The Lie-Group Bayesian Learning Rule
This work provides a more flexible framework for algorithm design in machine learning, though it appears incremental as an extension of existing Bayesian methods.
The authors tackled difficulties in applying the Bayesian Learning Rule by extending it with Lie-group theory, which simplifies parameterization, gradient computation, and manifold constraints. They demonstrated this by deriving a deep learning algorithm that learns sparse features with biologically-plausible attributes.
The Bayesian Learning Rule provides a framework for generic algorithm design but can be difficult to use for three reasons. First, it requires a specific parameterization of exponential family. Second, it uses gradients which can be difficult to compute. Third, its update may not always stay on the manifold. We address these difficulties by proposing an extension based on Lie-groups where posteriors are parametrized through transformations of an arbitrary base distribution and updated via the group's exponential map. This simplifies all three difficulties for many cases, providing flexible parametrizations through group's action, simple gradient computation through reparameterization, and updates that always stay on the manifold. We use the new learning rule to derive a new algorithm for deep learning with desirable biologically-plausible attributes to learn sparse features. Our work opens a new frontier for the design of new algorithms by exploiting Lie-group structures.