[Paper Review] Assumption-free confidence intervals for groups of variables in sparse high-dimensional regression

Nicolai Meinshausen|arXiv (Cornell University)|Sep 13, 2013

Optimal Experimental Design Methods7 citations

TL;DR

This paper proposes assumption-free confidence intervals for groups of variables in high-dimensional sparse regression by leveraging linear programming to derive lower bounds on regression coefficients without requiring restrictive design matrix assumptions. It demonstrates that group effects can be detected even when individual variables appear non-significant, under weaker conditions than those needed for individual variable inference.

ABSTRACT

It is in general challenging to provide confidence intervals for individual variables in high-dimensional regression without making strict or unverifiable assumptions on the design matrix. We show here that a group-bound confidence interval can be derived without making any assumptions on the design matrix. The lower bound for the regression coefficient of individual variables can be derived via linear programming. The idea also generalises naturally to groups of variables, where we can derive a one-sided confidence interval for the joint effect of a group. While the confidence intervals of individual variables are by the nature of the problem often very wide, it is shown to be possible to detect the contribution of groups of highly correlated predictor variables even when no variable individually shows a significant effect. The assumptions necessary to detect the effect of groups of variables are shown to be weaker than the weakest known assumptions to detect the effect of individual variables.

Motivation & Objective

• To address the challenge of constructing valid confidence intervals for high-dimensional regression coefficients without strong assumptions on the design matrix.
• To develop a method for deriving confidence intervals for groups of variables that are robust to design matrix dependencies.
• To show that group effects can be detected even when individual variables are not significant, under weaker assumptions than required for individual inference.
• To provide a computationally feasible approach using linear programming to compute lower bounds on group regression coefficients.

Proposed method

• Uses linear programming to compute lower bounds on regression coefficients without assumptions on the design matrix.
• Derives one-sided confidence intervals for groups of variables by formulating the problem as a constrained optimization task.
• Generalizes the approach to handle groups of highly correlated predictors, focusing on joint effects rather than individual contributions.
• Relies on the sparsity of the regression model to ensure computational tractability and statistical validity.
• Employs a dual formulation to derive bounds that are valid under minimal regularity conditions on the design matrix.

Experimental results

Research questions

• RQ1 Can confidence intervals for groups of variables be constructed without assuming specific properties of the design matrix?
• RQ2 Under what conditions can group effects be detected when individual variables are not significant?
• RQ3 How do the assumptions required to detect group effects compare to those needed for individual variable detection?
• RQ4 Can linear programming be effectively used to derive valid lower bounds on group regression coefficients?

Key findings

• The proposed method constructs valid confidence intervals for groups of variables without any assumptions on the design matrix, ensuring robustness.
• Group-bound confidence intervals are shown to be effective in detecting joint effects even when individual variables appear non-significant.
• The assumptions required to detect group effects are weaker than the weakest known assumptions for detecting individual variable effects.
• The method leverages sparsity and linear programming to produce computationally feasible and statistically valid inference.