Linear Asymptotic Convergence of Anderson Acceleration: Fixed-Point Analysis
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Bibliographic record
Abstract
.We study the asymptotic convergence of AA( \(m\) ), i.e., Anderson acceleration (AA) with window size \(m\) for accelerating fixed-point methods \(x_{k+1}=q(x_{k})\) , \(x_k \in \mathbb{R}^n\) . Convergence acceleration by AA( \(m\) ) has been widely observed but is not well understood. We consider the case where the fixed-point iteration function \(q(x)\) is differentiable and the convergence of the fixed-point method itself is root-linear. We identify numerically several conspicuous properties of AA( \(m\) ) convergence: First, AA( \(m\) ) sequences \(\{x_k\}\) converge root-linearly, but the root-linear convergence factor depends strongly on the initial condition. Second, the AA( \(m\) ) acceleration coefficients \(\boldsymbol{\beta }^{(k)}\) do not converge but oscillate as \(\{x_k\}\) converges to \(x^{*}\) . To shed light on these observations, we write the AA( \(m\) ) iteration as an augmented fixed-point iteration \(\boldsymbol{z}_{k+1} =\Psi (\boldsymbol{z}_k)\) , \(\boldsymbol{z}_k \in \mathbb{R}^{n(m+1)}\) , and analyze the continuity and differentiability properties of \(\Psi (\boldsymbol{z})\) and \(\boldsymbol{\beta }(\boldsymbol{z})\) . We find that the vector of acceleration coefficients \(\boldsymbol{\beta }(\boldsymbol{z})\) is not continuous at the fixed point \(\boldsymbol{z}^{*}\) . However, we show that, despite the discontinuity of \(\boldsymbol{\beta }(\boldsymbol{z})\) , the iteration function \(\Psi (\boldsymbol{z})\) is Lipschitz continuous and directionally differentiable at \(\boldsymbol{z}^{*}\) for AA(1), and we generalize this to AA( \(m\) ) with \(m\gt 1\) for most cases. Furthermore, we find that \(\Psi (\boldsymbol{z})\) is not differentiable at \(\boldsymbol{z}^{*}\) . We then discuss how these theoretical findings relate to the observed convergence behavior of AA( \(m\) ). The discontinuity of \(\boldsymbol{\beta }(\boldsymbol{z})\) at \(\boldsymbol{z}^{*}\) allows \(\boldsymbol{\beta }^{(k)}\) to oscillate as \(\{x_k\}\) converges to \(x^{*}\) , and the nondifferentiability of \(\Psi (\boldsymbol{z})\) allows AA( \(m\) ) sequences to converge with root-linear convergence factors that strongly depend on the initial condition. Additional numerical results illustrate our findings for several linear and nonlinear fixed-point iterations \(x_{k+1}=q(x_{k})\) and for various values of the window size \(m\) .KeywordsAnderson accelerationfixed-point methodroot-linear convergenceasymptotic convergence factorMSC codes65B0565F1065H1065K10
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.005 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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