发表机构
NEC Corporation（日本电气株式会社）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出小梯度跳过机制，应用于在线逆整数线性优化，在均匀分离条件下跳过无错误轮次的更新，实现常数遗憾和有限错误，并去除 $\log T$ 因子。
AI 中文摘要
在在线逆线性优化中，学习者在每一轮预测一个权重，观察智能体的最优行动，并更新其预测。在一般设置中，遗憾上界 $O(d \log T)$ 与下界 $Omega(d)$ 之间的 $\log T$ 差距尚未解决（这里 $T$ 是总轮数，$d$ 是维度）。当行动集是 M-凸时，遗憾已知被 $O(d \log d)$ 所界定，但实现该界的方法在每一轮都计算重心。因此，本文提出了小梯度跳过（SGS），一种在正确行动与其他候选行动均匀分离的情况下，在没有错误的轮次跳过更新的机制，并将其应用于在线梯度下降、在线牛顿步和 MetaGrad。对于这三种方法，错误次数被一个独立于 $T$ 的量所界定；对于在线牛顿步和带 SGS 的 MetaGrad，当正向问题是整数线性规划时，遗憾的维度依赖性变为 $O(d^2)$，即去除了 $\log T$ 因子。此外，当行动集是 M-凸时，遗憾可以在不计算重心的情况下被高效界定。
英文摘要
In online inverse linear optimization, the learner predicts a weight at each round, observes the optimal action of the agent, and updates its prediction. In the general setting, the gap of $\log T$ between the regret upper bound $O(d \log T)$ and the lower bound $Ω(d)$ is unresolved (here $T$ is the total number of rounds and $d$ is the dimension). When the action set is M-convex, the regret is known to be bounded by $O(d \log d)$, but the method attaining it computes a center of gravity at every round. This paper therefore proposes Small-Gradient Skipping (SGS), a mechanism that skips the update at rounds without a mistake, and applies it to online gradient descent, the online Newton step, and MetaGrad. When the forward problem is an integer linear program with a unique optimal solution, the number of mistakes is bounded, for all three, by a quantity independent of $T$; and for the online Newton step and for MetaGrad with SGS, the dimension dependence of the regret becomes $O(d^2)$, that is, the factor $\log T$ is removed. Moreover, when the action set is M-convex, the regret is bounded efficiently without computing a center of gravity.
Comments56 pages
