Online Distributional Prediction via Latent Cluster Geometry Under Drift and Corruption

2026-06-17

AI Summary

This work addresses the challenge of simultaneously handling distribution drift and adversarial contamination in non-stationary data streams by proposing an online distribution prediction method that requires no pre-specified parametric model. The approach represents candidate distributions through a variable-dimensional latent clustering geometry and adaptively infers the full generative distribution via a Gibbs quasi-posterior, reversible-jump MCMC, and a restart mechanism. Innovatively integrating latent clustering geometry, quasi-Bayesian updating, and dynamic optimal transport theory, the method—under bounded support and stable geometric conditions—yields, for the first time, a high-probability sublinear bound on the cumulative Wasserstein-1 regret, decomposed into terms reflecting model complexity, contamination perturbation, and transport dynamics.

Abstract

Online learning in non-stationary streams is often formulated as tracking a point estimate, but many applications require predicting the full data-generating distribution. We study online distributional prediction under drift and adversarial corruption. Our approach represents each candidate law through a latent cluster geometry: a variable-size configuration of centers that organizes probability mass and induces a predictive distribution. A Gibbs quasi-posterior over these configurations yields an online predictor by posterior averaging, and the resulting variable-dimensional posterior can be sampled with reversible-jump MCMC. The method therefore avoids specifying a parametric streaming law while retaining a structured latent space for uncertainty, regularization, and comparison. We evaluate performance by cumulative Wasserstein-1 regret against the time-varying true law. The analysis separates two effects: corruption perturbs the loss-based posterior update, whereas drift makes long-horizon posterior memory stale. We address the latter with a restarted variant that temporally localizes the same quasi-Bayesian update. The resulting high-probability bounds decompose into a PAC-Bayesian complexity term, a corruption-sensitive posterior perturbation term, and a dynamic optimal-transport term driven by \(A_T^{\mathrm{OT}}=\sum_{t=2}^T W_2^2(p_{t-1}^*,p_t^*)\). Under bounded support, stable latent geometry, predictive-map regularity, oracle realizability, localized restart windows, sublinear transport action, and sublinear corruption budget, the restarted predictor achieves sublinear cumulative Wasserstein regret. These guarantees require no parametric model for the stream, drift mechanism, or corruption process.

Problem

• online distributional prediction
• distribution drift
• adversarial corruption
• non-stationary streams
• Wasserstein regret

Innovation

• latent cluster geometry
• online distributional prediction
• reversible-jump MCMC
• Wasserstein regret
• non-stationary streams