Gradient Descent Convergence Guaranteed for General Neural Networks

This research provides significant theoretical guarantees for the convergence of gradient descent when training general feedforward neural networks. Unlike previous work that often required specific initialization schemes or dataset properties, this study applies to networks of arbitrary width and depth, making it broadly applicable. The key innovation lies in identifying a "generalized Lipschitz smoothness" condition. This condition demonstrates that the Lipschitz properties of activation functions, even when composed repeatedly, are partially preserved. This leads to an upper bound on the change in gradient that depends polynomially on the parameter norms at both ends of the change. Based on this insight, the authors prove a descent lemma: the loss function decreases as long as the learning rate is kept sufficiently small relative to the norms of the network parameters. By ensuring that these parameter norms do not grow uncontrollably, the paper concludes that the minimum squared gradient norm converges to zero over T iterations at a rate of O(1/T^(1/L)) for an L-layer network.