On the Asymptotic Linear Convergence Speed of Anderson Acceleration, Nesterov Acceleration, and Nonlinear GMRES
Bibliographic record
Abstract
We consider nonlinear convergence acceleration methods for fixed-point iteration $x_{k+1}=q(x_k)$, including Anderson acceleration (AA), nonlinear GMRES (NGMRES), and Nesterov-type acceleration (corresponding to AA with window size one). We focus on fixed-point methods that converge asymptotically linearly with convergence factor $ρ<1$ and that solve an underlying fully smooth and non-convex optimization problem. It is often observed that AA and NGMRES substantially improve the asymptotic convergence behavior of the fixed-point iteration, but this improvement has not been quantified theoretically. We investigate this problem under simplified conditions. First, we consider stationary versions of AA and NGMRES, and determine coefficients that result in optimal asymptotic convergence factors, given knowledge of the spectrum of $q'(x)$ at the fixed point $x^*$. This allows us to understand and quantify the asymptotic convergence improvement that can be provided by nonlinear convergence acceleration, viewing $x_{k+1}=q(x_k)$ as a nonlinear preconditioner for AA and NGMRES. Second, for the case of infinite window size, we consider linear asymptotic convergence bounds for GMRES applied to the fixed-point iteration linearized about $x^*$. Since AA and NGMRES are equivalent to GMRES in the linear case, one may expect the GMRES convergence factors to be relevant for AA and NGMRES as $x_k \rightarrow x^*$. Our results are illustrated numerically for a class of test problems from canonical tensor decomposition, comparing steepest descent and alternating least squares (ALS) as the fixed-point iterations that are accelerated by AA and NGMRES. Our numerical tests show that both approaches allow us to estimate asymptotic convergence speed for nonstationary AA and NGMRES with finite window size.
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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 | 0.004 | 0.024 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 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".