Revisiting Marginal Regression

TL;DR

This paper compares the performance of the traditional marginal regression method with the modern lasso in high-dimensional regression, analyzing their effectiveness across different noise and design scenarios.

Contribution

It provides a comprehensive theoretical comparison of marginal regression and lasso, including conditions for exact reconstruction and phase diagram analysis.

Findings

Marginal regression is computationally feasible in very high dimensions.

Conditions for exact reconstruction differ between the two methods.

A new phase diagram partitioning illustrates when each method is effective.

Abstract

The lasso has become an important practical tool for high dimensional regression as well as the object of intense theoretical investigation. But despite the availability of efficient algorithms, the lasso remains computationally demanding in regression problems where the number of variables vastly exceeds the number of data points. A much older method, marginal regression, largely displaced by the lasso, offers a promising alternative in this case. Computation for marginal regression is practical even when the dimension is very high. In this paper, we study the relative performance of the lasso and marginal regression for regression problems in three different regimes: (a) exact reconstruction in the noise-free and noisy cases when design and coefficients are fixed, (b) exact reconstruction in the noise-free case when the design is fixed but the coefficients are random, and (c) reconstruction in the noisy case where performance is measured by the number of coefficients whose sign is incorrect. In the first regime, we compare the conditions for exact reconstruction of the two procedures, find examples where each procedure succeeds while the other fails, and characterize the advantages and disadvantages of each. In the second regime, we derive conditions under which marginal regression will provide exact reconstruction with high probability. And in the third regime, we derive rates of convergence for the procedures and offer a new partitioning of the ``phase diagram,'' that shows when exact or Hamming reconstruction is effective.