On the Stability of Graph Convolutional Neural Networks under Edge Rewiring

Graph neural networks are experiencing a surge of popularity within the machine learning community due to their ability to adapt to non-Euclidean domains and instil inductive biases. Despite this, their stability, i.e., their robustness to small perturbations in the input, is not yet well understood. Although there exists some results showing the stability of graph neural networks, most take the form of an upper bound on the magnitude of change due to a perturbation in the graph topology. However, these existing bounds tend to be expressed in terms of uninterpretable variables, limiting our understanding of the model robustness properties. In this work, we develop an interpretable upper bound elucidating that graph neural networks are stable to rewiring between high degree nodes. This bound and further research in bounds of similar type provide further understanding of the stability properties of graph neural networks.

Kernel-based Graph Learning from Smooth Signals: A Functional Viewpoint

The problem of graph learning concerns the construction of an explicit topological structure revealing the relationship between nodes representing data entities, which plays an increasingly important role in the success of many graph-based representations and algorithms in the field of machine learning and graph signal processing. In this paper, we propose a novel graph learning framework that incorporates the node-side and observation-side information, and in particular the covariates that help to explain the dependency structures in graph signals. To this end, we consider graph signals as functions in the reproducing kernel Hilbert space associated with a Kronecker product kernel, and integrate functional learning with smoothness-promoting graph learning to learn a graph representing the relationship between nodes. The functional learning increases the robustness of graph learning against missing and incomplete information in the graph signals. In addition, we develop a novel graph-based regularisation method which, when combined with the Kronecker product kernel, enables our model to capture both the dependency explained by the graph and the dependency due to graph signals observed under different but related circumstances, e.g. different points in time. The latter means the graph signals are free from the i.i.d. assumptions required by the classical graph learning models. Experiments on both synthetic and real-world data show that our methods outperform the state-of-the-art models in learning a meaningful graph topology from graph signals, in particular under heavy noise, missing values, and multiple dependency.

Graph signal processing for machine learning: A review and new perspectives

The effective representation, processing, analysis, and visualization of large-scale structured data, especially those related to complex domains such as networks and graphs, are one of the key questions in modern machine learning. Graph signal processing (GSP), a vibrant branch of signal processing models and algorithms that aims at handling data supported on graphs, opens new paths of research to address this challenge. In this article, we review a few important contributions made by GSP concepts and tools, such as graph filters and transforms, to the development of novel machine learning algorithms. In particular, our discussion focuses on the following three aspects: exploiting data structure and relational priors, improving data and computational efficiency, and enhancing model interpretability. Furthermore, we provide new perspectives on future development of GSP techniques that may serve as a bridge between applied mathematics and signal processing on one side, and machine learning and network science on the other. Cross-fertilization across these different disciplines may help unlock the numerous challenges of complex data analysis in the modern age.

Neural Architecture Search using Bayesian Optimisation with Weisfeiler-Lehman Kernel

Bayesian optimisation (BO) has been widely used for hyperparameter optimisation but its application in neural architecture search (NAS) is limited due to the non-continuous, high-dimensional and graph-like search spaces. Current approaches either rely on encoding schemes, which are not scalable to large architectures and ignore the implicit topological structure of architectures, or use graph neural networks, which require additional hyperparameter tuning and a large amount of observed data, which is particularly expensive to obtain in NAS. We propose a neat BO approach for NAS, which combines the Weisfeiler-Lehman graph kernel with a Gaussian process surrogate to capture the topological structure of architectures, without having to explicitly define a Gaussian process over high-dimensional vector spaces. We also harness the interpretable features learnt via the graph kernel to guide the generation of new architectures. We demonstrate empirically that our surrogate model is scalable to large architectures and highly data-efficient; competing methods require 3 to 20 times more observations to achieve equally good prediction performance as ours. We finally show that our method outperforms existing NAS approaches to achieve state-of-the-art results on NAS datasets.

Gaussian Processes on Graphs via Spectral Kernel Learning

We propose a graph spectrum-based Gaussian process for prediction of signals defined on nodes of the graph. The model is designed to capture various graph signal structures through a highly adaptive kernel that incorporates a flexible polynomial function in the graph spectral domain. Unlike most existing approaches, we propose to learn such a spectral kernel, where the polynomial setup enables learning without the need for eigen-decomposition of the graph Laplacian. In addition, this kernel has the interpretability of graph filtering achieved by a bespoke maximum likelihood learning algorithm that enforces the positivity of the spectrum. We demonstrate the interpretability of the model in synthetic experiments from which we show the various ground truth spectral filters can be accurately recovered, and the adaptability translates to superior performances in the prediction of real-world graph data of various characteristics.

A Maximum Entropy approach to Massive Graph Spectra

Graph spectral techniques for measuring graph similarity, or for learning the cluster number, require kernel smoothing. The choice of kernel function and bandwidth are typically chosen in an ad-hoc manner and heavily affect the resulting output. We prove that kernel smoothing biases the moments of the spectral density. We propose an information theoretically optimal approach to learn a smooth graph spectral density, which fully respects the moment information. Our method's computational cost is linear in the number of edges, and hence can be applied to large networks, with millions of nodes. We apply our method to the problems to graph similarity and cluster number learning, where we outperform comparable iterative spectral approaches on synthetic and real graphs.

Laplacian-regularized graph bandits: Algorithms and theoretical analysis

We study contextual multi-armed bandit problems in the case of multiple users, where we exploit the structure in the user domain to reduce the cumulative regret. Specifically, we model user relation as a graph, and assume that the parameters (preferences) of users form smooth signals on the graph. This leads to a graph Laplacian-regularized estimator, for which we propose a novel bandit algorithm whose performance depends on a notion of local smoothness on the graph. We provide a closed-form solution to the estimator, enabling a theoretical analysis on the convergence property of the estimator as well as single-user upper confidence bound (UCB) and cumulative regret of the proposed bandit algorithm. Furthermore, we show that the regret scales linearly with the local smoothness measure, which approaches zero for densely connected graph. The single-user UCB also allows us to further propose an extension of the bandit algorithm, whose computational complexity scales linearly with the number of users. We support theoretical claims with empirical evidences, and demonstrate the advantage of the proposed algorithm in comparison with state-of-the-art graph-based bandit algorithms on both synthetic and real-world datasets.

Error Analysis on Graph Laplacian Regularized Estimator

We provide a theoretical analysis of the representation learning problem aimed at learning the latent variables (design matrix) $\Theta$ of observations $Y$ with the knowledge of the coefficient matrix $X$. The design matrix is learned under the assumption that the latent variables $\Theta$ are smooth with respect to a (known) topological structure $\mathcal{G}$. To learn such latent variables, we study a graph Laplacian regularized estimator, which is the penalized least squares estimator with penalty term proportional to a Laplacian quadratic form. This type of estimators has recently received considerable attention due to its capability in incorporating underlying topological graph structure of variables into the learning process. While the estimation problem can be solved efficiently by state-of-the-art optimization techniques, its statistical consistency properties have been largely overlooked. In this work, we develop a non-asymptotic bound of estimation error under the classical statistical setting, where sample size is larger than the ambient dimension of the latent variables. This bound illustrates theoretically the impact of the alignment between the data and the graph structure as well as the graph spectrum on the estimation accuracy. It also provides theoretical evidence of the advantage, in terms of convergence rate, of the graph Laplacian regularized estimator over classical ones (that ignore the graph structure) in case of a smoothness prior. Finally, we provide empirical results of the estimation error to corroborate the theoretical analysis.

Learning Quadratic Games on Networks

Individuals, or organizations, cooperate with or compete against one another in a wide range of practical situations. Such strategic interactions may be modeled as games played on networks, where an individual's payoff depends not only on her action but also that of her neighbors. The current literature has predominantly focused on analyzing the characteristics of network games in the scenario where the structure of the network, which is represented by a graph, is known beforehand. It is often the case, however, that the actions of the players are readily observable while the underlying interaction network remains hidden. In this paper, we propose two novel frameworks for learning, from the observations on individual actions, network games with linear-quadratic payoffs, and in particular the structure of the interaction network. Our frameworks are based on the Nash equilibrium of such games and involve solving a joint optimization problem for the graph structure and the individual marginal benefits. We test the proposed frameworks in synthetic settings and further study several factors that affect their learning performance. Moreover, with experiments on three real-world examples, we show that our methods can effectively and more accurately learn the games than the baselines. The proposed approach is among the first of its kind for learning quadratic games, and have both theoretical and practical implications for understanding strategic interactions in a network environment.

Learning Graphs from Data: A Signal Representation Perspective

The construction of a meaningful graph topology plays a crucial role in the effective representation, processing, analysis and visualization of structured data. When a natural choice of the graph is not readily available from the datasets, it is thus desirable to infer or learn a graph topology from the data. In this tutorial overview, we survey solutions to the problem of graph learning, including classical viewpoints from statistics and physics, and more recent approaches that adopt a graph signal processing (GSP) perspective. We further emphasize the conceptual similarities and differences between classical and GSP graph inference methods and highlight the potential advantage of the latter in a number of theoretical and practical scenarios. We conclude with several open issues and challenges that are keys to the design of future signal processing and machine learning algorithms for learning graphs from data.