投影自由在线凸优化的尖锐Oracle-遗憾权衡

中文摘要 AI 辅助

我们刻画了当可行集的访问仅限于精确线性优化Oracle时，在线凸优化中可达到的遗憾。学习器被给予一个内切球和直径界限，并且必须在每个一致实例上保持可行。对于凸的$G$-Lipschitz损失，直径至多$D$，总调用次数配额为$Q$，每轮严格限制为$B$次调用，无维度最小最大期望遗憾为$\Theta(GD\max\{\sqrt T,T/(1+\min\{Q,BT\})^{1/4}\})$。下界适用于任意随机化学习器。通用可行性首先迫使每个动作进入所给球体与先前Oracle回复的凸包中。然后，一个固定体构造将新鲜阶段方向耦合到共享单纯形上，使得即使所有损失具有共同的最小值点，有用的回复也反复代价高昂。一种带有交错块的计数近似梯度方法达到了匹配的速率。总预算和每轮严格保证作为特殊情况出现，包括每轮一次调用时的$T^{3/4}$速率，以及实现$\sqrt T$遗憾所需的二次总预算。对于指定的光滑度$\beta$，一种解析构造产生了依赖于曲率的下界，并确定了通用刻画保持尖锐的阈值。

英文摘要

We characterize the regret attainable in online convex optimization when access to the feasible set is limited to an exact linear optimization oracle. The learner is given an inscribed ball and a diameter bound and must remain feasible on every consistent instance. For convex $G$-Lipschitz losses, diameter at most $D$, a total allowance of $Q$ oracle calls, and a strict limit of $B$ calls per round, the dimension-free minimax expected regret is $Θ(GD\max\{\sqrt T,T/(1+\min\{Q,BT\})^{1/4}\})$. The lower bound applies to arbitrary randomized learners. Universal feasibility first forces each action into the hull of the supplied ball and the preceding oracle replies. A fixed-body construction then couples fresh phase directions to a shared simplex, making useful replies costly repeatedly even though all losses have a common minimizer. A counted approximate-gradient method with interleaved blocks attains the matching rate. Total-budget and strict per-round guarantees follow as special cases, including the $T^{3/4}$ rate with one call per round and the quadratic total budget needed for $\sqrt T$ regret. For prescribed smoothness $β$, an analytic construction yields a curvature-dependent lower bound and identifies the threshold above which the general characterization remains sharp.

发表机构

• Purdue University（普渡大学）

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