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 $A_T^{\mathrm{OT}}={\sum }_{t=2}^T{W}_2^2(p_{t-1}^{\ast },p_t^{\ast })$. 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, ${\epsilon }_T=o(1)$, and fixed confidence or $\log (1/{\delta }_0)=T^{o(1)}$. Sliding-window and discounted guarantees additionally require the stated localized-memory condition.