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.Comment: 11 pages, 14 figure

Discovery of low-dimensional structure in high-dimensional inference problems

Many learning and inference problems involve high-dimensional data such as images, video or genomic data, which cannot be processed efficiently using conventional methods due to their dimensionality. However, high-dimensional data often exhibit an inherent low-dimensional structure, for instance they can often be represented sparsely in some basis or domain. The discovery of an underlying low-dimensional structure is important to develop more robust and efficient analysis and processing algorithms. The first part of the dissertation investigates the statistical complexity of sparse recovery problems, including sparse linear and nonlinear regression models, feature selection and graph estimation. We present a framework that unifies sparse recovery problems and construct an analogy to channel coding in classical information theory. We perform an information-theoretic analysis to derive bounds on the number of samples required to reliably recover sparsity patterns independent of any specific recovery algorithm. In particular, we show that sample complexity can be tightly characterized using a mutual information formula similar to channel coding results. Next, we derive major extensions to this framework, including dependent input variables and a lower bound for sequential adaptive recovery schemes, which helps determine whether adaptivity provides performance gains. We compute statistical complexity bounds for various sparse recovery problems, showing our analysis improves upon the existing bounds and leads to intuitive results for new applications. In the second part, we investigate methods for improving the computational complexity of subgraph detection in graph-structured data, where we aim to discover anomalous patterns present in a connected subgraph of a given graph. This problem arises in many applications such as detection of network intrusions, community detection, detection of anomalous events in surveillance videos or disease outbreaks. Since optimization over connected subgraphs is a combinatorial and computationally difficult problem, we propose a convex relaxation that offers a principled approach to incorporating connectivity and conductance constraints on candidate subgraphs. We develop a novel nearly-linear time algorithm to solve the relaxed problem, establish convergence and consistency guarantees and demonstrate its feasibility and performance with experiments on real networks