Fast Causal Discovery by Approximate Kernel-based Generalized Score Functions with Linear Computational Complexity

Score-based causal discovery methods can effectively identify causal relationships by evaluating candidate graphs and selecting the one with the highest score. One popular class of scores is kernel-based generalized score functions, which can adapt to a wide range of scenarios and work well in practice because they circumvent assumptions about causal mechanisms and data distributions. Despite these advantages, kernel-based generalized score functions pose serious computational challenges in time and space, with a time complexity of $\mathcal{O}(n^3)$ and a memory complexity of $\mathcal{O}(n^2)$, where $n$ is the sample size. In this paper, we propose an approximate kernel-based generalized score function with $\mathcal{O}(n)$ time and space complexities by using low-rank technique and designing a set of rules to handle the complex composite matrix operations required to calculate the score, as well as developing sampling algorithms for different data types to benefit the handling of diverse data types efficiently. Our extensive causal discovery experiments on both synthetic and real-world data demonstrate that compared to the state-of-the-art method, our method can not only significantly reduce computational costs, but also achieve comparable accuracy, especially for large datasets.

Flexible Differentially Private Vertical Federated Learning with Adaptive Feature Embeddings

The emergence of vertical federated learning (VFL) has stimulated concerns about the imperfection in privacy protection, as shared feature embeddings may reveal sensitive information under privacy attacks. This paper studies the delicate equilibrium between data privacy and task utility goals of VFL under differential privacy (DP). To address the generality issue of prior arts, this paper advocates a flexible and generic approach that decouples the two goals and addresses them successively. Specifically, we initially derive a rigorous privacy guarantee by applying norm clipping on shared feature embeddings, which is applicable across various datasets and models. Subsequently, we demonstrate that task utility can be optimized via adaptive adjustments on the scale and distribution of feature embeddings in an accuracy-appreciative way, without compromising established DP mechanisms. We concretize our observation into the proposed VFL-AFE framework, which exhibits effectiveness against privacy attacks and the capacity to retain favorable task utility, as substantiated by extensive experiments.