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A homogeneous-network KKT point that is not globally optimal

Example

For the single training example $x_{0} = 1$ with label $y_{0} = + 1$ , take two hidden units with square activation, trainable input weights $θ_{1}, θ_{2}$ , zero biases, and fixed output weights $1, 4$ . The resulting degree-two homogeneous score is $q (θ) = θ_{1}^{2} + 4 θ_{2}^{2}$ . The feasible point $θ_{*} = (1, 0)$ is a Clarke KKT point of minimizing $\frac{1}{2} ∥ θ ∥_{2}^{2}$ subject to $q (θ) \geq 1$ , but it is not a global minimizer: $(0, \frac{1}{2})$ is feasible and has strictly smaller norm.

Example references

Finite feedforward graph evaluation convention

Parameter homogeneity and margin convention

Margin-form Clarke KKT conditions

Kaifeng Lyu and Jian Li, Gradient Descent Maximizes the Margin of Homogeneous Neural Networks, discussion following Theorem 4.4

Matus Telgarsky, Deep Learning Theory lecture notes, §10.2

Facts & Assumptions

Given: The degree-two homogeneous margin and the margin-form Clarke KKT definition.

[F1]

Homogeneous classifier margins and exponential training risk: a margin of degree $L$ satisfies $q (c θ) = c^{L} q (θ)$ .

[F2]

KKT conditions and MFCQ for margin constraints: a feasible point is Clarke KKT if nonnegative multipliers and Clarke subgradients satisfy stationarity and complementary slackness.

[F3]

Finite feedforward computation graphs and layered MLPs: each non-input node evaluates an activation at a weighted sum of earlier node values and a bias.

Proof

technique · compute the gradient, the KKT multiplier, and the objective values at the two feasible points

1.1F1F2F3givenalgebra

The stated two-node graph evaluates the hidden values $θ_{1}^{2}$ and $θ_{2}^{2}$ and combines them with its fixed output weights, so its score is $q (θ) = θ_{1}^{2} + 4 θ_{2}^{2}$ and $q (c θ) = c^{2} q (θ)$ for every $c > 0$ . At $θ_{*} = (1, 0)$ , $q (θ_{*}) = 1$ and $\nabla q (θ_{*}) = (2, 0)$ , so $\nabla q (θ_{*}) \in \partial^{\circ} q (θ_{*})$ and the multiplier $λ = \frac{1}{2}$ gives $θ_{*} = λ \nabla q (θ_{*})$ . Complementary slackness holds because the sole constraint is active. Thus [F2] gives the KKT property.

2.1F2step 1.1algebra∎

The point $(0, \frac{1}{2})$ satisfies $q (0, \frac{1}{2}) = 1$ and is feasible. The objective at $θ_{*}$ is $\frac{1}{2}$ , whereas at $(0, \frac{1}{2})$ it is $\frac{1}{2} (\frac{1}{4}) = \frac{1}{8} < \frac{1}{2}$ . Therefore $θ_{*}$ is not a global minimizer.

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Sources

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