A gradient method in a Hilbert space with an optimized inner product: achieving a Newton-like convergence
Bibliographic record
Abstract
In this paper we introduce a new gradient method which attains quadratic convergence in a certain sense. Applicable to infinite-dimensional unconstrained minimization problems posed in a Hilbert space $H$, the approach consists in finding the energy gradient $g(λ)$ defined with respect to an optimal inner product selected from an infinite family of equivalent inner products $(\cdot,\cdot)_λ$ in the space $H$. The inner products are parameterized by a space-dependent weight function $λ$. At each iteration of the method, where an approximation to the minimizer is given by an element $u\in H$, an optimal weight $\hlambda$ is found as a solution of a nonlinear minimization problem in the space of weights $Λ$. It turns out that the projection of $κg(\hlambda)$, where $0<κ\ll 1$ is a fixed step size, onto a certain finite-dimensional subspace generated by the method is consistent with Newton's step $h$, in the sense that $P_u(κg(\hlambda))=P_u(h)$, where $P_u$ is an operator describing the projection onto the subspace. As demonstrated by rigorous analysis, this property ensures that thus constructed gradient method attains quadratic convergence for error components contained in these subspaces, in addition to the linear convergence typical of the standard gradient method. We propose a numerical implementation of this new approach and analyze its complexity. Computational results obtained based on a simple model problem confirm the theoretically established convergence properties, demonstrating that the proposed approach performs much better than the standard steepest-descent method based on Sobolev gradients. The presented results offer an explanation of a number of earlier empirical observations concerning the convergence of Sobolev-gradient methods.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".