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

Gradient descent (GD) with large learning rates can successfully optimize neural networks even though classical smoothness-based learning rate conditions are violated. Yet theory for this regime largely covers linear models or linearly separable data. We analyze a minimal setting outside both regimes. Specifically, a two-neuron ReLU network trained with logistic loss on 1D non-linearly separable data. The objective is nonconvex and nonsmooth, and GD exhibits edge-of-stability (EoS) behavior. We identify an active cone in parameter space that contains every perfect classifier, where the problem becomes piecewise logistic regression with a varying effective dataset. Within this cone, we prove a finite-time loss bound for constant learning rates and show that GD converges in direction to an explicit max-margin vector, while the orthogonal component converges to the unique minimizer of a non-strongly convex potential. We further establish catapult results where exponentially small loss is reached after a single loss spike. Furthermore, we propose an adaptive schedule that lets learning rates grow to infinity without leaving the active cone. We prove that this schedule achieves a loss rate of $O(1/(T\log{T})$, which is faster than the standard $O(1/T)$ rate. Our theoretical results extend to local convergence on more general network and are validated on MNIST dataset. Overall, we show that this setup provides a tractable neural-network model for studying large and adaptive step sizes beyond linear predictors and separable data.

In this paper, we study the gradient descent dynamics for jointly training both layers of a one-hidden-layer ReLU network to fit a linear target function. Concretely, we consider a realizable setting where inputs are drawn i.i.d. from a Gaussian distribution and labels follow a planted linear model. This stylized framework captures salient features of end-to-end training in inverse problems and certain auto-encoder models. Despite its apparent simplicity, the dynamics remain poorly understood, in part because the loss landscape contains multiple non-strict saddle points, making it unclear why gradient descent from random initialization reliably escapes bad stationary regions. We provide a detailed characterization of the optimization landscape and prove that gradient descent from a moderately small random initialization-simultaneously training both layers-converges to a global minimizer at a linear rate with order-wise optimal sample complexity. Our analysis tracks the trajectory through three phases: an alignment phase in which hidden weights progressively align with the planted direction while the output weights maintain the correct sign pattern; a growth phase in which the norms of both layers increase while preserving alignment; and a local refinement phase in which the aligned neurons rapidly converge to the planted direction, yielding fast local convergence. To rigorously show that GD avoids non-strict saddles, we develop trajectory-level control arguments for the end-to-end dynamics. In addition, we establish novel uniform concentration results that hold along the entire trajectory, and are essential for obtaining order-wise optimal sample complexity. We corroborate our theory with extensive experiments across a range of configurations.

A surprising phenomenon in the training of neural networks is the ability of gradient descent to find global minimizers of the training loss despite its non-convexity. Following earlier works, we investigate this behavior for wide shallow networks. Existing results essentially cover the case of ReLU activations and the case of sigmoid activations with scalar output weights. We study a large class of models that includes multi-head attention layers and two-layer sigmoid networks with vector output weights. Building upon [Chizat and Bach, 2018], we prove that all non-global minimizers of the training loss are unstable under gradient descent dynamics. Thus, when the initial distribution of the parameters has full support (which includes the popular Gaussian case), and in the many hidden neurons or attention heads limit, continuous-time gradient descent can only converge to global minimizers. Establishing the instability of non-global minimizers corresponds to the construction of an ``escaping active set'' -- we complete the proof of [Chizat and Bach, 2018] to construct this set for models with bounded nonlinearities and scalar output weights. We also extend this construction to new cases for models with vector output weights. Finally, we show the well-posedness and the stability with respect to discretization of the mean field training dynamic for sub-Gaussian initializations.