Most popular optimizers for deep learning can be broadly categorized as adaptive methods (e.g. Adam) and accelerated schemes (e.g. stochastic gradient descent (SGD) with momentum). For many models such as convolutional neural networks (CNNs), adaptive methods typically converge faster but generalize worse compared to SGD; for complex settings such as generative adversarial networks (GANs), adaptive methods are typically the default because of their stability.We propose AdaBelief to simultaneously achieve three goals: fast convergence as in adaptive methods, good generalization as in SGD, and training stability. The intuition for AdaBelief is to adapt the stepsize according to the "belief" in the current gradient direction. Viewing the exponential moving average (EMA) of the noisy gradient as the prediction of the gradient at the next time step, if the observed gradient greatly deviates from the prediction, we distrust the current observation and take a small step; if the observed gradient is close to the prediction, we trust it and take a large step. We validate AdaBelief in extensive experiments, showing that it outperforms other methods with fast convergence and high accuracy on image classification and language modeling. Specifically, on ImageNet, AdaBelief achieves comparable accuracy to SGD. Furthermore, in the training of a GAN on Cifar10, AdaBelief demonstrates high stability and improves the quality of generated samples compared to a well-tuned Adam optimizer. Code is available at https://github.com/juntang-zhuang/Adabelief-Optimizer

Neural ordinary differential equations (NODEs) have recently attracted increasing attention; however, their empirical performance on benchmark tasks (e.g. image classification) are significantly inferior to discrete-layer models. We demonstrate an explanation for their poorer performance is the inaccuracy of existing gradient estimation methods: the adjoint method has numerical errors in reverse-mode integration; the naive method directly back-propagates through ODE solvers, but suffers from a redundantly deep computation graph when searching for the optimal stepsize. We propose the Adaptive Checkpoint Adjoint (ACA) method: in automatic differentiation, ACA applies a trajectory checkpoint strategy which records the forward-mode trajectory as the reverse-mode trajectory to guarantee accuracy; ACA deletes redundant components for shallow computation graphs; and ACA supports adaptive solvers. On image classification tasks, compared with the adjoint and naive method, ACA achieves half the error rate in half the training time; NODE trained with ACA outperforms ResNet in both accuracy and test-retest reliability. On time-series modeling, ACA outperforms competing methods. Finally, in an example of the three-body problem, we show NODE with ACA can incorporate physical knowledge to achieve better accuracy. We provide the PyTorch implementation of ACA: \url{https://github.com/juntang-zhuang/torch-ACA}.