On the Convergence of Step Decay Step-Size for Stochastic Optimization
This work addresses a theoretical gap for practitioners using step decay in machine learning, offering rigorous convergence proofs that validate its empirical success.
The paper tackled the lack of theoretical understanding for step decay step-size schedules in stochastic optimization, providing convergence guarantees with rates like O(ln T/√T) for non-convex problems and O(ln T/T) for strongly convex smooth cases, and demonstrated practical efficiency in large-scale neural network training.
The convergence of stochastic gradient descent is highly dependent on the step-size, especially on non-convex problems such as neural network training. Step decay step-size schedules (constant and then cut) are widely used in practice because of their excellent convergence and generalization qualities, but their theoretical properties are not yet well understood. We provide the convergence results for step decay in the non-convex regime, ensuring that the gradient norm vanishes at an $\mathcal{O}(\ln T/\sqrt{T})$ rate. We also provide the convergence guarantees for general (possibly non-smooth) convex problems, ensuring an $\mathcal{O}(\ln T/\sqrt{T})$ convergence rate. Finally, in the strongly convex case, we establish an $\mathcal{O}(\ln T/T)$ rate for smooth problems, which we also prove to be tight, and an $\mathcal{O}(\ln^2 T /T)$ rate without the smoothness assumption. We illustrate the practical efficiency of the step decay step-size in several large scale deep neural network training tasks.