High-dimensional regression with potential prior information on variable importance

TL;DR

This paper introduces a computationally efficient method for high-dimensional regression that leverages prior information on variable importance, improving model selection and performance in settings like missing data and time series.

Contribution

It proposes a simple, fast scheme for fitting a sequence of models based on variable importance orderings, with theoretical guarantees and practical effectiveness demonstrated.

Findings

Efficient model fitting with no additional computational cost for ridge regression.

Theoretical bound showing only a logarithmic penalty in model selection.

Effective application to missing data, corrupted data, and time series.

Abstract

There are a variety of settings where vague prior information may be available on the importance of predictors in high-dimensional regression settings. Examples include ordering on the variables offered by their empirical variances (which is typically discarded through standardisation), the lag of predictors when fitting autoregressive models in time series settings, or the level of missingness of the variables. Whilst such orderings may not match the true importance of variables, we argue that there is little to be lost, and potentially much to be gained, by using them. We propose a simple scheme involving fitting a sequence of models indicated by the ordering. We show that the computational cost for fitting all models when ridge regression is used is no more than for a single fit of ridge regression, and describe a strategy for Lasso regression that makes use of previous fits to greatly speed up fitting the entire sequence of models. We propose to select a final estimator by cross-validation and provide a general result on the quality of the best performing estimator on a test set selected from among a number $M$ of competing estimators in a high-dimensional linear regression setting. Our result requires no sparsity assumptions and shows that only a $\log M$ price is incurred compared to the unknown best estimator. We demonstrate the effectiveness of our approach when applied to missing or corrupted data, and time series settings. An R package is available on github.