Online structured prediction is a task of sequentially predicting outputs with complex structures based on inputs and past observations, encompassing online classification. Recent studies showed that in the full-information setting, we can achieve finite bounds on the \textit{surrogate regret}, \textit{i.e.,}~the extra target loss relative to the best possible surrogate loss. In practice, however, full-information feedback is often unrealistic as it requires immediate access to the whole structure of complex outputs. Motivated by this, we propose algorithms that work with less demanding feedback, \textit{bandit} and \textit{delayed} feedback. For bandit feedback, by using a standard inverse-weighted gradient estimator, we achieve a surrogate regret bound of $O(\sqrt{KT})$ for the time horizon $T$ and the size of the output set $K$. However, $K$ can be extremely large when outputs are highly complex, resulting in an undesirable bound. To address this issue, we propose another algorithm that achieves a surrogate regret bound of $O(T^{2/3})$, which is independent of $K$. This is achieved with a carefully designed pseudo-inverse matrix estimator. Furthermore, we numerically compare the performance of these algorithms, as well as existing ones. Regarding delayed feedback, we provide algorithms and regret analyses that cover various scenarios, including full-information and bandit feedback, as well as fixed and variable delays.

翻译：在线结构化预测是一项基于输入与历史观测结果、顺序预测具有复杂结构输出的任务，其涵盖在线分类问题。近期研究表明，在完全信息设定下，我们能够获得关于\textit{代理遗憾}（即相对于最优可能代理损失的额外目标损失）的有限界。然而在实际应用中，完全信息反馈往往不切实际，因其要求即时获取复杂输出的完整结构。受此启发，我们提出了适用于需求更低的反馈机制——\textit{bandit}反馈与\textit{延迟}反馈的算法。针对bandit反馈，通过使用标准的逆加权梯度估计器，我们在时间范围$T$与输出集规模$K$下实现了$O(\sqrt{KT})$的代理遗憾界。但当输出结构高度复杂时，$K$值可能极大，导致遗憾界不理想。为解决此问题，我们提出另一种算法，通过精心设计的伪逆矩阵估计器实现了与$K$无关的$O(T^{2/3})$代理遗憾界。此外，我们对这些算法及现有算法进行了数值性能比较。关于延迟反馈，我们提供了涵盖多种场景（包括完全信息与bandit反馈，以及固定延迟与可变延迟）的算法设计与遗憾分析。