Improving Kernel-Based Nonasymptotic Simultaneous Confidence Bands
Balázs Csanád Csáji, Bálint Horváth
TL;DR
The paper tackles nonparametric, nonasymptotic, distribution-free simultaneous confidence bands for a band-limited regression function within the Paley-Wiener RKHS framework. Building on a Paley-Wiener kernel-based construction, it introduces three refinements: (i) replacing the symmetric-noise assumption with a distributional invariance principle (e.g., permutation invariance); (ii) a convex-duality based refinement to compute a tighter upper bound on the kernel norm $\|f_* abla\|_{\mathcal{H}}^2$; and (iii) replacing box-style confidence intervals with ellipsoidal constraints to obtain sharper intervals at query inputs. The refinements preserve the original method's distribution-free guarantees and are supported by theoretical analysis and numerical experiments, including non-symmetric noise scenarios. Overall, the work enhances robustness and efficiency of nonparametric simultaneous confidence bands under known input distributions and Paley-Wiener structure, with practical implications for risk-sensitive regression settings.
Abstract
The paper studies the problem of constructing nonparametric simultaneous confidence bands with nonasymptotic and distribition-free guarantees. The target function is assumed to be band-limited and the approach is based on the theory of Paley-Wiener reproducing kernel Hilbert spaces. The starting point of the paper is a recently developed algorithm to which we propose three types of improvements. First, we relax the assumptions on the noises by replacing the symmetricity assumption with a weaker distributional invariance principle. Then, we propose a more efficient way to estimate the norm of the target function, and finally we enhance the construction of the confidence bands by tightening the constraints of the underlying convex optimization problems. The refinements are also illustrated through numerical experiments.