The Critical Line Algorithm and the Constrained LASSO: One Curve, Two Literatures
Abstract
Many statistical procedures compute an entire solution path rather than a single estimator. For one class the path is piecewise linear, traced corner to corner by an active-set homotopy. Two of its members coincide exactly: under Σ=X⊤ X and μ=X⊤ y the gross-exposure-constrained mean--variance program and the constrained LASSO trace the same piecewise-linear curve, and they keep doing so under arbitrary linear equality and inequality constraints. Which parameter is swept, a leverage budget or a return tilt, is not a choice the curve notices: under a homogeneous mandate the two sweeps differ by a scalar. A mandate that holds the portfolio invested costs a radial rescaling instead, and the efficient frontier is that curve rescaled; its corners pass unnoticed because the frontier is continuously differentiable where they fall. We give the map between the two parametrisations and the single way it degenerates. That mean--variance selection and the LASSO instantiate one parametric quadratic program is prior art (Gärtner, Jaggi and Maria, 2012). The Critical Line Algorithm itself lies off that route, and the correspondence established here is new. The identity is one of curves, not of statistical experiments. The map between the parametrisations is computed from the data, so the geometry of the path transfers while quantities averaged over the response do not. We cross that boundary deliberately and give the degrees of freedom of the constrained fit under arbitrary linear constraints. In the long-only, fully invested case it is the expected number of holdings away from a bound, less one.
Related
- CLASSO: a practical primal path-following algorithm for the constrained LASSO
- Square-Root Clustered Lasso for High-Dimensional Linear Regression with Finite-Sample Theory and Efficient Algorithm Implementation
- An Intersection Cutting Plane Algorithm for Chance-Constrained Programs with Finite Support
- Critical Curve for Three-Dimensional Radial Wave Systems with Distinct Speeds
- A Spatial Benders Algorithm for Delay Constrained Routing
BibTeX
@misc{schmelzer2026critical,
title = {{The Critical Line Algorithm and the Constrained LASSO: One Curve, Two Literatures}},
author = {Thomas Schmelzer and Trevor Hastie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25704},
eprint = {2609.25704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25704},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Critical Line Algorithm and the Constrained LASSO: One Curve, Two Literatures
AU - Thomas Schmelzer
AU - Trevor Hastie
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2609.25704
UR - https://arxiv.org/abs/2609.25704
AB - Many statistical procedures compute an entire solution path rather than a single estimator. For one class the path is piecewise linear, traced corner to corner by an active-set homotopy. Two of its members coincide exactly: under Σ=X^⊤ X and μ=X^⊤ y the gross-exposure-constrained mean--variance program and the constrained LASSO trace the same piecewise-linear curve, and they keep doing so under arbitrary linear equality and inequality constraints. Which parameter is swept, a leverage budget or a return tilt, is not a choice the curve notices: under a homogeneous mandate the two sweeps differ by a scalar. A mandate that holds the portfolio invested costs a radial rescaling instead, and the efficient frontier is that curve rescaled; its corners pass unnoticed because the frontier is continuously differentiable where they fall. We give the map between the two parametrisations and the single way it degenerates. That mean--variance selection and the LASSO instantiate one parametric quadratic program is prior art (Gärtner, Jaggi and Maria, 2012). The Critical Line Algorithm itself lies off that route, and the correspondence established here is new. The identity is one of curves, not of statistical experiments. The map between the parametrisations is computed from the data, so the geometry of the path transfers while quantities averaged over the response do not. We cross that boundary deliberately and give the degrees of freedom of the constrained fit under arbitrary linear constraints. In the long-only, fully invested case it is the expected number of holdings away from a bound, less one.
ER -