On the Rate of Convergence of GD in Non-linear Neural Networks: An Adversarial Robustness Perspective

The paper analyzes Gradient Descent (GD) on a minimal binary classification problem with a two‑neuron ReLU network trained on two instances. It proves that GD converges to an optimal robustness margin—maximizing the distance between the decision boundary and training points—but does so at a very slow rate, specifically θ(1/ln(t)). Empirical simulations confirm this slow convergence across various initializations, and the authors provide a rigorous analysis of GD trajectories across different activation patterns to establish the first explicit lower bound on robustness convergence in a non‑linear model.

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