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一个或多个有界性假设：域有界、损失是Lipschitz连续的，或两者兼具。在本文中，我们针对无界域和非Lipschitz损失的情景提出了一种新的在线学习设定。在该设定下，我们给出了一种算法，对于任何满足次梯度条件$\|g_{t}\|\le G+L\|w_{t}\|$的问题，保证其遗憾界为$R_{T}(u)\le \tilde O(G\|u\|\sqrt{T}+L\|u\|^{2}\sqrt{T})$，并证明该界在没有进一步假设的情况下不可改进。我们利用该算法开发了新的鞍点优化算法，使其在无界域中即便缺乏有效曲率时也能在对偶间隙上收敛。最后，我们提供了首个在无界域中针对非Lipschitz损失实现非平凡动态遗憾的算法，并给出了匹配的下界。当损失函数光滑时，该动态遗憾算法的遗憾自动提升至一种新颖的$L^{*}$界。