$\u2113_p$几何中最优的无参数梯度最小化
Optimal Parameter-Free Gradient Minimization in $\ell_p$ Geometry
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中文总结 AI 辅助
本研究解决了$\u2113_p$几何下无参数梯度最小化的一阶预言机复杂度问题，在非退化割线初始化下无需已知光滑度、距离等参数即可得到满足梯度精度要求的点，并给出了不同$p$值对应的复杂度界。
中文摘要 AI 辅助
我们研究了在$\u2113_p$几何中寻找具有小梯度的查询点的一阶预言机复杂度，特别关注适应未知光滑度和距离尺度所需的信息。在严格计数的局部值-梯度模型中，若无非退化局部尺度观测，不存在仅依赖于$LR/\u03b5$的有限复杂度界：一个一维构造保持$LR/\u03b5=4$，同时能突破任何预设的有限查询预算。\n我们解决了Diakonikolas针对每个固定$1<p<\u221e$的通用$\u2113_p$无参数扩展问题。在非退化割线初始化下，该方法既不知道光滑度常数$L$、初始解距离$R$，也不知道$f^*$，并返回一个查询点$\u0302x$，满足$\u2016\u2207f(\u0302x)\u2016_q\u2264\u03b5$。对于固定的有限$p>2$，我们首先建立了$K=LR/\u03b5$中无维度的确定性已知参数上指数$p/(p+2)$，与已发表的在其范围和维度限定下的多项式下界指数相匹配。有限局部例程采用相同的可观测尺度-半径过程，因此在不知道$L$或$R$的情况下该指数得以保留。记$\u0305K=\u03bcax\{1,LR/\u03b5\}$，初始化后的双预言机复杂度在$1<p<2$时为$O_p(\u0305K^{1/2})$，在$p=2$时为$O(\u0305K^{1/2})$，在$p>2$时为$O_p(\u0305K^{p/(p+2)})$，且在所有情形下都带有加性校准代价$O_p(\u03c2og(e+L/M_0))$。
英文摘要
We study the first-order oracle complexity of finding a queried point with small gradient in $\ell_p$ geometry, with particular attention to the information needed to adapt the unknown smoothness and distance scales. In the strict counted local value--gradient model, no finite complexity bound can depend only on $LR/\eps$ without a nondegenerate local scale observation: a one-dimensional construction keeps $LR/\eps=4$ while defeating every prescribed finite query budget. We resolve Diakonikolas's general-$\ell_p$ parameter-free extension question for every fixed $1<p<\infty$. Under a nondegenerate secant initialization, the method knows neither the smoothness constant $L$, the initial solution distance $R$, nor $f^*$, and returns a queried point $\widehat x$ with $\|\nabla f(\widehat x)\|_q\le\eps$. For fixed finite $p>2$, we first establish the dimension-free deterministic known-parameter upper exponent $p/(p+2)$ in $K=LR/\eps$, matching the published lower polynomial exponent under its horizon and dimension qualifications. The finite local routine fits the same observable scale--radius procedure, so this exponent is preserved without knowing $L$ or $R$. Writing $\Kbar=\max\{1,LR/\eps\}$, the post-initialization pair-oracle complexity is $O_p(\Kbar^{1/2})$ for $1<p<2$, $O(\Kbar^{1/2})$ for $p=2$, and $O_p(\Kbar^{p/(p+2)})$ for $p>2$, together with the additive calibration cost $O_p(\log(e+L/M_0))$ in every regime.
