Optimal Confidence Band for Kernel Gradient Flow Estimator

• URL: http://arxiv.org/abs/2605.05768v1
• Date: Thu, 07 May 2026 07:00:35 GMT
• Title: Optimal Confidence Band for Kernel Gradient Flow Estimator
• Abstract summary: We first establish convergence rates for the supremum norm generalization error of both continuous and discrete kernel gradient flows.<n>We then construct simultaneous confidence bands for both continuous and discrete kernel gradient flows.
• Score: 15.574125189708257
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we investigate the supremum-norm generalization error and the uniform inference for a specific class of kernel regression methods, namely the kernel gradient flows. Under the widely adopted capacity-source condition framework in the kernel regression literature, we first establish convergence rates for the supremum norm generalization error of both continuous and discrete kernel gradient flows under the source condition $s>α_0$, where $α_0\in(0,1)$ denotes the embedding index of the kernel function. Moreover, we show that these rates match the minimax optimal rates. Building on this result, we then construct simultaneous confidence bands for both continuous and discrete kernel gradient flows. Notably, the widths of the proposed confidence bands are also optimal, in the sense that their shrinkage rates are greater than, while can be arbitrarily close to, the minimax optimal rates.

Related papers

• Generalization Bound of Gradient Flow through Training Trajectory and Data-dependent Kernel [55.82768375605861]
  We establish a generalization bound for gradient flow that aligns with the classical Rademacher complexity for kernel methods.<n>Unlike static kernels such as NTK, the LPK captures the entire training trajectory, adapting to both data and optimization dynamics.
  arXiv  Detail & Related papers  (2025-06-12T23:17:09Z)
• Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows [40.884514919698596]
  Flow matching has emerged as a powerful framework for generative modeling.<n>We prove that higher-order flow matching preserves worst case optimality as a distribution estimator.
  arXiv  Detail & Related papers  (2025-03-12T05:07:07Z)
• Gradient Normalization Provably Benefits Nonconvex SGD under Heavy-Tailed Noise [60.92029979853314]
  We investigate the roles of gradient normalization and clipping in ensuring the convergence of Gradient Descent (SGD) under heavy-tailed noise. Our work provides the first theoretical evidence demonstrating the benefits of gradient normalization in SGD under heavy-tailed noise. We introduce an accelerated SGD variant incorporating gradient normalization and clipping, further enhancing convergence rates under heavy-tailed noise.
  arXiv  Detail & Related papers  (2024-10-21T22:40:42Z)
• A Bound on the Maximal Marginal Degrees of Freedom [0.0]
  This paper addresses low rank approximations and surrogates for kernel ridge regression.<n>We demonstrate that the computational cost of the most popular low rank approach, which is the Nystr"om method, is almost linear in the sample size.<n>The result builds on a thorough theoretical analysis of the approximation of elementary kernel functions by elements in the range of the associated integral operator.
  arXiv  Detail & Related papers  (2024-02-20T10:25:44Z)
• Kernel Ridge Regression Inference [5.981623867626376]
  We construct valid and sharp confidence sets that shrink at nearly the minimax rate, allowing nonstandard regressors.<n>We use the procedure to develop a test for match effects, i.e. whether students benefit more from the schools they rank highly.
  arXiv  Detail & Related papers  (2023-02-13T18:26:36Z)
• Coefficient-based Regularized Distribution Regression [4.21768682940933]
  We consider the coefficient-based regularized distribution regression which aims to regress from probability measures to real-valued responses over a kernel reproducing Hilbert space (RKHS) Asymptotic behaviors of the algorithm in different regularity ranges of the regression function are comprehensively studied. We get the optimal rates under some mild conditions, which matches the one-stage sampled minimax optimal rate.
  arXiv  Detail & Related papers  (2022-08-26T03:46:14Z)
• Meta-Learning Hypothesis Spaces for Sequential Decision-making [79.73213540203389]
  We propose to meta-learn a kernel from offline data (Meta-KeL) Under mild conditions, we guarantee that our estimated RKHS yields valid confidence sets. We also empirically evaluate the effectiveness of our approach on a Bayesian optimization task.
  arXiv  Detail & Related papers  (2022-02-01T17:46:51Z)
• High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails [55.561406656549686]
  We consider non- Hilbert optimization using first-order algorithms for which the gradient estimates may have tails. We show that a combination of gradient, momentum, and normalized gradient descent convergence to critical points in high-probability with best-known iteration for smooth losses.
  arXiv  Detail & Related papers  (2021-06-28T00:17:01Z)
• Scalable Variational Gaussian Processes via Harmonic Kernel Decomposition [54.07797071198249]
  We introduce a new scalable variational Gaussian process approximation which provides a high fidelity approximation while retaining general applicability. We demonstrate that, on a range of regression and classification problems, our approach can exploit input space symmetries such as translations and reflections. Notably, our approach achieves state-of-the-art results on CIFAR-10 among pure GP models.
  arXiv  Detail & Related papers  (2021-06-10T18:17:57Z)
• Towards Unbiased Random Features with Lower Variance For Stationary Indefinite Kernels [26.57122949130266]
  Our algorithm achieves lower variance and approximation error compared with the existing kernel approximation methods. With better approximation to the originally selected kernels, improved classification accuracy and regression ability is obtained.
  arXiv  Detail & Related papers  (2021-04-13T13:56:50Z)
• How rotational invariance of common kernels prevents generalization in high dimensions [8.508198765617196]
  Kernel ridge regression is well-known to achieve minimax optimal rates in low-dimensional settings. Recent work establishes consistency for kernel regression under certain assumptions on the ground truth function and the distribution of the input data.
  arXiv  Detail & Related papers  (2021-04-09T08:27:37Z)
• Flow-based Kernel Prior with Application to Blind Super-Resolution [143.21527713002354]
  Kernel estimation is generally one of the key problems for blind image super-resolution (SR) This paper proposes a normalizing flow-based kernel prior (FKP) for kernel modeling. Experiments on synthetic and real-world images demonstrate that the proposed FKP can significantly improve the kernel estimation accuracy.
  arXiv  Detail & Related papers  (2021-03-29T22:37:06Z)
• Optimal Rates for Averaged Stochastic Gradient Descent under Neural Tangent Kernel Regime [50.510421854168065]
  We show that the averaged gradient descent can achieve the minimax optimal convergence rate. We show that the target function specified by the NTK of a ReLU network can be learned at the optimal convergence rate.
  arXiv  Detail & Related papers  (2020-06-22T14:31:37Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.