W} \mid \bm{U})$. We develop a more general single-proxy negative control method called Partial Identification via Sum-of-ratios Fractional Programming (PI-SFP). It is a global optimization algorithm based on the branch-and-bound strategy, aiming to provide the valid bound of the causal effect. In the simulation, PI-SFP provides promising numerical results and fills in the blank spots that can not be handled in the previous literature, such as we have partial information of $P(\bm{W} \mid \bm{U})$

Adaptive exact penalty DC algorithms for nonsmooth DC optimization problems with equality and inequality constraints

We propose and study two DC (difference of convex functions) algorithms based on exact penalty functions for solving nonsmooth DC optimization problems with nonsmooth DC equality and inequality constraints. Both methods employ adaptive penalty updating strategies to improve their performance. The first method is based on exact penalty functions with individual penalty parameter for each constraint (i.e. multidimensional penalty parameter) and utilizes a primal-dual approach to penalty updates. The second method is based on the so-called steering exact penalty methodology and relies on solving some auxiliary convex subproblems to determine a suitable value of the penalty parameter. We present a detailed convergence analysis of both methods and give several simple numerical examples highlighting peculiarites of two different penalty updating strategies studied in this paper

DC Semidefinite Programming and Cone Constrained DC Optimization

In the first part of this paper we discuss possible extensions of the main ideas and results of constrained DC optimization to the case of nonlinear semidefinite programming problems (i.e. problems with matrix constraints). To this end, we analyse two different approaches to the definition of DC matrix-valued functions (namely, order-theoretic and componentwise), study some properties of convex and DC matrix-valued functions and demonstrate how to compute DC decompositions of some nonlinear semidefinite constraints appearing in applications. We also compute a DC decomposition of the maximal eigenvalue of a DC matrix-valued function, which can be used to reformulate DC semidefinite constraints as DC inequality constrains. In the second part of the paper, we develop a general theory of cone constrained DC optimization problems. Namely, we obtain local optimality conditions for such problems and study an extension of the DC algorithm (the convex-concave procedure) to the case of general cone constrained DC optimization problems. We analyse a global convergence of this method and present a detailed study of a version of the DCA utilising exact penalty functions. In particular, we provide two types of sufficient conditions for the convergence of this method to a feasible and critical point of a cone constrained DC optimization problem from an infeasible starting point

DC Programming and DCA for General DC Programs

International audienc

Advances in knowledge discovery and data mining Part II

19th Pacific-Asia Conference, PAKDD 2015, Ho Chi Minh City, Vietnam, May 19-22, 2015, Proceedings, Part II</p