Nicola Bariletto

Khai Nguyen, Nicola Bariletto, Nhat Ho

Monte Carlo (MC) integration has been employed as the standard approximation method for the Sliced Wasserstein (SW) distance, whose analytical expression involves an intractable expectation. However, MC integration is not optimal in terms of absolute approximation error. To provide a better class of empirical SW, we propose quasi-sliced Wasserstein (QSW) approximations that rely on Quasi-Monte Carlo (QMC) methods. For a comprehensive investigation of QMC for SW, we focus on the 3D setting, specifically computing the SW between probability measures in three dimensions. In greater detail, we empirically evaluate various methods to construct QMC point sets on the 3D unit-hypersphere, including the Gaussian-based and equal area mappings, generalized spiral points, and optimizing discrepancy energies. Furthermore, to obtain an unbiased estimator for stochastic optimization, we extend QSW to Randomized Quasi-Sliced Wasserstein (RQSW) by introducing randomness in the discussed point sets. Theoretically, we prove the asymptotic convergence of QSW and the unbiasedness of RQSW. Finally, we conduct experiments on various 3D tasks, such as point-cloud comparison, point-cloud interpolation, image style transfer, and training deep point-cloud autoencoders, to demonstrate the favorable performance of the proposed QSW and RQSW variants.

Nicola Bariletto, Stephen G. Walker

We introduce a novel framework for uncertainty quantification in clustering. By combining the martingale posterior paradigm with density-based clustering, uncertainty in the estimated density is naturally propagated to the clustering structure. The approach scales effectively to high-dimensional and irregularly shaped data by leveraging modern neural density estimators and GPU-friendly parallel computation. We establish frequentist consistency guarantees and validate the methodology on synthetic and real data.

Nicola Bariletto, Huy Nguyen, Nhat Ho et al.

Mixture-of-experts models provide a flexible framework for learning complex probabilistic input-output relationships by combining multiple expert models through an input-dependent gating mechanism. These models have become increasingly prominent in modern machine learning, yet their theoretical properties in the Bayesian framework remain largely unexplored. In this paper, we study Bayesian mixture-of-experts models, focusing on the ubiquitous softmax-based gating mechanism. Specifically, we investigate the asymptotic behavior of the posterior distribution for three fundamental statistical tasks: density estimation, parameter estimation, and model selection. First, we establish posterior contraction rates for density estimation, both in the regimes with a fixed, known number of experts and with a random learnable number of experts. We then analyze parameter estimation and derive convergence guarantees based on tailored Voronoi-type losses, which account for the complex identifiability structure of mixture-of-experts models. Finally, we propose and analyze two complementary strategies for selecting the number of experts. Taken together, these results provide one of the first systematic theoretical analyses of Bayesian mixture-of-experts models with softmax gating, and yield several theory-grounded insights for practical model design.

Nicola Bariletto, Nhat Ho

Training machine learning and statistical models often involves optimizing a data-driven risk criterion. The risk is usually computed with respect to the empirical data distribution, but this may result in poor and unstable out-of-sample performance due to distributional uncertainty. In the spirit of distributionally robust optimization, we propose a novel robust criterion by combining insights from Bayesian nonparametric (i.e., Dirichlet process) theory and a recent decision-theoretic model of smooth ambiguity-averse preferences. First, we highlight novel connections with standard regularized empirical risk minimization techniques, among which Ridge and LASSO regressions. Then, we theoretically demonstrate the existence of favorable finite-sample and asymptotic statistical guarantees on the performance of the robust optimization procedure. For practical implementation, we propose and study tractable approximations of the criterion based on well-known Dirichlet process representations. We also show that the smoothness of the criterion naturally leads to standard gradient-based numerical optimization. Finally, we provide insights into the workings of our method by applying it to a variety of tasks based on simulated and real datasets.

Nicola Bariletto, Khai Nguyen, Nhat Ho

We develop an analytical synthesis that bridges data-driven Distributionally Robust Optimization (DRO) and Economic Decision Theory under Ambiguity (DTA). By reinterpreting standard regularization and DRO techniques as data-driven counterparts of ambiguity-averse decision models, we provide a unified framework that clarifies their intrinsic connections. Building on this synthesis, we propose a novel DRO approach that leverages a popular DTA model of smooth ambiguity-averse preferences together with tools from Bayesian nonparametric statistics. Our baseline framework employs Dirichlet Process (DP) posteriors, which naturally extend to heterogeneous data sources via Hierarchical Dirichlet Processes (HDPs), and can be further refined to induce outlier robustness through a procedure that selectively filters poorly-fitting observations during training. Theoretical performance guarantees and convergence results, together with extensive simulations and real-data experiments, illustrate the method's favorable performance in terms of prediction accuracy and stability.