Optimal minimax rate of learning nonlocal interaction kernels
Bibliographic record
Abstract
Nonparametric estimation of nonlocal interaction kernels is crucial in various applications involving interacting particle systems. The inference challenge, situated at the nexus of statistical learning and inverse problems, arises from the nonlocal dependency. A central question is whether the optimal minimax rate of convergence for this problem aligns with the rate of $M^{-\frac{2β}{2β+1}}$ in classical nonparametric regression, where $M$ is the sample size and $β$ represents the regularity index of the radial kernel. Our study confirms this alignment for systems with a finite number of particles. We introduce a tamed least squares estimator (tLSE) that achieves the optimal convergence rate when $β\geq 1/4$ for a broad class of exchangeable distributions by leveraging random matrix theory and Sobolev embedding. The upper minimax rate relies on fourth-moment bounds for normal vectors and nonasymptotic bounds for the left tail probability of the smallest eigenvalue of the normal matrix. The lower minimax rate is derived using the Fano-Tsybakov hypothesis testing method. Our tLSE method offers a straightforward approach for establishing the optimal minimax rate for models with either local or nonlocal dependency.
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 | 0.014 | 0.066 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.004 |
| 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".