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Homogeneous margin constraints satisfy MFCQ

Statement

For homogeneous sample margins $q_{i}$ of degree $L > 0$ , every feasible point of the problem $\min_{θ} \frac{1}{2} ∥ θ ∥_{2}^{2}$ subject to $q_{i} (θ) \geq 1$ satisfies MFCQ in the margin convention.

Statement references

Margin-form MFCQ definition

Homogeneous sample-margin convention

Clarke Euler identity

Kaifeng Lyu and Jian Li, Gradient Descent Maximizes the Margin of Homogeneous Neural Networks, Appendix C.2, Lemma C.7 (its MFCQ sign convention is inconsistent)

Sign-consistent local MFCQ convention

Facts & Assumptions

Given: The degree- $L$ homogeneous classifier margins and a feasible point $θ$ .

[F1]

Homogeneous classifier margins and exponential training risk: each sample margin satisfies $q_{i} (c θ) = c^{L} q_{i} (θ)$ for every $c > 0$ , with $L > 0$ .

[F2]

Euler identity for Clarke subgradients of homogeneous margins: for $h_{i} \in \partial^{\circ} q_{i} (θ)$ , $⟨ θ, h_{i} ⟩ = L q_{i} (θ)$ when $θ \neq 0$ .

[F3]

KKT conditions and MFCQ for margin constraints: at a feasible point, MFCQ in margin form asks for one direction whose pairing with every Clarke subgradient of every active margin is strictly positive.

Proof

technique · use the parameter vector itself as the strict feasible direction

1.1F1givenalgebra

By [F1], $q_{i} (0) = q_{i} (2 \cdot 0) = 2^{L} q_{i} (0)$ . Since $2^{L} > 1$ , this forces $q_{i} (0) = 0$ for every $i$ . Thus a feasible point cannot be $θ = 0$ , because feasibility requires every margin to be at least one.

2.1F2F3step 1.1givenchoosealgebra∎

Choose $v = θ$ . If no constraint is active, the universal condition in [F3] is vacuous. Otherwise, for each active index $i$ , $q_{i} (θ) = 1$ and every $h_{i} \in \partial^{\circ} q_{i} (θ)$ satisfies $⟨ h_{i}, v ⟩ = ⟨ θ, h_{i} ⟩ = L q_{i} (θ) = L > 0$ by [F2]. Hence the same direction $v$ satisfies MFCQ for all active margins.

Depends on

Used by

Homogeneous gradient-flow limit points satisfy margin KKT conditions Theorem

Dependency tree · two levels

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Sources

Kaifeng Lyu and Jian Li, Gradient Descent Maximizes the Margin of Homogeneous Neural Networks, Appendix C.2, Lemma C.7 (standard reference, not scraped)