光滑凸优化的近最优确定性精确值复杂度

中文摘要 AI 辅助

我们研究了当算法仅接收精确函数值时，光滑凸优化的确定性预言机复杂度。目标函数是全局$\beta$-光滑凸函数，所有查询和最终输出限制在半径为$R$的欧几里得球内，唯一极小点位于半径为$R/2$的球内。我们利用坐标有限差分结合误差鲁棒的加速投影方法，建立了$O(d\sqrt{\beta R^2/\epsilon})$的上界。我们的主要贡献是匹配的下界，直至构造的高精度饱和：任何确定性自适应值预言机算法需要$\Omega\\!\left(d\min\{\sqrt{\beta R^2/\epsilon},(d/\log(ed))^{1/3}\}\right)$次查询。因此，在中等精度区间$\beta R^2(\log(ed)/d)^{2/3}\leq\epsilon\leq c\beta R^2$（其中$c>0$为通用常数）内，极小极大预言机复杂度为$\Theta(d\sqrt{\beta R^2/\epsilon})$。下界必须考虑单个精确实数值可以编码任意多信息的事实。为克服这一困难，我们使用Moreau光滑的偏置最大链、精确前缀屏蔽机制和批量延迟旋转构造了一个固定的光滑凸困难实例。这些技术保持了与完整自适应记录的一致性，并为确定性有界查询算法建立了平方根复杂度分支的最优性。

英文摘要

We study the deterministic oracle complexity of smooth convex optimization when the algorithm receives only exact function values. The objective is a globally $β$-smooth convex function, all queries and the final output are restricted to the Euclidean ball of radius $R$, and the unique minimizer lies in the ball of radius $R/2$. We establish an upper bound of $O(d\sqrt{βR^2/ε})$ using coordinate finite differences together with an error-robust accelerated projected method. Our main contribution is a matching lower bound, up to the high-accuracy saturation of the construction: any deterministic adaptive value-oracle algorithm requires $Ω\!\left(d\min\{\sqrt{βR^2/ε},(d/\log(ed))^{1/3}\}\right)$ queries. Consequently, the minimax oracle complexity is $Θ(d\sqrt{βR^2/ε})$ throughout the moderate-accuracy regime $βR^2(\log(ed)/d)^{2/3}\leqε\leq cβR^2$ for a universal constant $c>0$. The lower bound must account for the fact that a single exact real value can encode arbitrarily much information. To overcome this difficulty, we construct a single fixed smooth convex hard instance using a Moreau-smoothed biased max chain, an exact prefix-shielding mechanism, and batched delayed rotations. These techniques preserve consistency with the full adaptive transcript and establish the optimality of the square-root complexity branch for deterministic bounded-query algorithms.

发表机构

• Peking University（北京大学）

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