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The one-coordinate absolute-value quadratic minimizer

Statement

Let $a > 0$ , $b \in R$ , and $τ > 0$ . The scalar convex function $q (u) = \frac{a}{2} u^{2} - b u + τ ∣ u ∣ (u \in R)$ has the unique global minimizer $u^{*} = S_{τ} (b) / a$ , where $S_{τ}$ is the soft-threshold map of Soft-thresholding map. At $b = τ$ and $b = - τ$ , the unique minimizer is $0$ .

Facts & Assumptions

Given: The parameters in the Statement.

[F1]

Soft-thresholding map: $S_{τ} (b) = b - τ$ for $b > τ$ , $0$ for $- τ \leq b \leq τ$ , and $b + τ$ for $b < - τ$ .

[F2]

The triangle inequality and Basic properties of the absolute value: absolute value is subadditive and positively homogeneous.

Proof

1.1givenF2algebra

For $0 \leq s \leq 1$ , the square identity $((1 - s) u + s v)^{2} = (1 - s) u^{2} + s v^{2} - s (1 - s) (u - v)^{2}$ and $a > 0$ make the quadratic term convex; [F2] makes the absolute-value term convex, and the linear term is affine. Thus $q$ is convex. On $u \geq 0$ , completing the square gives $q (u) = \frac{a}{2} (u - (b - τ) / a)^{2} - \frac{(b - τ)^{2}}{2 a}$ , so the unique minimizer on that half-line is $u_{+} = \max {(b - τ) / a, 0}$ .

1.2givenalgebra

On $u \leq 0$ , $q (u) = \frac{a}{2} (u - (b + τ) / a)^{2} - \frac{(b + τ)^{2}}{2 a}$ , so the unique minimizer on that half-line is $u_{-} = \min {(b + τ) / a, 0}$ .

2.1F1step 1.1step 1.2algebra∎

If $b > τ$ , then $u_{+} = (b - τ) / a > 0$ and $q (u_{+}) = - \frac{(b - τ)^{2}}{2 a} < 0 = q (0)$ , while the negative-half-line minimum is $0$ ; hence this is the unique global minimizer. If $b < - τ$ , then $u_{-} = (b + τ) / a < 0$ and $q (u_{-}) = - \frac{(b + τ)^{2}}{2 a} < 0 = q (0)$ , while the positive-half-line minimum is $0$ ; hence this is the unique global minimizer. If $- τ \leq b \leq τ$ , then $u_{+} = u_{-} = 0$ , the unique minimizer on each half-line by steps 1.1–1.2. These cases equal $S_{τ} (b) / a$ by [F1], including the two kink values.

Remarks

For the finite sanity case $b = 3$ and $τ = 1$ , the formula gives $S_{1} (3) = 2$ and the minimizer of $\frac{1}{2} u^{2} - 3 u + ∣ u ∣$ is $u = 2$ .

Depends on

Used by

An R-only proximal coordinate rate fails for nonsmooth penalties Counterexample
Nonunique Lasso coefficients with a unique fitted value Example
Exact Lasso coordinate update, including a zero column Lemma

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Clément Royer, Regularized, Large-Scale and Distributed Optimization, §2.2.3 (standard reference, not scraped)
Stephen J. Wright, Coordinate Descent Algorithms, §2 (standard reference, not scraped)