Convergence of stochastic gradient descent for learning linear neural networks

Stochastic gradient descent (SGD) is the cornerstone of modern deep learning, yet its theoretical convergence properties remain poorly understood in non-convex settings. To address this gap, we study SGD dynamics in deep linear neural networks - simplified models that retain key non-convex characteristics while remaining analytically tractable. Building on recent results for gradient descent, we extend convergence guarantees to SGD under regularized squared loss. We establish almost sure boundedness of SGD iterates. Using the framework of asymptotic pseudotrajectories from stochastic approximation theory, our analysis establishes then, for both deterministic and adaptive step-size choices, convergence of SGD to the set of critical points under mild conditions (without relying on Lipschitz properties or whitened data). To our knowledge, this is the first work providing almost sure convergence guarantees for SGD in deep linear networks, bridging the gap between theoretical optimization and practical stochastic training.