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We present new algorithms for online convex optimization over unboundeddomains that obtain parameter-free regret in high-probability given access onlyto potentially heavy-tailed subgradient estimates. Previous work in unboundeddomains considers only in-expectation results for sub-exponential subgradients.Unlike in the bounded domain case, we cannot rely on straight-forwardmartingale concentration due to exponentially large iterates produced by thealgorithm. We develop new regularization techniques to overcome these problems.Overall, with probability at most $\delta$, for all comparators $\mathbf{u}$our algorithm achieves regret $\tilde{O}(\| \mathbf{u} \| T^{1/\mathfrak{p}}\log (1/\delta))$ for subgradients with bounded $\mathfrak{p}^{th}$ moments forsome $\mathfrak{p} \in (1, 2]$.

Parameter-free Regret in High Probability with Heavy Tails