Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses
Steffen Dereich, Arnulf Jentzen, Adrian Riekert
TL;DR
The paper tackles the fundamental issue that SGD and common adaptive methods can diverge or stall when learning rates do not decay to zero. It introduces a learning-rate-adaptive framework, including a variant of Adam, that adjusts step sizes based on empirical objective estimates and incorporates a practical LR grid search. The authors provide a rigorous convergence analysis via invariant-measure techniques and a general theory for SGD with random, predictable learning rates, complemented by a detailed quadratic-case result. Numerically, the adaptive LR approach accelerates convergence across high-dimensional PDE solvers (DKM), PINNs, and the Deep Ritz Method, often outperforming fixed LR baselines. Overall, the work links theoretical convergence guarantees with practical gains in deep learning-based PDE solvers, offering a principled method for LR control in complex optimization landscapes.
Abstract
It is known that the standard stochastic gradient descent (SGD) optimization method, as well as accelerated and adaptive SGD optimization methods such as the Adam optimizer fail to converge if the learning rates do not converge to zero (as, for example, in the situation of constant learning rates). Numerical simulations often use human-tuned deterministic learning rate schedules or small constant learning rates. The default learning rate schedules for SGD optimization methods in machine learning implementation frameworks such as TensorFlow and Pytorch are constant learning rates. In this work we propose and study a learning-rate-adaptive approach for SGD optimization methods in which the learning rate is adjusted based on empirical estimates for the values of the objective function of the considered optimization problem (the function that one intends to minimize). In particular, we propose a learning-rate-adaptive variant of the Adam optimizer and implement it in case of several neural network learning problems, particularly, in the context of deep learning approximation methods for partial differential equations such as deep Kolmogorov methods, physics-informed neural networks, and deep Ritz methods. In each of the presented learning problems the proposed learning-rate-adaptive variant of the Adam optimizer faster reduces the value of the objective function than the Adam optimizer with the default learning rate. For a simple class of quadratic minimization problems we also rigorously prove that a learning-rate-adaptive variant of the SGD optimization method converges to the minimizer of the considered minimization problem. Our convergence proof is based on an analysis of the laws of invariant measures of the SGD method as well as on a more general convergence analysis for SGD with random but predictable learning rates which we develop in this work.