-
作者：He, Ziyu; Liu, Junyi; Pang, Jong-Shi
作者单位：University of Southern California; University of Southern California; Tsinghua University
摘要：This paper explores Logarithmic Integral Optimization (LIO) problems, providing a unified computational framework for various tasks in computational statistics. Key among these are Maximum Likelihood Estimation (MLE) and Maximum a Posteriori (MAP) inference for probabilistic models. Specifically, we investigate scenarios where the model consists of conditional density functions with intractable normalizers. This feature can pose substantial computational challenges for the associated LIO, espe...
-
作者：Lejeune, Miguel; Romeijnders, Ward; Krokhmal, Pavlo
作者单位：George Washington University; University of Groningen; University of Arizona
-
作者：Carrasco, Pablo; Munoz, Gonzalo
作者单位：Universidad de Chile
摘要：The non-convex nature of trained neural networks has created significant obstacles in their incorporation into optimization models. In this context, Anderson et al. (2020) provided a framework to obtain the convex hull of the graph of a piecewise linear convex activation function composed with an affine function; this effectively convexifies activations such as the ReLU together with the affine transformation that precedes it. In this article, we contribute to this line of work by developing a...
-
作者：Luo, Honglin; Wang, Xianfu; Wang, Ziyuan; Yang, Xinmin
作者单位：Chongqing Normal University
摘要：Level proximal subdifferential was introduced by Rockafellar recently for studyingproximal mappings of possibly nonconvex functions. In this paper a systematic studyof level proximal subdifferential is given. We characterize variational convexity of afunction by local firm nonexpansiveness of proximal mappings or local relative mono-tonicity of level proximal subdifferential, and use them to study local convergence ofproximal gradient method and others for variationally convex functions. Varia...
-
作者：Chen, Renjie; Xu, Huifu; Zahle, Henryk
作者单位：Chinese University of Hong Kong; Saarland University
摘要：Finding an approximation of the inverse of the covariance matrix, also known as precision matrix, of a random vector with empirical data is widely discussed in finance and engineering. In data-driven problems, empirical data may be contaminated. This raises the question as to whether the approximate precision matrix is reliable from a statistical point of view. In this paper, we concentrate on a much-noticed sparse estimator of the precision matrix and investigate the issue from the perspectiv...
-
作者：Carmon, Yair; Hinder, Oliver
作者单位：Pennsylvania Commonwealth System of Higher Education (PCSHE); University of Pittsburgh
摘要：We prove impossibility results for adaptivity in non-smooth stochastic convex optimization. Given a set of problem parameters we wish to adapt to, we define a price of adaptivity (PoA) that, roughly speaking, measures the multiplicative increase in suboptimality due to uncertainty in these parameters. When the initial distance to the optimum is unknown but a gradient norm bound is known, we show that the PoA is at least logarithmic for expected suboptimality, and double-logarithmic for median ...
-
作者：Aliev, Iskander; Celaya, Marcel; Henk, Martin
作者单位：Cardiff University; Technical University of Berlin
摘要：We obtain new transference bounds that connect the additive integrality gap and sparsity of solutions for integer linear programs. Specifically, we consider the integer programs min{cx:x is an element of P boolean AND Zn}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min \{{\varvec{c}}\cdot {\varvec{x}}: {\varvec...
-
作者：Arakcheev, Aleksandr; Bauschke, Heinz H.
作者单位：University of British Columbia
摘要：Opial's Lemma is a fundamental result in the convergence analysis of sequences generated by optimization algorithms in real Hilbert spaces. We introduce the concept of Opial sequences-sequences for which the limit of the distance to each point in a given set exists. We systematically derive properties of Opial sequences, contrasting them with the well-studied Fej & eacute;r monotone sequences, and establish conditions for weak and strong convergence. Key results include characterizations of we...
-
作者：Yang, Yan; Gao, Bin; Yuan, Ya-xiang
作者单位：Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; Chinese Academy of Sciences; University of Chinese Academy of Sciences, CAS
摘要：Imposing additional constraints on low-rank optimization has garnered growing interest. However, the geometry of coupled constraints hampers the well-developed low-rank structure and makes the problem intricate. To this end, we propose a space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints. The space-decoupling is reflected in several ways. We show that the tangent cone of coupled constraints is the intersection of tangent cones of each c...
-
作者：Cartis, Coralia; Zhu, Wenqi
作者单位：University of Oxford
摘要：There has been growing interest in high-order tensor methods for nonconvex optimization, with adaptive regularization, as they possess better/optimal worst-case evaluation complexity globally and faster convergence asymptotically. These algorithms crucially rely on repeatedly minimizing nonconvex multivariate Taylor-based polynomial sub-problems, at least locally. Finding efficient techniques for the solution of these sub-problems, beyond the second-order case, has been an open question. This ...
