The standard stochastic gradient descent (SGD) optimization method, as well as adaptive methods such as the Adam optimizer fail to converge if the learning rates do not converge to zero (particularly, in the situation of constant learning rates). In practice, human-tuned deterministic learning rate schedules or small constant learning rates are often used, and implementations in machine learning frameworks like Tensorflow and Pytorch typically employ constant learning rates. We propose a learning-rate-adaptive approach for SGD methods, adjusting the learning rate based on empirical estimates for the objective function values. Specifically, we propose a learning-rate-adaptive variant of the Adam optimizer and implement it for several machine learning problems, including deep learning methods for partial differential equations such as deep Kolmogorov methods, physics-informed neural networks, and deep Ritz methods. We refer to https://github.com/deeplearningmethods/adaptive-learning-rate for the Python source codes for the numerical simulations in this work. Our results show that the proposed adaptive Adam variant achieves faster reductions of the objective function value compared to Adam with default learning rates. For certain quadratic minimization problems, we rigorously prove that an adaptive SGD variant converges to the global minimizer. This proof uses properties of invariant measures of the SGD dynamics and a generalized convergence analysis for SGD with random predictable learning rates which we develop in this work.

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