AI 中文总结
研究近端点法的安德森加速中的问题，精确回答了自适应方法极小极大复杂度、谱相变及最优保护相关的三个问题，修正完善结构化问题理论，通过数值验证预测，给出了核心方法及主要贡献。
AI 中文摘要
我们研究了用于极大单调包含的近端点法（PPM）的残差多项式加速，以安德森加速（AA）作为典型的自适应方案。我们精确回答了三个问题。（i）所有自适应方法上的极小极大复杂度精确为每次\(K\)次预解式评估\(d_0/(K + 1)\)。上界由平均反射估计器达到；匹配的下界使用一个明确的斜自伴实例，其预解式特征值在\(u^{K + 1} = -1\)的根处且具有\(\csc^2\)分布的质量，在此每个\(K\)次多项式方法满足\(\|r(y_K)\|\geq 1/(K + 1)\)。最优多项式唯一地是费耶尔核，且同一实例证明了每步下限。（ii）一个尖锐的相变分隔了不同 regime：当谱下限\(s\)满足\(sK\to\infty\)时，杰克逊核多项式达到\(O(d_0/(K^2 s))\)，而在临界尺度\(s\asymp 1/K\)时，障碍恰好是\(1/(K + 1)\)。该图景扩展到正规算子和非线性族\(M = S + N_C\)。（iii）在线性问题上，AA - PPM不需要保护；在非线性问题上，\(O(1/k)\)包络的证明需要每次迭代恰好两次预言机评估，且这个因子是最优的。我们还修正并完善了结构化问题——仿射、强单调、分段仿射和赫尔德增长——的理论，并通过数值验证了所有预测。
英文摘要
\noindent We study residual-polynomial acceleration of the proximal point method (PPM) for maximal monotone inclusions, with Anderson acceleration (AA) as the prototypical adaptive scheme. We answer three questions exactly. (i)~The minimax complexity over all adaptive methods is precisely $d_0/(K+1)$ per $K$ resolvent evaluations. The upper bound is attained by the averaged-reflection estimator; the matching lower bound uses an explicit skew-adjoint instance with resolvent eigenvalues at the roots of $u^{K+1}=-1$ and $\csc^2$-distributed masses, on which every degree-$K$ polynomial method satisfies $\|r(y_K)\|\ge 1/(K+1)$. The optimal polynomial is uniquely the Fejér kernel, and the same instance certifies a per-step floor. (ii)~A sharp phase transition separates regimes: Jackson-kernel polynomials achieve $O(d_0/(K^2 s))$ when the spectral floor $s$ satisfies $sK\to\infty$, while at the critical scale $s\asymp 1/K$ the barrier is exactly $1/(K+1)$. The picture extends to normal operators and the nonlinear family $M=S+N_C$. (iii)~On linear problems AA-PPM needs no safeguarding; on nonlinear problems certification of the $O(1/k)$ envelope requires exactly two oracle evaluations per iteration, and this factor is optimal. We also correct and complete the theory for structured problems---affine, strongly monotone, piecewise-affine, and Hölderian growth---and confirm all predictions numerically.
Comments26 pages, 3 figures
