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Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning

Frederik Hoppe, Claudio Mayrink Verdun, Hannah Laus, Felix Krahmer, Holger Rauhut

TL;DR

The paper addresses the challenge of uncertainty quantification in high-dimensional learning by developing non-asymptotic confidence intervals around debiased estimators. It introduces a data-driven correction for the remainder term, yielding valid finite-sample CI radii that do not rely on asymptotic regimes, and also provides Gaussian-remainder-based intervals with provable coverage. The framework extends beyond classical sparse regression to model-based deep learning for inverse problems, including MRI reconstruction with unrolled networks, and is validated through extensive experiments showing improved coverage over traditional asymptotic intervals. This work thus bridges theory and practice for reliable UQ in high-dimensional, data-driven settings with important applications in medical imaging and beyond.

Abstract

Uncertainty quantification (UQ) is a crucial but challenging task in many high-dimensional regression or learning problems to increase the confidence of a given predictor. We develop a new data-driven approach for UQ in regression that applies both to classical regression approaches such as the LASSO as well as to neural networks. One of the most notable UQ techniques is the debiased LASSO, which modifies the LASSO to allow for the construction of asymptotic confidence intervals by decomposing the estimation error into a Gaussian and an asymptotically vanishing bias component. However, in real-world problems with finite-dimensional data, the bias term is often too significant to be neglected, resulting in overly narrow confidence intervals. Our work rigorously addresses this issue and derives a data-driven adjustment that corrects the confidence intervals for a large class of predictors by estimating the means and variances of the bias terms from training data, exploiting high-dimensional concentration phenomena. This gives rise to non-asymptotic confidence intervals, which can help avoid overestimating uncertainty in critical applications such as MRI diagnosis. Importantly, our analysis extends beyond sparse regression to data-driven predictors like neural networks, enhancing the reliability of model-based deep learning. Our findings bridge the gap between established theory and the practical applicability of such debiased methods.

Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning

TL;DR

The paper addresses the challenge of uncertainty quantification in high-dimensional learning by developing non-asymptotic confidence intervals around debiased estimators. It introduces a data-driven correction for the remainder term, yielding valid finite-sample CI radii that do not rely on asymptotic regimes, and also provides Gaussian-remainder-based intervals with provable coverage. The framework extends beyond classical sparse regression to model-based deep learning for inverse problems, including MRI reconstruction with unrolled networks, and is validated through extensive experiments showing improved coverage over traditional asymptotic intervals. This work thus bridges theory and practice for reliable UQ in high-dimensional, data-driven settings with important applications in medical imaging and beyond.

Abstract

Uncertainty quantification (UQ) is a crucial but challenging task in many high-dimensional regression or learning problems to increase the confidence of a given predictor. We develop a new data-driven approach for UQ in regression that applies both to classical regression approaches such as the LASSO as well as to neural networks. One of the most notable UQ techniques is the debiased LASSO, which modifies the LASSO to allow for the construction of asymptotic confidence intervals by decomposing the estimation error into a Gaussian and an asymptotically vanishing bias component. However, in real-world problems with finite-dimensional data, the bias term is often too significant to be neglected, resulting in overly narrow confidence intervals. Our work rigorously addresses this issue and derives a data-driven adjustment that corrects the confidence intervals for a large class of predictors by estimating the means and variances of the bias terms from training data, exploiting high-dimensional concentration phenomena. This gives rise to non-asymptotic confidence intervals, which can help avoid overestimating uncertainty in critical applications such as MRI diagnosis. Importantly, our analysis extends beyond sparse regression to data-driven predictors like neural networks, enhancing the reliability of model-based deep learning. Our findings bridge the gap between established theory and the practical applicability of such debiased methods.
Paper Structure (14 sections, 3 theorems, 21 equations, 9 figures, 3 tables, 2 algorithms)

This paper contains 14 sections, 3 theorems, 21 equations, 9 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Problem Formulation
  3. Data-Driven Confidence Intervals
  4. Confidence Intervals for Gaussian Remainders
  5. Numerical Experiments
  6. UQ for Classical Model-Based Regression
  7. UQ for MRI Reconstruction with Neural Networks
  8. Final Remarks
  9. Further Discussion of Main Result
  10. Proofs
  11. Further Numerical Evaluation
  12. UQ for Classical Model-Based Regression
  13. UQ for MRI Reconstruction with Neural Networks
  14. Distribution Visualization of Remainder Term

Key Result

Theorem 1

Let $x^{(1)},\hdots, x^{(l)}\in\mathbb{C}^N$ be i.i.d. data. Let $b^{(i)}=Ax^{(i)}+\varepsilon^{(i)}$ be a high-dimensional regression model with noise $\varepsilon^{(i)} \sim \mathcal{CN}(0, \sigma^2 I_{N \times N})$. With the data, derive, for a significance level $\alpha$, a confidence radius $r_

Figures (9)

  • Figure 1: Illustration of the confidence interval correction. Figs. \ref{['subfig:CI_comp_old']}, \ref{['subfig:CI_comp_gauss']}, \ref{['subfig:CI_comp_new']} show the construction of CIs with standard debiased techniques (w/o data adjustment) and with our proposed method (w/ Gaussian adjustment - Thm. \ref{['thm:remainder_dist_gaussian']} - in Fig. \ref{['subfig:CI_comp_gauss']} and data adjustment - Thm. \ref{['thm:main_stat_result']} - in Fig. \ref{['subfig:CI_comp_new']}), respectively. The red points represent the entries that are not captured by the CIs. Additionally, Fig. \ref{['subfig:CI_comp_box_all_old']} shows box plots of coverage over all components, and Fig. \ref{['subfig:CI_comp_box_S_old']} shows them on the support. In the last two plots, the left box refers to the asymptotic and the right to the non-asymptotic CI based on Gaussian adjustment of $500$ feature vectors. We solve a sparse regression problem $y=Ax+\varepsilon$ via the LASSO, where $A \in \mathbb{C}^{4000 \times 10000}$, $x \in \mathbb{C}^N$ is 200-sparse, and the noise level is $\approx 10\%$. The averaged coverage over $250$ vectors with significance level $\alpha=0.05$ of the asymptotic confidence intervals is $h^W(0.05)=0.9353$ and on the support $h^W_S(0.05)=0.8941$. Confidence intervals built with our proposed method yield for Gaussian adjustment $h^G(0.05)=0.9684$ and on the support $h_S^G(0.05)=0.9421$, and for data-driven adjustment $h(0.05)=h_S(0.05)=1$. For more details, cf. Section \ref{['subsec:classic regression']} and Appendix \ref{['sec:further_numerical']}.
  • Figure 2: Confidence intervals of asymptotic type \ref{['subfig:3comparison_old']}, with Gaussian adjustment \ref{['subfig:3comparison_gauss']} and data-driven adjustment \ref{['subfig:3comparision_new']} for one evaluation feature vector in the sparse regression setting described in Section \ref{['subsec:classic regression']}. Box plots of hit rates $h_j(0.05)$ and $h_j^G(0.05)$, $j=1,\hdots,N$, averaged over feature vectors $x^{(1)},\hdots, x^{(500)}$\ref{['subfig:gauss_regr_boxplot_all']} and hit rates $h_S(0.05)^{(i)}$ and $(h_S^G)^{(i)}(0.05)$, $i=1,\hdots,k$, averaged over components $j$\ref{['subfig:gauss_regr_boxplot_s']}.
  • Figure 3: Reconstruction obtained with the It-Net as described in \ref{['subsec:UQNN']}. Data-driven adjustment confidence intervals \ref{['subfig:CIsnew']} and asymptotic confidence intervals \ref{['subfig:CIsold']} for the region (50 pixels) in 320x320 knee image \ref{['subfig:imagewithbox']}; Box plots of hit rates \ref{['subfig:boxplot']} for $90 \%$ confidence level for the Gaussian adjusted and asymptotic confidence intervals.
  • Figure 4: Box plots for hit rates of sparse regression experiments. The settings are those described in Table \ref{['tab:results_gauss_regr']}. The first row presents the hit rates over all components, and the second the hit rates of the support, e.g., \ref{['subfig:1']} and \ref{['subfig:1s']} correspond to the first column of the table, \ref{['subfig:2']} and \ref{['subfig:2s']} to the second one and so forth. In each plot, the left box represents the asymptotic hit rates, and the right one the Gaussian-adjusted hit rates.
  • Figure 5: Knee MRI groundtruth image from fastMRI dataset \ref{['subfig:knee']}zbontar2019fastmrifastMRIdataset, radial sampling mask \ref{['subfig:mask']} and undersampled k-space data \ref{['subfig:k-space']}.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Theorem 1: Informal Version
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm:main_stat_result']}
  • proof : Proof of Theorem \ref{['thm:remainder_dist_gaussian']}