We present new algorithms for online convex optimization over unbounded domains that obtain parameter-free regret in high-probability given access only to potentially heavy-tailed subgradient estimates. Previous work in unbounded domains considers only in-expectation results for sub-exponential subgradients. Unlike in the bounded domain case, we cannot rely on straight-forward martingale concentration due to exponentially large iterates produced by the algorithm. 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 for some $\mathfrak{p} \in (1, 2]$.

翻译：我们提出了在无界域上进行在线凸优化的新算法，这些算法在仅能访问潜在重尾次梯度估计时，能以高概率实现无参数遗憾。先前在无界域上的工作仅考虑了次指数次梯度的期望结果。与有界域情况不同，由于算法会产生指数级增长的迭代值，我们无法依赖直接的鞅集中性。为克服这些问题，我们开发了新的正则化技术。总体而言，以最多$\delta$的概率，对于所有比较器$\mathbf{u}$，我们的算法在次梯度具有有界$\mathfrak{p}^{th}$阶矩（其中$\mathfrak{p} \in (1, 2]$）时实现了遗憾值$\tilde{O}(\| \mathbf{u} \| T^{1/\mathfrak{p}} \log (1/\delta))$。