Stochastic Flows and Geometric Optimization on the Orthogonal Group
This work addresses optimization problems in machine learning for researchers and practitioners, offering a novel geometric approach that connects to graph theory and shows competitive results, though it appears incremental in its method development.
The authors tackled optimization on the orthogonal group and related manifolds by developing a new class of stochastic, geometrically-driven algorithms, demonstrating broad applicability in machine learning tasks such as reinforcement learning and neural networks, with strong performance on challenging benchmarks like the Humanoid agent in OpenAI Gym.
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 $\mathrm{Humanoid}$ agent from $\mathrm{OpenAI}$ $\mathrm{Gym}$ and improving convolutional neural networks.