In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Example Input: The alternating direction method of multipliers (ADMM) has been recognized as a versatile approach for solving modern large-scale machine learning and signal processing problems efficiently. When the data size and/or the problem dimension is large, a distributed version of ADMM can be used, which is capable of distributing the computation load and the data set to a network of computing nodes. Unfortunately, a direct synchronous implementation of such algorithm does not scale well with the problem size, as the algorithm speed is limited by the slowest computing nodes. To address this issue, in a companion paper, we have proposed an asynchronous distributed ADMM (AD-ADMM) and studied its worst-case convergence conditions. In this paper, we further the study by characterizing the conditions under which the AD-ADMM achieves linear convergence. Our conditions as well as the resulting linear rates reveal the impact that various algorithm parameters, network delay and network size have on the algorithm performance. To demonstrate the superior time efficiency of the proposed AD-ADMM, we test the ADADMM on a high-performance computer cluster by solving a large-scale logistic regression problem. Keywords− Distributed optimization, ADMM, Asynchronous, Consensus optimization ⋆Tsung-Hui Chang is the corresponding author. Address: School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, China 518172, E-mail: tsunghui.chang@ieee.org. Wei-Cheng Liao is with Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA, E-mail: mhong@umn.edu Mingyi Hong is with Department of Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, 50011, USA, E-mail: mingyi@iastate.edu Xiangfeng Wang is with Shanghai Key Lab for Trustworthy Computing, Software Engineering Institute, East China Normal University, Shanghai, 200062, China, E-mail: xfwang@sei.ecnu.edu.cn September 10, 2015 DRAFT
Example Output: Asynchronous Distributed ADMM for Large-Scale Optimization- Part II: Linear Convergence Analysis and Numerical Performance

Example Input: We consider relative error low rank approximation of tensors with respect to the Frobenius norm. Namely, given an order-q tensor A ∈ R ∏q i=1 ni , output a rank-k tensor B for which ‖A − B‖F ≤ (1 + ) OPT, where OPT = infrank-k A′ ‖A − A‖F . Despite much success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that there may be no rank-k tensor Ak achieving the above infinum. Another, computational issue, is that an efficient relative error low rank approximation algorithm for tensors would allow one to compute the rank of a tensor, which is NP-hard. We bypass these two issues via (1) bicriteria and (2) parameterized complexity solutions: 1. We give an algorithm which outputs a rank k′ = O((k/ )q−1) tensor B for which ‖A − B‖F ≤ (1+ ) OPT in nnz(A)+n ·poly(k/ ) time in the real RAM model, whenever either Ak exists or OPT > 0. Here nnz(A) denotes the number of non-zero entries in A. If both Ak does not exist and OPT = 0, then B instead satisfies ‖A − B‖F < γ, where γ is any positive, arbitrarily small function of n. 2. We give an algorithm for any δ > 0 which outputs a rank k tensor B for which ‖A−B‖F ≤ (1+ ) OPT and runs in (nnz(A)+n poly(k/ )+exp(k/ )) ·nδ time in the unit cost RAM model, whenever OPT > 2−O(n ) and there is a rank-k tensor B = ∑k i=1 ui ⊗ vi ⊗ wi for which ‖A − B‖F ≤ (1 + /2) OPT and ‖ui‖2, ‖vi‖2, ‖wi‖2 ≤ 2 δ). If OPT ≤ 2−Ω(nδ), then B instead satisfies ‖A−B‖F ≤ 2−Ω(n δ). Our first result is polynomial time, and in fact input sparsity time, in n, k, and 1/ , for any k ≥ 1 and any 0 < < 1, while our second result is fixed parameter tractable in k and 1/ . For outputting a rank-k tensor, or even a bicriteria solution with rank-Ck for a certain constant C > 1, we show a 2 1−o(1)) time lower bound under the Exponential Time Hypothesis. Our results are based on an “iterative existential argument”, and give the first relative error low rank approximations for tensors for a large number of error measures for which nothing was known. In particular, we give the first relative error approximation algorithms on tensors for: column row and tube subset selection, entrywise `p-low rank approximation for 1 ≤ p < 2, low rank approximation with respect to sum of Euclidean norms of faces or tubes, weighted low rank approximation, and low rank approximation in distributed and streaming models. We also obtain several new results for matrices, such as nnz(A)-time CUR decompositions, improving the previous nnz(A) log n-time CUR decompositions, which may be of independent interest. ∗Work done while visiting IBM Almaden, and supported in part by UTCS TAship (CS361 Spring 17 Introduction to Computer Security). †Supported in part by Simons Foundation, and NSF CCF-1617955. ar X iv :1 70 4. 08 24 6v 1 [ cs .D S] 2 6 A pr 2 01 7
Example Output: Relative Error Tensor Low Rank Approximation

Example Input: Neural word representations have proven useful in Natural Language Processing (NLP) tasks due to their ability to efficiently model complex semantic and syntactic word relationships. However, most techniques model only one representation per word, despite the fact that a single word can have multiple meanings or ”senses”. Some techniques model words by using multiple vectors that are clustered based on context. However, recent neural approaches rarely focus on the application to a consuming NLP algorithm. Furthermore, the training process of recent word-sense models is expensive relative to single-sense embedding processes. This paper presents a novel approach which addresses these concerns by modeling multiple embeddings for each word based on supervised disambiguation, which provides a fast and accurate way for a consuming NLP model to select a sense-disambiguated embedding. We demonstrate that these embeddings can disambiguate both contrastive senses such as nominal and verbal senses as well as nuanced senses such as sarcasm. We further evaluate Part-of-Speech disambiguated embeddings on neural dependency parsing, yielding a greater than 8% average error reduction in unlabeled attachment scores across 6 languages.
Example Output:
A FAST AND ACCURATE METHOD FOR WORD SENSE DISAMBIGUATION IN NEURAL WORD EMBEDDINGS