Minimizing a sum of simple submodular functions of limited support is a special case of general submodular function minimization that has seen numerous applications in machine learning. We develop fast techniques for instances where components in the sum are cardinality-based, meaning they depend only on the size of the input set. This variant is one of the most widely applied in practice, encompassing, e.g., common energy functions arising in image segmentation and recent generalized hypergraph cut functions. We develop the first approximation algorithms for this problem, where the approximations can be quickly computed via reduction to a sparse graph cut problem, with graph sparsity controlled by the desired approximation factor. Our method relies on a new connection between sparse graph reduction techniques and piecewise linear approximations to concave functions. Our sparse reduction technique leads to significant improvements in theoretical runtimes, as well as substantial practical gains in problems ranging from benchmark image segmentation tasks to hypergraph clustering problems.
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Given the noisy pairwise measurements among a set of unknown group elements, how to recover them efficiently and robustly? This problem, known as group synchronization, has drawn tremendous attention in the scientific community. In this work, we focus on orthogonal group synchronization that has found many applications, including computer vision, robotics, and cryo-electron microscopy. One commonly used approach is the least squares estimation that requires solving a highly nonconvex optimization program. The past few years have witnessed considerable advances in tackling this challenging problem by convex relaxation and efficient first-order methods. However, one fundamental theoretical question remains to be answered: how does the recovery performance depend on the noise strength? To answer this question, we study a benchmark model: recovering orthogonal group elements from their pairwise measurements corrupted by Gaussian noise. We investigate the performance of convex relaxation and the generalized power method (GPM). By applying the novel~\emph{leave-one-out} technique, we prove that the GPM with spectral initialization enjoys linear convergence to the global optima to the convex relaxation that also matches the maximum likelihood estimator. Our result achieves a near-optimal performance bound on the convergence of the GPM and improves the state-of-the-art theoretical guarantees on the tightness of convex relaxation by a large margin.
We initiate a systematic study on $\mathit{dynamic}$ $\mathit{influence}$ $\mathit{maximization}$ (DIM). In the DIM problem, one maintains a seed set $S$ of at most $k$ nodes in a dynamically involving social network, with the goal of maximizing the expected influence spread while minimizing the amortized updating cost. We consider two evolution models. In the $\mathit{incremental}$ model, the social network gets enlarged over time and one only introduces new users and establishes new social links, we design an algorithm that achieves $(1-1/e-\epsilon)$-approximation to the optimal solution and has $k \cdot\mathsf{poly}(\log n, \epsilon^{-1})$ amortized running time, which matches the state-of-art offline algorithm with only poly-logarithmic overhead. In the $\mathit{fully}$ $\mathit{dynamic}$ model, users join in and leave, influence propagation gets strengthened or weakened in real time, we prove that under the Strong Exponential Time Hypothesis (SETH), no algorithm can achieve $2^{-(\log n)^{1-o(1)}}$-approximation unless the amortized running time is $n^{1-o(1)}$. On the technical side, we exploit novel adaptive sampling approaches that reduce DIM to the dynamic MAX-k coverage problem, and design an efficient $(1-1/e-\epsilon)$-approximation algorithm for it. Our lower bound leverages the recent developed distributed PCP framework.
Learning structural representations of node sets from graph-structured data is crucial for applications ranging from node-role discovery to link prediction and molecule classification. Graph Neural Networks (GNNs) have achieved great success in structural representation learning. However, most GNNs are limited by the 1-Weisfeiler-Lehman (WL) test and thus possible to generate identical representation for structures and graphs that are actually different. More powerful GNNs, proposed recently by mimicking higher-order-WL tests, only focus on entire-graph representations and cannot utilize sparsity of the graph structure to be computationally efficient. Here we propose a general class of structure-related features, termed Distance Encoding (DE), to assist GNNs in representing node sets with arbitrary sizes with strictly more expressive power than the 1-WL test. DE essentially captures the distance between the node set whose representation is to be learnt and each node in the graph, which includes important graph-related measures such as shortest-path-distance and generalized PageRank scores. We propose two general frameworks for GNNs to use DEs (1) as extra node attributes and (2) further as controllers of message aggregation in GNNs. Both frameworks may still utilize the sparse structure to keep scalability to process large graphs. In theory, we prove that these two frameworks can distinguish node sets embedded in almost all regular graphs where traditional GNNs always fail. We also rigorously analyze their limitations. Empirically, we evaluate these two frameworks on node structural roles prediction, link prediction and triangle prediction over six real networks. The results show that our models outperform GNNs without DEs by up-to 15% improvement in average accuracy and AUC. Our models also significantly outperform other SOTA baselines particularly designed for those tasks.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
Discovering causal structure among a set of variables is a fundamental problem in many empirical sciences. Traditional score-based casual discovery methods rely on various local heuristics to search for a Directed Acyclic Graph (DAG) according to a predefined score function. While these methods, e.g., greedy equivalence search, may have attractive results with infinite samples and certain model assumptions, they are usually less satisfactory in practice due to finite data and possible violation of assumptions. Motivated by recent advances in neural combinatorial optimization, we propose to use Reinforcement Learning (RL) to search for the DAG with the best scoring. Our encoder-decoder model takes observable data as input and generates graph adjacency matrices that are used to compute rewards. The reward incorporates both the predefined score function and two penalty terms for enforcing acyclicity. In contrast with typical RL applications where the goal is to learn a policy, we use RL as a search strategy and our final output would be the graph, among all graphs generated during training, that achieves the best reward. We conduct experiments on both synthetic and real datasets, and show that the proposed approach not only has an improved search ability but also allows a flexible score function under the acyclicity constraint.
We present a new clustering method in the form of a single clustering equation that is able to directly discover groupings in the data. The main proposition is that the first neighbor of each sample is all one needs to discover large chains and finding the groups in the data. In contrast to most existing clustering algorithms our method does not require any hyper-parameters, distance thresholds and/or the need to specify the number of clusters. The proposed algorithm belongs to the family of hierarchical agglomerative methods. The technique has a very low computational overhead, is easily scalable and applicable to large practical problems. Evaluation on well known datasets from different domains ranging between 1077 and 8.1 million samples shows substantial performance gains when compared to the existing clustering techniques.
The eigendeomposition of nearest-neighbor (NN) graph Laplacian matrices is the main computational bottleneck in spectral clustering. In this work, we introduce a highly-scalable, spectrum-preserving graph sparsification algorithm that enables to build ultra-sparse NN (u-NN) graphs with guaranteed preservation of the original graph spectrums, such as the first few eigenvectors of the original graph Laplacian. Our approach can immediately lead to scalable spectral clustering of large data networks without sacrificing solution quality. The proposed method starts from constructing low-stretch spanning trees (LSSTs) from the original graphs, which is followed by iteratively recovering small portions of "spectrally critical" off-tree edges to the LSSTs by leveraging a spectral off-tree embedding scheme. To determine the suitable amount of off-tree edges to be recovered to the LSSTs, an eigenvalue stability checking scheme is proposed, which enables to robustly preserve the first few Laplacian eigenvectors within the sparsified graph. Additionally, an incremental graph densification scheme is proposed for identifying extra edges that have been missing in the original NN graphs but can still play important roles in spectral clustering tasks. Our experimental results for a variety of well-known data sets show that the proposed method can dramatically reduce the complexity of NN graphs, leading to significant speedups in spectral clustering.
Hashing, or learning binary embeddings of data, is frequently used in nearest neighbor retrieval. In this paper, we develop learning to rank formulations for hashing, aimed at directly optimizing ranking-based evaluation metrics such as Average Precision (AP) and Normalized Discounted Cumulative Gain (NDCG). We first observe that the integer-valued Hamming distance often leads to tied rankings, and propose to use tie-aware versions of AP and NDCG to evaluate hashing for retrieval. Then, to optimize tie-aware ranking metrics, we derive their continuous relaxations, and perform gradient-based optimization with deep neural networks. Our results establish the new state-of-the-art for image retrieval by Hamming ranking in common benchmarks.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.
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