Algorithms for online learning typically require one or more boundedness assumptions: that the domain is bounded, that the losses are Lipschitz, or both. In this paper, we develop a new setting for online learning with unbounded domains and non-Lipschitz losses. For this setting we provide an algorithm which guarantees $R_{T}(u)\le \tilde O(G\|u\|\sqrt{T}+L\|u\|^{2}\sqrt{T})$ regret on any problem where the subgradients satisfy $\|g_{t}\|\le G+L\|w_{t}\|$, and show that this bound is unimprovable without further assumptions. We leverage this algorithm to develop new saddle-point optimization algorithms that converge in duality gap in unbounded domains, even in the absence of meaningful curvature. Finally, we provide the first algorithm achieving non-trivial dynamic regret in an unbounded domain for non-Lipschitz losses, as well as a matching lower bound. The regret of our dynamic regret algorithm automatically improves to a novel $L^{*}$ bound when the losses are smooth.

翻译：在线学习算法通常需要若干有界性假设：定义域有界、损失函数Lipschitz连续，或两者兼具。本文针对无界定义域和非Lipschitz损失函数，提出了一种新的在线学习框架。在此框架下，我们设计了一个算法，对于次梯度满足$\|g_{t}\|\le G+L\|w_{t}\|$的任意问题，可保证遗憾值$R_{T}(u)\le \tilde O(G\|u\|\sqrt{T}+L\|u\|^{2}\sqrt{T})$，并证明该界在无额外假设条件下不可改进。我们利用该算法开发了新的鞍点优化算法，使其能在无界定义域中收敛对偶间隙，即便在缺乏有效曲率的情况下。最后，我们首次给出了非Lipschitz损失下无界定义域中实现非平凡动态遗憾的算法，并提出了匹配的下界。当损失函数光滑时，该动态遗憾算法的遗憾值会自动改进为全新的$L^{*}$界。