[Paper Review] Group-bound: confidence intervals for groups of variables in sparse high-dimensional regression without assumptions on the design

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

Optimal Experimental Design Methods23 references5 citations

TL;DR

This paper introduces Group-bound, a method for constructing valid confidence intervals for groups of variables in high-dimensional sparse regression without requiring assumptions on the design matrix. By leveraging linear programming to derive lower bounds on group effects, it enables detection of jointly significant predictor groups even when individual variables appear non-significant, under weaker conditions than those needed for individual 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 confidence intervals for individual variables in high-dimensional regression under minimal assumptions.
• To develop a method that detects the joint effect of groups of highly correlated predictors when individual variables are not significant.
• To provide valid inference for group effects without relying on restrictive design conditions such as compatibility or irrepresentability.
• To extend the framework to allow one-sided confidence intervals for group norms and hypothesis tests on group coefficients.
• To demonstrate that group-level inference requires weaker assumptions than individual-level inference in sparse high-dimensional models.

Proposed method

• The method constructs a one-sided confidence interval for the group effect by solving a linear program to find the lower bound of the group coefficient norm.
• It uses the subgradient of the ℓ₁-norm at the true sparse solution to define a dual constraint that links estimation error to group effects.
• The approach relies on a primal-dual analysis where the dual variable is constrained to satisfy a compatibility condition with the design matrix.
• The confidence interval is derived using a concentration inequality that bounds the probability of estimation error exceeding a threshold.
• The method is designed to be valid for any design matrix, including those with high correlation among predictors.
• It generalizes to any ℓ_q norm of the group coefficients, with explicit solutions for q=1 via linear programming.

Experimental results

Research questions

• RQ1 Can valid confidence intervals be constructed for groups of variables in high-dimensional regression without assumptions on the design matrix?
• RQ2 Is it possible to detect the joint effect of a group of highly correlated predictors even when no individual variable in the group shows significance?
• RQ3 How do the assumptions required for detecting group effects compare to those needed for detecting individual variable effects?
• RQ4 Can the method provide one-sided confidence intervals for the ℓ₁-norm of group coefficients under minimal design assumptions?
• RQ5 Does the method maintain valid error control across all possible design matrices, including those violating standard compatibility conditions?

Key findings

• The method provides valid one-sided confidence intervals for group effects without any assumptions on the design matrix, making it universally applicable.
• Group-bound can detect the contribution of a group of variables even when no individual variable in the group is significant, due to high correlation.
• The assumptions required to detect group effects are weaker than those required to detect individual variable effects, offering a statistical advantage.
• The lower bound for the group coefficient norm is derived via linear programming, ensuring computational feasibility.
• The method achieves coverage probability at least 1−γ under mild regularity conditions, with explicit bounds on the error probability.
• The hierarchical monotonicity property ensures that if a group is rejected under the null hypothesis, all subgroups are also rejected, preserving logical consistency.