高阶导数不会加速不动点的计算
Higher-Order Derivatives Do Not Accelerate the Computation of Fixed Points
AI总结：
研究光滑压缩不动点计算中，查询导数的高阶方法能否加速。通过分析发现有限阶导数信息不能加速其最坏情况复杂度，与高阶光滑最小化不同，后者高阶导数可提高最坏情况速率。
摘要：
皮卡德迭代以线性速率\(q^N\)收敛到\(q\)-压缩算子的唯一不动点，仿射构造的下界表明仅查询算子值的确定性方法无法做得更好。但查询导数的高阶方法呢？一次雅可比矩阵求值就能完全揭示一个仿射映射，所以仿射构造对高阶方法没有说明。在这项工作中，我们表明有限阶导数信息仍不会加速光滑压缩不动点计算的最坏情况复杂度。这与高阶光滑最小化形成对比，在高阶光滑最小化中，高阶导数确实能提高凸和非凸最小化的最坏情况速率。
英文摘要：
The Picard iteration converges to the unique fixed point of a $q$-contractive operator at a linear rate $q^N$, and a lower bound with an affine construction shows that no deterministic method querying only operator values can do better. But what about higher-order methods that query derivatives? A single Jacobian evaluation reveals an affine map entirely, so the affine construction says nothing about higher-order methods. In this work, we show that finite-order derivative information still does not accelerate the worst-case complexity for smooth contractive fixed-point computation. This contrasts with higher-order smooth minimization, where higher-order derivatives do improve worst-case rates for convex and non-convex minimization.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
