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A redundant active margin with zero KKT multiplier

Example

Consider the two positive examples $x_{1} = (1, 0)$ and $x_{2} = (1, 1)$ in $R^{2}$ with bias-free score $w \mapsto ⟨ w, x ⟩$ . Their hard-margin constraints are $w_{1} \geq 1$ and $w_{1} + w_{2} \geq 1$ . The unique minimum-norm feasible vector is $w_{*} = (1, 0)$ ; both constraints are active, but the second active constraint has a KKT multiplier equal to zero.

Example references

Bias-free margins and hard-margin problem

Margin-form Clarke KKT conditions

Daniel Soudry et al., The Implicit Bias of Gradient Descent on Separable Data, Appendix B discussion of support vectors

Matus Telgarsky, Deep Learning Theory lecture notes, §10.1

Facts & Assumptions

Given: The two-point positive sample, its bias-free linear score, and the hard-margin and Clarke-KKT definitions.

[F1]

Bias-free linear margins and the hard-margin SVM: the signed margin is $m_{i} (w) = ⟨ w, x_{i} ⟩$ for a positive label, and the hard-margin problem minimizes $\frac{1}{2} ∥ w ∥_{2}^{2}$ subject to every margin being at least one.

[F2]

A strictly separable finite sample has a unique hard-margin vector: a finite strictly separable sample has a unique feasible minimum-norm hard-margin vector.

[F3]

KKT conditions and MFCQ for margin constraints: KKT stationarity represents $w_{*}$ as a nonnegative combination of active margin subgradients, with zero multipliers allowed for active constraints.

[F4]

Clarke subdifferential and gradient-flow trajectory: for a differentiable linear margin, its Clarke subdifferential is the singleton containing its ordinary gradient.

Proof

technique · verify strict separability, solve the two scalar inequalities, and exhibit a valid multiplier vector with a zero coordinate

1.1F1F2givenalgebra

The vector $(1, 0)$ gives both margins equal to one, so the sample is strictly separable. Feasibility forces $w_{1} \geq 1$ , hence $∥ w ∥_{2}^{2} = w_{1}^{2} + w_{2}^{2} \geq 1$ , with equality only for $w = (1, 0)$ . Thus $w_{*} = (1, 0)$ is the unique hard-margin minimizer, consistent with [F2].

2.1F1F3F4step 1.1algebra∎

Both margins at $w_{*}$ equal one. Their ordinary gradients, and hence their Clarke subgradients by [F4], are $(1, 0)$ and $(1, 1)$ . The multipliers $λ_{1} = 1$ and $λ_{2} = 0$ are nonnegative and satisfy $λ_{1} (1 - 1) = λ_{2} (1 - 1) = 0$ and $λ_{1} (1, 0) + λ_{2} (1, 1) = w_{*}$ . These are the KKT conditions in [F3], while the second constraint is active with zero multiplier.

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Sources

Daniel Soudry et al., The Implicit Bias of Gradient Descent on Separable Data, Appendix B discussion of support vectors (standard reference, not scraped)
Matus Telgarsky, Deep Learning Theory lecture notes, §10.1 (standard reference, not scraped)