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Homogeneous gradient-flow limit points satisfy margin KKT conditions

Statement

Let $N, p \geq 1$ and let $q_{1}, \dots, q_{N} : R^{p} \to R$ be locally Lipschitz functions that are positively homogeneous of common degree $L > 0$ and admit the Clarke chain rule along absolutely continuous arcs. Set $L (θ) = \sum_{i} e^{- q_{i} (θ)}$ . Suppose $θ : [0, \infty) \to R^{p}$ is a global Clarke gradient-flow trajectory for $L$ and $L (θ (t_{0})) < 1$ at some $t_{0} \geq 0$ . Then every limit point $u_{*}$ of $θ (t) / ∥ θ (t) ∥_{2}$ as $t \to \infty$ satisfies $q_{\min} (u_{*}) > 0$ , and there is an $α > 0$ such that $α u_{*}$ is a Clarke KKT point of minimizing $\frac{1}{2} ∥ θ ∥_{2}^{2}$ subject to $q_{i} (θ) \geq 1$ for all $i$ .

Statement references

Global Clarke trajectory convention

Margin-form Clarke KKT definition

Cluster points and tolerance-wise approximate KKT witnesses

Kaifeng Lyu and Jian Li, Gradient Descent Maximizes the Margin of Homogeneous Neural Networks, Theorem 4.4 and Appendix C.4

Matus Telgarsky, Deep Learning Theory lecture notes, §§9.1 and 10.2

Facts & Assumptions

Given: The homogeneous margin model, exponential loss, a global Clarke flow, and a time when the loss is below one.

[F1]

Homogeneous classifier margins and exponential training risk: each margin is degree- $L$ positively homogeneous and the empirical risk is $L = \sum_{i} e^{- q_{i}}$ .

[F2]

KKT conditions and MFCQ for margin constraints: a Clarke KKT point is feasible and has nonnegative multipliers, subgradients, stationarity, and complementary slackness; MFCQ is positivity of every active margin subgradient on one direction.

[F3]

Homogeneous Clarke flow produces approximate margin-KKT points: each normalized-direction cluster point has positive minimum margin and, for every tolerance, admits a nearby feasible rescaling with stationarity and complementarity errors at most that tolerance.

[F4]

Limits of approximate margin-KKT points under MFCQ: a feasible point satisfying MFCQ is KKT if, for every positive tolerance, it has a nearby feasible approximate KKT witness with both errors at most that tolerance.

[F5]

Homogeneous margin constraints satisfy MFCQ: every feasible point of these positive-degree homogeneous margin constraints satisfies MFCQ.

Proof

technique · approximate each directional limit by feasible approximate KKT points, then pass to an exact KKT point using MFCQ

1.1F1F3givenchoose

Fix a limit point $u_{*}$ of the normalized trajectory. By [F3], $q_{\min} (u_{*}) > 0$ and, for every $ε > 0$ , there is a feasible $(ε, ε)$ -approximate KKT point within $ε$ of $θ_{*} : = u_{*} / q_{\min} (u_{*})^{1 / L}$ .

2.1F2F4F5step 1.1algebra∎

The limit $θ_{*}$ is feasible and satisfies MFCQ by [F5]. Applying [F4] directly to the tolerance-wise witnesses from step 1.1 shows that $θ_{*}$ is a Clarke KKT point. Taking $α = q_{\min} (u_{*})^{- 1 / L} > 0$ gives $θ_{*} = α u_{*}$ , as required.

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Sources

Kaifeng Lyu and Jian Li, Gradient Descent Maximizes the Margin of Homogeneous Neural Networks, Theorem 4.4 and Appendix C.4 (standard reference, not scraped)
Matus Telgarsky, Deep Learning Theory lecture notes, §§9.1 and 10.2 (standard reference, not scraped)