Linear Asymptotic Convergence of Anderson Acceleration: Fixed-Point Analysis
Bibliographic record
Abstract
Abstract. We study the asymptotic convergence of AA([Formula: see text]), i.e., Anderson acceleration (AA) with window size [Formula: see text] for accelerating fixed-point methods [Formula: see text], [Formula: see text]. Convergence acceleration by AA([Formula: see text]) has been widely observed but is not well understood. We consider the case where the fixed-point iteration function [Formula: see text] is differentiable and the convergence of the fixed-point method itself is root-linear. We identify numerically several conspicuous properties of AA([Formula: see text]) convergence: First, AA([Formula: see text]) sequences [Formula: see text] converge root-linearly, but the root-linear convergence factor depends strongly on the initial condition. Second, the AA([Formula: see text]) acceleration coefficients [Formula: see text] do not converge but oscillate as [Formula: see text] converges to [Formula: see text]. To shed light on these observations, we write the AA([Formula: see text]) iteration as an augmented fixed-point iteration [Formula: see text], [Formula: see text], and analyze the continuity and differentiability properties of [Formula: see text] and [Formula: see text]. We find that the vector of acceleration coefficients [Formula: see text] is not continuous at the fixed point [Formula: see text]. However, we show that, despite the discontinuity of [Formula: see text], the iteration function [Formula: see text] is Lipschitz continuous and directionally differentiable at [Formula: see text] for AA(1), and we generalize this to AA([Formula: see text]) with [Formula: see text] for most cases. Furthermore, we find that [Formula: see text] is not differentiable at [Formula: see text]. We then discuss how these theoretical findings relate to the observed convergence behavior of AA([Formula: see text]). The discontinuity of [Formula: see text] at [Formula: see text] allows [Formula: see text] to oscillate as [Formula: see text] converges to [Formula: see text], and the nondifferentiability of [Formula: see text] allows AA([Formula: see text]) 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 [Formula: see text] and for various values of the window size [Formula: see text].
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".