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Stochastic Flows and Geometric Optimization on the Orthogonal Group

David Cheikhi
Jared Davis
Valerii Likhosherstov
Achille Nazaret
Achraf Bahamou
Xingyou Song
Mrugank Akarte
Jack Parker-Holder
Jacob Bergquist
Yuan Gao
Aldo Pacchiano
Adrian Weller
Thirty-seventh International Conference on Machine Learning (ICML 2020) (to appear)


We present a new class of stochastic, geometrically-driven optimization algorithms on the orthogonal group O(d) and naturally reductive homogeneous manifolds obtained from the action of the rotation group SO(d). We theoretically and experimentally demonstrate that our methods can be applied in various fields of machine learning including deep, convolutional and recurrent neural networks, reinforcement learning, normalizing flows and metric learning. We show an intriguing connection between efficient stochastic optimization on the orthogonal group and graph theory (e.g. matching problem, partition functions over graphs, graph-coloring). We leverage the theory of Lie groups and provide theoretical results for the designed class of algorithms. We demonstrate broad applicability of our methods by showing strong performance on the seemingly unrelated tasks of learning world models to obtain stable policies for the most difficult Humanoid agent from OpenAI Gym and improving convolutional neural networks.