Adaptive minimax optimality in statistical inverse problems via SOLIT -- Sharp Optimal Lepskii-Inspired Tuning
Housen Li, Frank Werner
TL;DR
This work addresses adaptive parameter selection in filter-based regularization for statistical inverse problems, introducing SOLIT, a Lepski-inspired tuning rule that relies only on data, the noise level, T, and the filter qα. The authors establish an oracle inequality for the MSE and prove that SOLIT attains adaptive minimax convergence rates across mildly and, under certain conditions, severely ill-posed settings, without incurring extra logarithmic penalties. They provide a practical implementation framework, including line-search-based candidate construction and quantile-based thresholds, and demonstrate through numerical experiments that SOLIT closely matches oracle performance across diverse ill-posedness regimes. The results suggest SOLIT as a robust, scalable, and tuning-free approach for adaptive inverse problem solving with broad applicability, while highlighting open questions for severe ill-posedness and non-Gaussian noise. Overall, SOLIT challenges the conventional belief that Lepski-type methods necessarily incur log-factor losses, delivering sharp adaptive rates in many scenarios and offering practical computational strategies for large-scale problems.
Abstract
We consider statistical linear inverse problems in separable Hilbert spaces and filter-based reconstruction methods of the form $\hat f_α= q_α\left(T^*T\right)T^*Y$, where $Y$ is the available data, $T$ the forward operator, $\left(q_α\right)_{α\in \mathcal A}$ an ordered filter, and $α> 0$ a regularization parameter. Whenever such a method is used in practice, $α$ has to be appropriately chosen. Typically, the aim is to find or at least approximate the best possible $α$ in the sense that mean squared error (MSE) $\mathbb E [\Vert \hat f_α- f^\dagger\Vert^2]$ w.r.t.~the true solution $f^\dagger$ is minimized. In this paper, we introduce the Sharp Optimal Lepskiĭ-Inspired Tuning (SOLIT) method, which yields an a posteriori parameter choice rule ensuring adaptive minimax rates of convergence. It depends only on $Y$ and the noise level $σ$ as well as the operator $T$ and the filter $\left(q_α\right)_{α\in \mathcal A}$ and does not require any problem-dependent tuning of further parameters. We prove an oracle inequality for the corresponding MSE in a general setting and derive the rates of convergence in different scenarios. By a careful analysis we show that no other a posteriori parameter choice rule can yield a better performance in terms of the order of the convergence rate of the MSE. In particular, our results reveal that the typical understanding of Lepski\uı-type methods in inverse problems leading to a loss of a log factor is wrong. In addition, the empirical performance of SOLIT is examined in simulations.