# Adaptive estimation of High-Dimensional Signal-to-Noise Ratios

arXiv: https://arxiv.org/abs/1602.08006 · HTML: https://mlsanity.com/paper/1602.08006/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Feb 2016.

## Built on by (most cited first, 5 of 5)

- [Offline Reinforcement Learning: Fundamental Barriers for Value Function Approximation](https://mlsanity.com/paper/2111.10919/) (2021)
- [Universal and data-adaptive algorithms for model selection in linear contextual bandits](https://mlsanity.com/paper/2111.04688/) (2021)
- [Variance function estimation in regression model via aggregation procedures](https://mlsanity.com/paper/2110.02715/) (2021)
- [Estimating Optimal Policy Value in General Linear Contextual Bandits](https://mlsanity.com/paper/2302.09451/) (2023)
- [Hold-Out Scoring for Efficient Gaussian DAG Learning](https://mlsanity.com/paper/2610.02785/) (2026)

## Abstract (arXiv)

We consider the equivalent problems of estimating the residual variance, the proportion of explained variance η and the signal strength in a high-dimensional linear regression model with Gaussian random design. Our aim is to understand the impact of not knowing the sparsity of the regression parameter and not knowing the distribution of the design on minimax estimation rates of η. Depending on the sparsity k of the regression parameter, optimal estimators of η either rely on estimating the regression parameter or are based on U-type statistics, and have minimax rates depending on k. In the important situation where k is unknown, we build an adaptive procedure whose convergence rate simultaneously achieves the minimax risk over all k up to a logarithmic loss which we prove to be non avoidable. Finally, the knowledge of the design distribution is shown to play a critical role. When the distribution of the design is unknown, consistent estimation of explained variance is indeed possible in much narrower regimes than for known design distribution.