Online Distributional Prediction via Latent Cluster Geometry Under Drift and Corruption

Abstract

Online learning in non-stationary streams is often formulated as tracking a point estimate, although many applications require predicting the full data-generating distribution. We study online distributional prediction under drift and bounded causal adversarial corruption. Candidate laws are represented by variable-size latent cluster configurations that induce predictive distributions. A Gibbs quasi-posterior over configurations yields an online predictor by posterior averaging and can be sampled using reversible-jump MCMC. The method uses no stream likelihood but specifies a kernel-mixture predictive family. We evaluate cumulative Wasserstein-1 regret against the time-varying true law and study periodic restart, sliding-window, and exponential-discounting memory against full history. The bounds separate PAC-Bayesian complexity, corruption-sensitive posterior perturbation, and drift through $A_T^{\mathrm{OT}}=\sum_{t=2}^T W_2^2(p_{t-1}^*,p_t^*)$. Under bounded support, latent quadratic growth, predictive-map regularity, and localized-comparator assumptions, restart has sublinear high-probability cumulative regret when $0\le a<1$, $q,\zeta,b<1$, $0\le\gamma<\beta<1$, and $0<h<(1-a)/2$. Here $a,q,\zeta,b$ control the polynomial growth of drift, predictive-family approximation error, off-order posterior mass, and oracle-order switch cost; $\lambda_s\asymp s^{-\beta}$ and $H\asymp T^h$. Corruptions satisfy the segment-local condition $\Lambda_{r,s}=O(s^\gamma)$, so a sublinear global budget alone is insufficient. The regime requires oracle-informed tuning, $\varepsilon_T=o(1)$, and fixed confidence or $\log(1/\delta_0)=T^{o(1)}$. Sliding-window and discounted guarantees additionally require the stated localized-memory condition.