在线凸优化中带指示器切换成本的动态遗憾
Dynamic Regret in Online Convex Optimization with Indicator Switching Costs
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中文总结 AI 辅助
针对带指示器切换成本的在线凸优化，提出元学习框架，实现动态遗憾的极小极大最优界限，无需先验知识。
中文摘要 AI 辅助
我们研究带有指示器切换成本的在线凸优化中的动态遗憾：每当两个连续决策不同时，就会产生一个固定惩罚。这捕捉了诸如服务器激活、模型部署和缓存更新等启动开销，并且在有界域上，它作为特殊情况恢复了基于范数的移动成本。现有的针对指示器成本的保证仅处理静态比较器。我们证明，将这些技术直接扩展到动态遗憾被证明是失败的，这促使我们采用不同的方法。我们提出一个元学习框架：一组随机化的懒惰FTRL基学习器在二进时间尺度上重启，由一个移动感知的主学习器聚合，该主学习器混合它们的提议密度，并通过最大耦合连续混合来采样动作。所得到的算法在期望上满足 $\mathcal{R}^{\mathbf{1}}_T \le \tilde{\mathcal{O}}(\min\{\sqrt{T(S_T{+}1)},T^{2/3}(P_T+1)^{1/3}\})$，其中 $\mathcal{R}^{\mathbf{1}}_T$ 是动态遗憾加上累积指示器切换成本，$S_T$ 统计比较器切换次数，$P_T$ 是比较器路径长度。该界限对所有序列同时成立，并且不需要事先知道 $S_T$ 或 $P_T$：对于跟踪分段常数比较器，它是极小极大最优的（直到对数因子），并且也捕获了总路径长度小的频繁移动比较器。
英文摘要
We study dynamic regret in online convex optimization with an \emph{indicator switching cost}: a fixed penalty incurred whenever two consecutive decisions differ. This captures startup overheads such as server activation, model deployment, and cache updates, and on a bounded domain it recovers norm-based movement costs as a special case. Existing guarantees for indicator costs handle only static comparators. We show that a direct extension of these techniques to dynamic regret provably fails, motivating a different approach. We propose a meta-learning framework: a set of randomized lazy FTRL base learners restarted at dyadic time scales, aggregated by a movement-aware master that mixes their proposal densities and samples actions via maximal coupling of consecutive mixtures. The resulting algorithm satisfies, in expectation, $\mathcal{R}^{\mathbf{1}}_T \le \tilde{\mathcal{O}}(\min\{\sqrt{T(S_T{+}1)},T^{2/3}(P_T+1)^{1/3}\})$, where $\mathcal{R}^{\mathbf{1}}_T$ is the dynamic regret plus the cumulative indicator switching cost, $S_T$ counts comparator switches, and $P_T$ is the comparator path length. The bound holds simultaneously for all sequences and requires no prior knowledge of $S_T$ or $P_T$: it is minimax-optimal (up to logarithmic factors) for tracking piecewise-constant comparators, and also captures frequently moving comparators with small total path length.
发表机构
- Faculty of Electrical Engineering, Mathematics and Computer Science（电气工程、数学与计算机科学学院）
- TU Delft（代尔夫特理工大学）
机构由 AI 辅助整理，请以论文原文为准。
