Parameter-Free Interval-Dynamic Regret under Heavy-Tailed Noise

We study online convex optimization with one unbiased stochastic subgradient per round and an unknown finite conditional pth noise moment, 1<p≤2. For every fixed interval I of length n and comparator path with LambdaI=1+PI/D, one learner achieves [ E[RegretI(u)]≤min(GDn, C[GD√n(LambdaI+log2(2T)) +σ Dn1/p(LambdaI+log2(2T))(p-1)/p]). ] The learner uses none of G,σ,p,I,PI, and the constant is universal. Interval adaptation adds to comparator complexity, preserving the distinct mean-gradient and noise exponents. The analysis controls calibration in expectation and limits the cost of observation-scale changes. Its general theorem compares to distributions over predictably available experts with relative-entropy dependence on a nonuniform prior. A common prior favors long windows and long restart lengths. With the statistics supplied, the interval cost becomes 1+log(T/n), including the optimal full-horizon static rate. A change-of-measure lower bound identifies the noise power of this logarithm for learners retaining a full-horizon optimal guarantee, under explicit conditions. Static comparisons and deterministic partitions follow from the same decisions.