arXiv subjects

Baojian Zhou

Publications and source records attributed to Baojian Zhou.

28 records · Page 2Linked to original sources

arXiv2020

Stochastic Hard Thresholding Algorithms for AUC Maximization

In this paper, we aim to develop stochastic hard thresholding algorithms for the important problem of AUC maximization in imbalanced classification. The main challenge is the pairwise loss involved in AUC maximization. We overcome this obstacle by reformulating the U-statistics objective function as an empirical risk minimization (ERM), from which a stochastic hard thresholding algorithm (\texttt{SHT-AUC}) is developed. To our best knowledge, this is the first attempt to provide stochastic hard thresholding algorithms for AUC maximization with a per-iteration cost $Ø(b d)$ where $d$ and $b$ are the dimension of the data and the minibatch size, respectively. We show that the proposed algorithm enjoys the linear convergence rate up to a tolerance error. In particular, we show, if the data is generated from the Gaussian distribution, then its convergence becomes slower as the data gets more imbalanced. We conduct extensive experiments to show the efficiency and effectiveness of the proposed algorithms.

cs.LG↗

arXiv2020

Online AUC Optimization for Sparse High-Dimensional Datasets

The Area Under the ROC Curve (AUC) is a widely used performance measure for imbalanced classification arising from many application domains where high-dimensional sparse data is abundant. In such cases, each $d$ dimensional sample has only $k$ non-zero features with $k \ll d$, and data arrives sequentially in a streaming form. Current online AUC optimization algorithms have high per-iteration cost $\mathcal{O}(d)$ and usually produce non-sparse solutions in general, and hence are not suitable for handling the data challenge mentioned above. In this paper, we aim to directly optimize the AUC score for high-dimensional sparse datasets under online learning setting and propose a new algorithm, \textsc{FTRL-AUC}. Our proposed algorithm can process data in an online fashion with a much cheaper per-iteration cost $\mathcal{O}(k)$, making it amenable for high-dimensional sparse streaming data analysis. Our new algorithmic design critically depends on a novel reformulation of the U-statistics AUC objective function as the empirical saddle point reformulation, and the innovative introduction of the "lazy update" rule so that the per-iteration complexity is dramatically reduced from $\mathcal{O}(d)$ to $\mathcal{O}(k)$. Furthermore, \textsc{FTRL-AUC} can inherently capture sparsity more effectively by applying a generalized Follow-The-Regularized-Leader (FTRL) framework. Experiments on real-world datasets demonstrate that \textsc{FTRL-AUC} significantly improves both run time and model sparsity while achieving competitive AUC scores compared with the state-of-the-art methods. Comparison with the online learning method for logistic loss demonstrates that \textsc{FTRL-AUC} achieves higher AUC scores especially when datasets are imbalanced.

cs.LG↗

arXiv2019

Dual Averaging Method for Online Graph-structured Sparsity

Online learning algorithms update models via one sample per iteration, thus efficient to process large-scale datasets and useful to detect malicious events for social benefits, such as disease outbreak and traffic congestion on the fly. However, existing algorithms for graph-structured models focused on the offline setting and the least square loss, incapable for online setting, while methods designed for online setting cannot be directly applied to the problem of complex (usually non-convex) graph-structured sparsity model. To address these limitations, in this paper we propose a new algorithm for graph-structured sparsity constraint problems under online setting, which we call \textsc{GraphDA}. The key part in \textsc{GraphDA} is to project both averaging gradient (in dual space) and primal variables (in primal space) onto lower dimensional subspaces, thus capturing the graph-structured sparsity effectively. Furthermore, the objective functions assumed here are generally convex so as to handle different losses for online learning settings. To the best of our knowledge, \textsc{GraphDA} is the first online learning algorithm for graph-structure constrained optimization problems. To validate our method, we conduct extensive experiments on both benchmark graph and real-world graph datasets. Our experiment results show that, compared to other baseline methods, \textsc{GraphDA} not only improves classification performance, but also successfully captures graph-structured features more effectively, hence stronger interpretability.

cs.LG↗

arXiv2019

Stochastic Iterative Hard Thresholding for Graph-structured Sparsity Optimization

Stochastic optimization algorithms update models with cheap per-iteration costs sequentially, which makes them amenable for large-scale data analysis. Such algorithms have been widely studied for structured sparse models where the sparsity information is very specific, e.g., convex sparsity-inducing norms or $\ell^0$-norm. However, these norms cannot be directly applied to the problem of complex (non-convex) graph-structured sparsity models, which have important application in disease outbreak and social networks, etc. In this paper, we propose a stochastic gradient-based method for solving graph-structured sparsity constraint problems, not restricted to the least square loss. We prove that our algorithm enjoys a linear convergence up to a constant error, which is competitive with the counterparts in the batch learning setting. We conduct extensive experiments to show the efficiency and effectiveness of the proposed algorithms.

cs.LG↗

arXiv2018

A Generic Framework for Interesting Subspace Cluster Detection in Multi-attributed Networks

Detection of interesting (e.g., coherent or anomalous) clusters has been studied extensively on plain or univariate networks, with various applications. Recently, algorithms have been extended to networks with multiple attributes for each node in the real-world. In a multi-attributed network, often, a cluster of nodes is only interesting for a subset (subspace) of attributes, and this type of clusters is called subspace clusters. However, in the current literature, few methods are capable of detecting subspace clusters, which involves concurrent feature selection and network cluster detection. These relevant methods are mostly heuristic-driven and customized for specific application scenarios. In this work, we present a generic and theoretical framework for detection of interesting subspace clusters in large multi-attributed networks. Specifically, we propose a subspace graph-structured matching pursuit algorithm, namely, SG-Pursuit, to address a broad class of such problems for different score functions (e.g., coherence or anomalous functions) and topology constraints (e.g., connected subgraphs and dense subgraphs). We prove that our algorithm 1) runs in nearly-linear time on the network size and the total number of attributes and 2) enjoys rigorous guarantees (geometrical convergence rate and tight error bound) analogous to those of the state-of-the-art algorithms for sparse feature selection problems and subgraph detection problems. As a case study, we specialize SG-Pursuit to optimize a number of well-known score functions for two typical tasks, including detection of coherent dense and anomalous connected subspace clusters in real-world networks. Empirical evidence demonstrates that our proposed generic algorithm SG-Pursuit performs superior over state-of-the-art methods that are designed specifically for these two tasks.

cs.LG↗

arXiv2017

Can Self-Censorship in News Media be Detected Algorithmically? A Case Study in Latin America

Censorship in social media has been well studied and provides insight into how governments stifle freedom of expression online. Comparatively less (or no) attention has been paid to detecting (self) censorship in traditional media (e.g., news) using social media as a bellweather. We present a novel unsupervised approach that views social media as a sensor to detect censorship in news media wherein statistically significant differences between information published in the news media and the correlated information published in social media are automatically identified as candidate censored events. We develop a hypothesis testing framework to identify and evaluate censored clusters of keywords, and a new near-linear-time algorithm (called GraphDPD) to identify the highest scoring clusters as indicators of censorship. We outline extensive experiments on semi-synthetic data as well as real datasets (with Twitter and local news media) from Mexico and Venezuela, highlighting the capability to accurately detect real-world self censorship events.

cs.SI↗

arXiv2016

Technical Report: A Generalized Matching Pursuit Approach for Graph-Structured Sparsity

Sparsity-constrained optimization is an important and challenging problem that has wide applicability in data mining, machine learning, and statistics. In this paper, we focus on sparsity-constrained optimization in cases where the cost function is a general nonlinear function and, in particular, the sparsity constraint is defined by a graph-structured sparsity model. Existing methods explore this problem in the context of sparse estimation in linear models. To the best of our knowledge, this is the first work to present an efficient approximation algorithm, namely, Graph-structured Matching Pursuit (Graph-Mp), to optimize a general nonlinear function subject to graph-structured constraints. We prove that our algorithm enjoys the strong guarantees analogous to those designed for linear models in terms of convergence rate and approximation accuracy. As a case study, we specialize Graph-Mp to optimize a number of well-known graph scan statistic models for the connected subgraph detection task, and empirical evidence demonstrates that our general algorithm performs superior over state-of-the-art methods that are designed specifically for the task of connected subgraph detection.

cs.LG↗

arXiv2016

Technical Report: Graph-Structured Sparse Optimization for Connected Subgraph Detection

Structured sparse optimization is an important and challenging problem for analyzing high-dimensional data in a variety of applications such as bioinformatics, medical imaging, social networks, and astronomy. Although a number of structured sparsity models have been explored, such as trees, groups, clusters, and paths, connected subgraphs have been rarely explored in the current literature. One of the main technical challenges is that there is no structured sparsity-inducing norm that can directly model the space of connected subgraphs, and there is no exact implementation of a projection oracle for connected subgraphs due to its NP-hardness. In this paper, we explore efficient approximate projection oracles for connected subgraphs, and propose two new efficient algorithms, namely, Graph-IHT and Graph-GHTP, to optimize a generic nonlinear objective function subject to connectivity constraint on the support of the variables. Our proposed algorithms enjoy strong guarantees analogous to several current methods for sparsity-constrained optimization, such as Projected Gradient Descent (PGD), Approximate Model Iterative Hard Thresholding (AM-IHT), and Gradient Hard Thresholding Pursuit (GHTP) with respect to convergence rate and approximation accuracy. We apply our proposed algorithms to optimize several well-known graph scan statistics in several applications of connected subgraph detection as a case study, and the experimental results demonstrate that our proposed algorithms outperform state-of-the-art methods.

cs.AI↗

arXiv2016

A Quadratic Programming Relaxation Approach to Compute-and-Forward Network Coding Design

Using physical layer network coding, compute-and-forward is a promising relaying scheme that effectively exploits the interference between users and thus achieves high rates. In this paper, we consider the problem of finding the optimal integer-valued coefficient vector for a relay in the compute-and-forward scheme to maximize the computation rate at that relay. Although this problem turns out to be a shortest vector problem, which is suspected to be NP-hard, we show that it can be relaxed to a series of equality-constrained quadratic programmings. The solutions of the relaxed problems serve as real-valued approximations of the optimal coefficient vector, and are quantized to a set of integer-valued vectors, from which a coefficient vector is selected. The key to the efficiency of our method is that the closed-form expressions of the real-valued approximations can be derived with the Lagrange multiplier method. Numerical results demonstrate that compared with the existing methods, our method offers comparable rates at an impressively low complexity.

cs.IT↗

arXiv2016

An Efficient Algorithm for Optimally Solving a Shortest Vector Problem in Compute-and-Forward Protocol Design

We consider the problem of finding the optimal coefficient vector that maximizes the computation rate at a relay in the compute-and-forward scheme. Based on the idea of sphere decoding, we propose a highly efficient algorithm that finds the optimal coefficient vector. First, we derive a novel algorithm to transform the original quadratic form optimization problem into a shortest vector problem (SVP) using the Cholesky factorization. Instead of computing the Cholesky factor explicitly, the proposed algorithm realizes the Cholesky factorization with only $\bigO(n)$ flops by taking advantage of the structure of the Gram matrix in the quadratic form. Then, we propose some conditions that can be checked with $\bigO(n)$ flops, under which a unit vector is the optimal coefficient vector. Finally, by taking into account some useful properties of the optimal coefficient vector, we modify the Schnorr-Euchner search algorithm to solve the SVP. We show that the estimated average complexity of our new algorithm is $\bigO(n^{1.5}P^{0.5})$ flops for i.i.d. Gaussian channel entries with SNR $P$ based on the Gaussian heuristic. Simulations show that our algorithm is not only much more efficient than the existing ones that give the optimal solution, but also faster than some best known suboptimal methods. Besides, we show that our algorithm can be readily adapted to output a list of $L$ best candidate vectors for use in the compute-and-forward design. The estimated average complexity of the resultant list-output algorithm is $\bigO\left(n^{1.5}P^{0.5}\log L + nL\right)$ flops for i.i.d. Gaussian channel entries.

cs.IT↗