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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The zero-feature-column Lasso update

Example

In the fixed-design Lasso of Lasso estimator for fixed design, if column $X_{j} = 0$ and $τ > 0$ , the exact coordinate minimizer is $θ_{j} = 0$ for every residual formed from the other columns. The formula for a nonzero column is not evaluated with a zero denominator.

Facts & Assumptions

Given: A fixed design $X$ , response $y$ , penalty $τ > 0$ , and one zero column $X_{j} = 0$ .

[F1]

Cyclic exact block minimization: an exact coordinate update minimizes the objective over the selected coordinate with the others fixed.

[F2]

Lasso estimator for fixed design: the penalty contribution for coordinate $j$ is $2 τ ∣ θ_{j} ∣$ .

[F3]

Exact Lasso coordinate update, including a zero column: for a zero design column the unique minimizing coordinate is zero.

[F4]

The Lasso objective has a nonempty compact convex minimizer set: a global Lasso minimizer exists for every fixed design and $τ > 0$ .

Proof

1.1F1F2givenalgebra

With $X_{j} = 0$ , changing $θ_{j}$ leaves $y - X θ$ unchanged and leaves all other penalty terms unchanged. The only varying part of the objective is $2 τ ∣ θ_{j} ∣$ , which is nonnegative and vanishes only at zero because $τ > 0$ .

2.1F3F4step 1.1algebra∎

Therefore every exact coordinate minimization has the unique update $θ_{j} = 0$ , independent of the residual from the other columns. This agrees with the zero-column branch of [F3]; [F4] ensures the overall Lasso problem is not being treated as an empty minimization problem.

Depends on

Used by

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Dependency tree · two levels

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Sources

Stephen J. Wright, Coordinate Descent Algorithms, §2 (standard reference, not scraped)
Clément Royer, Regularized, Large-Scale and Distributed Optimization, Lecture 3, §3.3 (standard reference, not scraped)