The Constrained Lasso
by Professor Gareth James
Abstract: Motivated by applications in areas as diverse as finance, image reconstruction, and curve estimation, we introduce the Constrained Lasso problem, where the underlying parameters satisfy a collection of linear constraints. We show that many statistical methods, such as the fused lasso, monotone curve estimation and the Generalized Lasso, are all special cases of the Constrained Lasso. Computing the Constrained Lasso poses some technical challenges but we develop an efficient algorithm for fitting it over a grid of tuning parameters. Error bounds are developed which suggest that the Constrained Lasso should outperform the Lasso in situations where the true parameters satisfy the underlying constraints. Extensive numerical experiments show that our method performs well, both computationally and statistically. Finally, we apply our method to a real data set to estimate the demand curve for loans as a function of interest rate, and demonstrate that it can outperform more standard approaches.
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