Generalized Gradient Descent is a Hypergraph Functor
Tyler Hanks Matthew Klawonn James Fairbanks
Cartesian reverse derivative categories (CRDCs) provide an axiomatic
generalization of the reverse derivative, which allows generalized analogues of
classic optimization algorithms such as gradient descent to be applied to a
broad class of problems. In this paper, we show that generalized gradient
descent with respect to a given CRDC induces a hypergraph functor from a
hypergraph category of optimization problems to a hypergraph category of
dynamical systems. The domain of this functor consists of objective functions
that are 1) general in the sense that they are defined with respect to an
arbitrary CRDC, and 2) open in that they are decorated spans that can be
composed with other such objective functions via variable sharing. The codomain
is specified analogously as a category of general and open dynamical systems
for the underlying CRDC. We describe how the hypergraph functor induces a
distributed optimization algorithm for arbitrary composite problems specified
in the domain. To illustrate the kinds of problems our framework can model, we
show that parameter sharing models in multitask learning, a prevalent machine
learning paradigm, yield a composite optimization problem for a given choice of
CRDC. We then apply the gradient descent functor to this composite problem and
describe the resulting distributed gradient descent algorithm for training
parameter sharing models.