Machine Learning (stat.ML)

Abstract: Combining predictions from different models is a central problem in Bayesian inference and machine learning more broadly. Currently, these predictive distributions are almost exclusively combined using linear mixtures such as Bayesian model averaging, Bayesian stacking, and mixture of experts. Such linear mixtures impose idiosyncrasies that might be undesirable for some applications, such as multi-modality. While there exist alternative strategies (e.g. geometric bridge or superposition), optimising their parameters usually involves computing an intractable normalising constant repeatedly. We present two novel Bayesian model combination tools. These are generalisations of model stacking, but combine posterior densities by log-linear pooling (locking) and quantum superposition (quacking). To optimise model weights while avoiding the burden of normalising constants, we investigate the Hyvarinen score of the combined posterior predictions. We demonstrate locking with an illustrative example and discuss its practical application with importance sampling.

Abstract: In recent years, different types of distributed learning schemes have received increasing attention for their strong advantages in handling large-scale data information. In the information era, to face the big data challenges which stem from functional data analysis very recently, we propose a novel distributed gradient descent functional learning (DGDFL) algorithm to tackle functional data across numerous local machines (processors) in the framework of reproducing kernel Hilbert space. Based on integral operator approaches, we provide the first theoretical understanding of the DGDFL algorithm in many different aspects in the literature. On the way of understanding DGDFL, firstly, a data-based gradient descent functional learning (GDFL) algorithm associated with a single-machine model is proposed and comprehensively studied. Under mild conditions, confidence-based optimal learning rates of DGDFL are obtained without the saturation boundary on the regularity index suffered in previous works in functional regression. We further provide a semi-supervised DGDFL approach to weaken the restriction on the maximal number of local machines to ensure optimal rates. To our best knowledge, the DGDFL provides the first distributed iterative training approach to functional learning and enriches the stage of functional data analysis.

Abstract: Regression methods assume that accurate labels are available for training. However, in certain scenarios, obtaining accurate labels may not be feasible, and relying on multiple specialists with differing opinions becomes necessary. Existing approaches addressing noisy labels often impose restrictive assumptions on the regression function. In contrast, this paper presents a novel, more flexible approach. Our method consists of two steps: estimating each labeler's expertise and combining their opinions using learned weights. We then regress the weighted average against the input features to build the prediction model. The proposed method is formally justified and empirically demonstrated to outperform existing techniques on simulated and real data. Furthermore, its flexibility enables the utilization of any machine learning technique in both steps. In summary, this method offers a simple, fast, and effective solution for training regression models with noisy labels derived from diverse expert opinions.

Abstract: Originally introduced as a neural network for ensemble learning, mixture of experts (MoE) has recently become a fundamental building block of highly successful modern deep neural networks for heterogeneous data analysis in several applications, including those in machine learning, statistics, bioinformatics, economics, and medicine. Despite its popularity in practice, a satisfactory level of understanding of the convergence behavior of Gaussian-gated MoE parameter estimation is far from complete. The underlying reason for this challenge is the inclusion of covariates in the Gaussian gating and expert networks, which leads to their intrinsically complex interactions via partial differential equations with respect to their parameters. We address these issues by designing novel Voronoi loss functions to accurately capture heterogeneity in the maximum likelihood estimator (MLE) for resolving parameter estimation in these models. Our results reveal distinct behaviors of the MLE under two settings: the first setting is when all the location parameters in the Gaussian gating are non-zeros while the second setting is when there exists at least one zero-valued location parameter. Notably, these behaviors can be characterized by the solvability of two different systems of polynomial equations. Finally, we conduct a simulation study to verify our theoretical results.

Abstract: Attention-based graph neural networks (GNNs), such as graph attention networks (GATs), have become popular neural architectures for processing graph-structured data and learning node embeddings. Despite their empirical success, these models rely on labeled data and the theoretical properties of these models have yet to be fully understood. In this work, we propose a novel attention-based node embedding framework for graphs. Our framework builds upon a hierarchical kernel for multisets of subgraphs around nodes (e.g. neighborhoods) and each kernel leverages the geometry of a smooth statistical manifold to compare pairs of multisets, by "projecting" the multisets onto the manifold. By explicitly computing node embeddings with a manifold of Gaussian mixtures, our method leads to a new attention mechanism for neighborhood aggregation. We provide theoretical insights into genralizability and expressivity of our embeddings, contributing to a deeper understanding of attention-based GNNs. We propose efficient unsupervised and supervised methods for learning the embeddings, with the unsupervised method not requiring any labeled data. Through experiments on several node classification benchmarks, we demonstrate that our proposed method outperforms existing attention-based graph models like GATs. Our code is available at https://github.com/BorgwardtLab/fisher_information_embedding.