One of the primary reasons behind the success of neural networks has been the emergence of an array of new, highly-successful optimizers, perhaps most importantly the Adam optimizer. It is wiedely used for training neural networks, yet notoriously hard to interpret. Lacking a clear physical intuition, Adam is difficult to generalize to manifolds. Some attempts have been made to directly apply parts of the Adam algorithm to manifolds or to find an underlying structure, but a full generalization has remained elusive. In this work a new approach is presented that leverages the special structure of the manifolds which are relevant for optimization of neural networks, such as the Stiefel manifold, the symplectic Stiefel manifold, the Grassmann manifold and the symplectic Grassmann manifold: all of these are homogeneous spaces and as such admit a global tangent space representation. This global tangent space representation is used to perform all of the steps in the Adam optimizer. The resulting algorithm is then applied to train a transformer for which orthogonality constraints are enforced up to machine precision and we observe significant speed-ups in the training process. Optimization of neural networks where they weights do not lie on a manifold is identified as a special case of the presented framkework. This allows for a flexible implementation in which the learning rate is adapted simultaneously for all parameters, irrespective of whether they are an element of a general manifold or a vector space.

翻译：神经网络成功的主要原因之一，是一系列高效新优化器的涌现，其中最重要的或许是Adam优化器。它被广泛用于训练神经网络，却以难以解释著称。由于缺乏清晰的物理直觉，Adam很难被推广到流形上。已有一些尝试将Adam算法的部分内容直接应用于流形，或寻找其底层结构，但完整的推广仍难以实现。本文提出了一种新方法，利用了与神经网络优化相关的特殊流形结构，如Stiefel流形、辛Stiefel流形、Grassmann流形和辛Grassmann流形：所有这些流形都是齐次空间，因此具有全局切空间表示。我们利用这种全局切空间表示来执行Adam优化器的所有步骤。然后，将所得算法应用于训练一个满足机器精度正交约束的Transformer，并观察到训练过程中显著的加速效果。当神经网络权值不位于流形上时的优化问题，被视为本文框架的一个特例。这允许一种灵活的实现方式，其中学习率同时自适应调整所有参数，无论这些参数是来自一般流形还是向量空间。