光滑单调变分不等式的高阶预言机复杂度匹配

中文摘要 AI 辅助

我们为光滑单调变分不等式建立了近最优的高阶预言机界。对于固定的 $p\ge2$，设 $F$ 在直径至多为 $D$ 的已知紧凸集 $X$ 上单调，且满足 $\operatorname{Lip}(D^{p-1}F)\le L_p$。每次可行查询返回完整的喷射 $(F,DF,\ldots,D^{p-1}F)$，目标是找到 $x$ 使得切向残差 $\operatorname{dist}(0,F(x)+N_X(x))\le\varepsilon$。记 $Q=L_pD^p/\varepsilon$，我们通过一个维度无关的确定性算法，将 Chen 等人的 $\widetilde O_p(Q^{1/p})$ 上界改进为 $\widetilde O_p(Q^{2/(3p-1)})$，该算法返回显式的切向残差证书。我们证明了匹配的 $\Omega_p(Q^{2/(3p-1)})$ 下界，适用于任意自适应确定性算法以及每次实例成功概率至少为 $2/3$ 的随机算法，且不限制跨度或张量更新。因此，高维最坏情况预言机复杂度为 $\widetilde\Theta_p((L_pD^p/\varepsilon)^{2/(3p-1)})$。同样的方法适用于光滑凸-凹极小极大问题，将固定几何精度指数从 $4/(3p+1)$ 改进为 $2/(3p-1)$，并匹配已知的下界指数。

英文摘要

We establish near-optimal higher-order oracle bounds for smooth monotone variational inequalities. For fixed $p\ge2$, let $F$ be monotone on a known compact convex set $X$ of diameter at most $D$, with $\operatorname{Lip}(D^{p-1}F)\le L_p$. Each feasible query returns the complete jet $(F,DF,\ldots,D^{p-1}F)$, and the goal is to find $x$ with tangent residual $\operatorname{dist}(0,F(x)+N_X(x))\le\varepsilon$. Writing $Q=L_pD^p/\varepsilon$, we improve the $\widetilde O_p(Q^{1/p})$ upper bound of Chen et al. to $\widetilde O_p(Q^{2/(3p-1)})$ via a dimension-independent deterministic algorithm that returns an explicit tangent-residual certificate. We prove a matching $Ω_p(Q^{2/(3p-1)})$ lower bound for arbitrary adaptive deterministic algorithms and randomized algorithms with per-instance success probability at least $2/3$, without span or tensor-update restrictions. Hence the high-dimensional worst-case oracle complexity is $\widetildeΘ_p((L_pD^p/\varepsilon)^{2/(3p-1)})$. The same method applies to smooth convex--concave minimax problems, improving the fixed-geometry accuracy exponent from $4/(3p+1)$ to $2/(3p-1)$ and matching the known lower-bound exponent.

发表机构

• Peking University（北京大学）

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