On The Rate Of Convergence Of GD In Non Linear Neural Networks: An Adversarial Robustness Perspective

We study the convergence dynamics of Gradient Descent (GD) in a minimal binary classification setting, consisting of a two-neuron ReLU network and two training instances. We prove that while GD successfully converges to an optimal robustness margin, effectively maximizing the distance between the decision boundary and the training points, this convergence occurs at a prohibitively slow rate, scaling strictly as $\Theta (1/\ln (t))$. To the best of our knowledge, this establishes the first explicit lower bound on the convergence rate of the robustness margin in a non-linear model. Through empirical simulations, we further demonstrate that this inherent failure mode is present beyond our theoretical setting, exhibiting qualitatively similar slow convergence across a range of settings, including different network initializations, the Adam optimizer, and higher-dimensional inputs with wider networks. Our theoretical guarantees are derived via a rigorous analysis of the GD trajectories across the distinct activation patterns of the model. Specifically, we develop tight control over the system's dynamics to bound the trajectory of the decision boundary, overcoming the primary technical challenge introduced by the non-linear nature of the architecture.