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An R-only proximal coordinate rate fails for nonsmooth penalties

Statement refuted

The claim that the uniform randomized proximal update for every proper closed convex separable penalty satisfies $E [F (x^{k}) - F^{*}] \leq \frac{2 n L_{\max} R^{2}}{k + 4}$ under only a bounded initial sublevel radius $R$ and positive coordinate majorizers $L_{i}$ is false without an error term or other dependence on the penalty.

Witness

Let $n = 2$ , $f (x) = 0$ , $g_{1} (u) = g_{2} (u) = 4 ∣ u ∣$ , $L_{1} = L_{2} = L_{\max} = 1$ , and $x^{0} = (1, 1)$ . The objective is $F (x) = 4 (∣ x_{1} ∣ + ∣ x_{2} ∣)$ and $F^{*} = 0$ at $x^{*} = (0, 0)$ . The initial sublevel is the closed diamond $∣ x_{1} ∣ + ∣ x_{2} ∣ \leq 2$ , whose radius about $x^{*}$ is $R = 2$ .

Facts & Assumptions

Given: The proximal update in Randomized proximal coordinate descent with certified inner solves and the scalar minimizer in The one-coordinate absolute-value quadratic minimizer.

[F1]

Randomized proximal coordinate descent with certified inner solves: for selected block $i$ the exact update minimizes $⟨ \nabla_{i} f (x), u - x_{i} ⟩ + (M_{i} / 2) ∣ u - x_{i} ∣^{2} + g_{i} (u)$ .

[F2]

The one-coordinate absolute-value quadratic minimizer: $\frac{a}{2} u^{2} - b u + τ ∣ u ∣$ has unique minimizer $S_{τ} (b) / a$ when $a, τ > 0$ .

[F3]

Proper closed convex extended-real functions on $R^{n}$ : a block penalty is proper, closed and convex when its effective domain is nonempty, its epigraph is closed, and it satisfies the segment inequality.

[F4]

Absolute value in an ordered field, Basic properties of the absolute value and The triangle inequality: absolute value is nonnegative, positively homogeneous, and subadditive.

[F5]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: the two-coordinate Euclidean norm is a norm and its triangle inequality gives $∥ (x_{1}, x_{2}) ∥_{2} \leq ∣ x_{1} ∣ + ∣ x_{2} ∣$ by decomposing into coordinate vectors.

[A1]

The proximal coordinate algorithm uses common model majorizers $M_{1} = M_{2} = 1$ , samples uniformly, and runs at least one update with no stopping tolerance.

Proof

1.1A1F1F3F4F5givenalgebra

The assumptions hold: $f = 0$ is convex and continuously differentiable and has zero gradient, so $L_{i} = M_{i} = 1$ is a valid positive block-gradient bound and majorizer. Each $g_{i} (u) = 4 ∣ u ∣$ is finite everywhere and hence proper; [F4] gives convexity; and $epi g_{i} = {(u, t) : t \geq 4 u, t \geq - 4 u}$ is an intersection of closed half-planes, so it is closed in the epigraph convention [F3]. Thus $F (x) = 4 (∣ x_{1} ∣ + ∣ x_{2} ∣)$ and its minimum is $F^{*} = 0$ at $x^{*} = (0, 0)$ . The initial value is $F (x^{0}) = 8$ and its sublevel is $∣ x_{1} ∣ + ∣ x_{2} ∣ \leq 2$ . There $∥ x - x^{*} ∥_{2} = \sqrt{x_{1}^{2} + x_{2}^{2}} \leq ∣ x_{1} ∣ + ∣ x_{2} ∣ \leq 2$ by [F4]--[F5], with equality at $(2, 0)$ ; hence $R = 2$ .

2.1F1F2step 1.1algebra

For either selected coordinate the subproblem at its current value $1$ is $4 ∣ u ∣ + \frac{1}{2} (u - 1)^{2}$ , which differs by a constant from $\frac{1}{2} u^{2} - u + 4 ∣ u ∣$ . By [F2] its unique minimizer is $S_{4} (1) = 0$ . The other coordinate remains $1$ , so after one update $F (x^{1}) = 4$ for either draw and $E [F (x^{1}) - F^{*}] = 4$ .

3.1step 1.1step 2.1algebra∎

The proposed upper bound at $k = 1$ is $2 \cdot 2 \cdot 1 \cdot 2^{2} / (1 + 4) = 16 / 5 < 4$ , contradicting step 2.1. Therefore the claimed penalty-independent composite bound is false.

Remarks

The controlling plan's $O (1 / k)$ rate is for the smooth coordinate-gradient branch $g_{i} = 0$ . This witness does not refute that branch.

Depends on

Used by

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Sources

Stephen J. Wright, Coordinate Descent Algorithms, Theorem 3.1 and §3.7 (standard reference, not scraped)
Clément Royer, Regularized, Large-Scale and Distributed Optimization, Theorem 3.2.1 (standard reference, not scraped)