凸-凹极小极大优化的近最优高阶预言复杂度

中文摘要 AI 辅助

对于光滑的凸-凹极小极大优化，Chen等人（2026）的高阶下界适用于具有规定正则化泰勒模型更新的受限张量算法类。我们为任意自适应确定性和随机化算法建立了相同的下界，在对数因子范围内匹配Zhang等人（2026）的上界。固定整数$p\ge 2$，设$L_p>0$为直径至多$D_Z>0$的紧凸乘积域上目标函数$p$阶导数的Lipschitz常数上界。每次可行查询返回目标值及直至$p$阶的所有导数。对于精度$\epsilon>0$，设$Q_{\mathrm{tan}}=L_pD_Z^p/\epsilon$（切向残差）和$Q_{\mathrm{gap}}=L_pD_Z^{p+1}/\epsilon$（鞍点间隙）。设$T_E^{\mathrm{det}}(\epsilon)$和$T_E^{\mathrm{rand}}(\epsilon)$为准则$E\in\{\mathrm{tan},\mathrm{gap}\}$的高维极小极大查询复杂度，其中随机化算法在每个实例上的成功概率至少为$2/3$。我们的下界和现有上界给出：对于足够大的$Q_E$，$c_pQ_E^{2/(3p-1)}\le T_E^{\mathrm{rand}}(\epsilon)\le T_E^{\mathrm{det}}(\epsilon)\le C_pQ_E^{2/(3p-1)}[1+\log(3+Q_E)]^{6(p-1)}$，其中$c_p,C_p>0$仅依赖于$p$。因此，相同的精度指数适用于超越张量更新规则的情形，甚至适用于随机化查询和任意可行输出。证明构造了一个标量凸-凹链，其具有完全平坦的门以隐藏完整的导数信息。直接乘积域误差见证和自适应转录论证建立了两种准则的下界。

英文摘要

For smooth convex--concave minimax optimization, the higher-order lower bound of Chen et al. (2026) applies to a restricted tensor-algorithm class with prescribed regularized Taylor-model updates. We establish the same bound for arbitrary adaptive deterministic and randomized algorithms, matching, up to logarithmic factors, the upper bound of Zhang et al. (2026). Fix an integer $p\ge 2$ and let $L_p>0$ bound the Lipschitz constant of the objective's $p$-th derivative on a compact convex product domain of diameter at most $D_Z>0$. Each feasible query returns the objective value and all derivatives through order $p$. For accuracy $ε>0$, set $Q_{\mathrm{tan}}=L_pD_Z^p/ε$ for tangent residual and $Q_{\mathrm{gap}}=L_pD_Z^{p+1}/ε$ for saddle gap. Let $T_E^{\mathrm{det}}(ε)$ and $T_E^{\mathrm{rand}}(ε)$ denote the high-dimensional minimax query complexities for criterion $E\in\{\mathrm{tan},\mathrm{gap}\}$, with randomized success probability at least $2/3$ on every instance. Our lower bounds and the existing upper bound give $c_pQ_E^{2/(3p-1)}\le T_E^{\mathrm{rand}}(ε)\le T_E^{\mathrm{det}}(ε)\le C_pQ_E^{2/(3p-1)}[1+\log(3+Q_E)]^{6(p-1)}$ for sufficiently large $Q_E$, where $c_p,C_p>0$ depend only on $p$. Thus the same accuracy exponent holds beyond tensor update rules, even for randomized queries and arbitrary feasible outputs. The proof constructs a scalar convex--concave chain with exactly flat gates that hide complete derivative information. Direct product-domain error witnesses and adaptive transcript arguments establish the lower bounds for both criteria.

发表机构

• Peking University（北京大学）

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