We study online convex optimization with one unbiased stochastic subgradient per round and an unknown finite conditional $p$th noise moment, $1
For every fixed interval $I$ of length $n$ and comparator path with $Λ_I=1+P_I/D$, one learner achieves \[ E[Regret_I(u)]\le\min(GDn, C[GD\sqrt{n(Λ_I+\log^2(2T))} +σDn^{1/p}(Λ_I+\log^2(2T))^{(p-1)/p}]). \]
The learner uses none of $G,σ,p,I,P_I$, 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.