Comparator-Uniform Dynamic Regret under Heavy-Tailed Noise

Existing high-probability guarantees for online convex optimization with heavy-tailed gradients fix the comparator before the interaction, may pay an adaptation logarithm in the horizon, and require the horizon as an input. We address these three issues on a bounded convex set with adaptively chosen, non-clairvoyant convex Lipschitz losses. The learner receives one stochastic gradient per round, and the noise is assumed only to have a bounded conditional $p$-th central moment for $p\in (1,2]$, so its variance may be infinite. Our algorithm aggregates clipped online-gradient-descent experts on a grid of learning rates anchored at the static statistical scale. Each expert uses a threshold tied to its step size. A per-expert bias offset preserves the supermartingale property of the aggregate wealth, while an auxiliary descent sequence used only in the analysis handles the noise residuals. This sequence turns the comparator-dependent noise term into a martingale along a predictable direction without requiring an additional sample. Consequently, one event of probability $1-\delta$ covers every prefix and every comparator path, including paths chosen from the complete feedback record. The resulting regret combines a square-root term in the horizon and path-confidence complexity with a heavy-tail term scaling as $T^{1/p}$ and the same complexity to exponent $(p-1)/p$. The bound contains no additional horizon, dimension, or path-adaptation logarithm. A doubling wrapper gives an anytime version with an iterated-logarithm cost, while the noiseless case recovers the deterministic path-length rate. Self-contained lower bounds match the heavy-tail exponents in path length and confidence and identify the remaining gap in the deterministic term.