Scalable and Privacy-Aware Online Learning of Nonlinear Structural Equation Models

An online topology estimation algorithm for nonlinear structural equation models (SEM) is proposed in this paper, addressing the nonlinearity and the non-stationarity of real-world systems. The nonlinearity is modeled using kernel formulations, and the curse of dimensionality associated with the kernels is mitigated using random feature approximation. The online learning strategy uses a group-lasso-based optimization framework with a prediction-corrections technique that accounts for the model evolution. The proposed approach has three properties of interest. First, it enjoys node-separable learning, which allows for scalability in large networks. Second, it offers privacy in SEM learning by replacing the actual data with node-specific random features. Third, its performance can be characterized theoretically via a dynamic regret analysis, showing that it is possible to obtain a linear dynamic regret bound under mild assumptions. Numerical results with synthetic and real data corroborate our findings and show competitive performance w.r.t. state-of-the-art alternatives.

Introduction

Structural Equation Models (SEM) are a prevalent tool to model interactions in real-world networks due to their simplicity and ability to express instantaneous directed relationships between interacting entities [1], [2], [3]. The advantages of SEM over simple correlation-based models lie in leveraging the directionality, which is key to many applications, such as modeling the functional connectivity between brain regions [4] and interactions in financial networks [5], to name a few. SEM modeling and its topology estimation are challenging because real-life networks are large, dynamic, and comprise nonlinear interactions, as well as leveraging directly node-specific data may raise privacy concerns [1].

Although SEM-based topology est