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Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning
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Proposes a non-asymptotic, data-driven correction of debiased-LASSO and unrolled-network confidence intervals by estimating mean and variance of the remainder term from training/estimation data, with a Gaussian-adjusted variant for sparse regression and MRI reconstruction.
Problem
Debiased LASSO confidence intervals are asymptotic and ignore the remainder (bias) term, so they are too narrow in finite-sample high-dimensional regimes; no rigorous UQ exists for unrolled/learned estimators like neural networks.
Approach
Decompose the debiased estimation error into a Gaussian term and a remainder; estimate the remainder's mean and variance from data and use an empirical Chebyshev inequality (plus a Gaussian adjustment when the remainder is approximately Gaussian) to obtain non-asymptotic confidence radii that apply to any estimator, including neural networks, without assumptions on the estimator's quality.
Key results
- On the sparse Gaussian setting (N=10000, alpha=0.05), the data-driven adjustment gives hit rate 1.0 on all components and support, while the asymptotic CIs give 0.8692 overall and 0.6783 on support.
- For It-Net MRI reconstruction at alpha=0.1, the asymptotic CI hit rate is 0.9502 overall and 0.87 on support versus 0.9999 overall and 0.9998 on support for data-driven adjustment.
- Gaussian adjustment gives overall coverage near nominal level (e.g., 0.9691 at alpha=0.05 in sparse Gaussian setting) with tighter radii than the fully data-driven adjustment.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add data-driven remainder adjustment instead of neglecting the remainder | Debiased LASSO · Sparse regression, Gaussian, N=10000, 60% undersampling | hit rate h(0.05): 0.8692 -> 1 | Helped |
| Add Gaussian remainder adjustment instead of neglecting the remainder | Debiased LASSO · Sparse regression, Gaussian, N=10000, 60% undersampling | support hit rate: 0.6783 -> 0.8948 | Helped |
| Add data-driven remainder adjustment instead of neglecting the remainder | It-Net · fastMRI knee, 60% undersampling, 10% noise | support hit rate at 0.1 significance: 0.87 -> 1.0 | Helped |
| Add Gaussian remainder adjustment instead of neglecting the remainder | It-Net · fastMRI knee, 60% undersampling, 10% noise | hit rate at 0.1 significance: 0.9502 -> 0.9752 | Helped |
| Add Gaussian remainder adjustment instead of neglecting the remainder | U-Net · fastMRI knee, 30% undersampling, 7% noise | support hit rate at 0.05 significance: 0.695 -> 0.995 | Helped |
| Add data-driven remainder adjustment instead of neglecting the remainder | U-Net · fastMRI knee, 30% undersampling, 7% noise | support hit rate at 0.05 significance: 0.695 -> 1.0 | Helped |
| Replace Gaussian adjustment with data-driven remainder adjustment | Debiased LASSO · Sparse regression, Gaussian, N=10000, 60% undersampling | hit rate h(0.05): 0.9691 -> 1 | Helped |
| Use data-driven rather than Gaussian remainder adjustment | It-Net · fastMRI knee, 60% undersampling, 10% noise | hit rate at 0.1 significance: 0.9752 -> 0.9999 | Helped |
| Use data-driven rather than Gaussian remainder adjustment | U-Net · fastMRI knee, 30% undersampling, 7% noise | hit rate at 0.05 significance: 0.9943 -> 1.0 | Helped |
Surprising findings
- In the sparse Gaussian setting, the remainder's l-infinity norm exceeds the Gaussian term's (ratio 1.1581), so the remainder cannot be neglected even in classical LASSO settings.
- Asymptotic confidence intervals fall below the nominal 95% level on the support for several MRI and sparse settings.
Limitations
- Wider intervals when the remainder is large; larger corrections are needed.
- Accuracy depends on how well mean and variance of the remainder are estimated from limited data.
- Requires the estimator to be independent of the estimation data and the noise level to be fixed and known.
Experimental setup
Datasets: fastMRI single-coil knee synthetic complex Gaussian and Fourier sparse regression data
Benchmarks: fastMRI single-coil knee synthetic sparse regression (Gaussian and Fourier matrices)
Metrics: hit rate (coverage) confidence radius length R/W norm ratio
Baselines: asymptotic debiased LASSO confidence intervals LASSO U-Net
Builds on: debiased LASSO LISTA It-Net empirical Chebyshev inequality
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