Sparse Hierarchical Regression with Polynomials

TL;DR

This paper introduces an exact hierarchical sparse polynomial regression method that efficiently identifies relevant inputs and monomials in high-dimensional data, balancing model complexity and prediction accuracy.

Contribution

It presents a novel two-step approach combining input ranking heuristics and cutting plane optimization to achieve exact sparse polynomial regression.

Findings

Method accurately identifies relevant features and monomials.

Phase transition observed in feature selection performance.

Scales to datasets with approximately 10,000 observations and 1,000 inputs.

Abstract

We present a novel method for exact hierarchical sparse polynomial regression. Our regressor is that degree $r$ polynomial which depends on at most $k$ inputs, counting at most $\ell$ monomial terms, which minimizes the sum of the squares of its prediction errors. The previous hierarchical sparse specification aligns well with modern big data settings where many inputs are not relevant for prediction purposes and the functional complexity of the regressor needs to be controlled as to avoid overfitting. We present a two-step approach to this hierarchical sparse regression problem. First, we discard irrelevant inputs using an extremely fast input ranking heuristic. Secondly, we take advantage of modern cutting plane methods for integer optimization to solve our resulting reduced hierarchical $(k, \ell)$-sparse problem exactly. The ability of our method to identify all $k$ relevant inputs and all $\ell$ monomial terms is shown empirically to experience a phase transition. Crucially, the same transition also presents itself in our ability to reject all irrelevant features and monomials as well. In the regime where our method is statistically powerful, its computational complexity is interestingly on par with Lasso based heuristics. The presented work fills a void in terms of a lack of powerful disciplined nonlinear sparse regression methods in high-dimensional settings. Our method is shown empirically to scale to regression problems with $n\approx 10,000$ observations for input dimension $p\approx 1,000$.