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The optimization with orthogonality has been shown useful in training deepneural networks (DNNs). To impose orthogonality on DNNs, both computationalefficiency and stability are important. However, existing methods utilizingRiemannian optimization or hard constraints can only ensure stability whilethose using soft constraints can only improve efficiency. In this paper, wepropose a novel method, named Feedback Gradient Descent (FGD), to ourknowledge, the first work showing high efficiency and stability simultaneously.FGD induces orthogonality based on the simple yet indispensable Eulerdiscretization of a continuous-time dynamical system on the tangent bundle ofthe Stiefel manifold. In particular, inspired by a numerical integration methodon manifolds called Feedback Integrators, we propose to instantiate it on thetangent bundle of the Stiefel manifold for the first time. In the extensiveimage classification experiments, FGD comprehensively outperforms the existingstate-of-the-art methods in terms of accuracy, efficiency, and stability.

Feedback Gradient Descent: Efficient and Stable Optimization with Orthogonality for DNNs