Gradient flows on graphons: existence, convergence, continuity equations
This work addresses the need for continuum limits in optimization problems involving large graphs, such as in multi-layer neural networks, by providing a theoretical framework for gradient flows on graphons.
The paper tackles the problem of extending Wasserstein gradient flows to large graphs by showing that Euclidean gradient flows on edge weights converge to a novel continuum limit on graphons, described as a curve of maximal slope, with examples including homomorphism functions and scalar entropy.
Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction involving a gradient-type potential. However, in many problems, such as in multi-layer neural networks, the so-called particles are edge weights on large graphs whose nodes are exchangeable. Such large graphs are known to converge to continuum limits called graphons as their size grow to infinity. We show that the Euclidean gradient flow of a suitable function of the edge-weights converges to a novel continuum limit given by a curve on the space of graphons that can be appropriately described as a gradient flow or, more technically, a curve of maximal slope. Several natural functions on graphons, such as homomorphism functions and the scalar entropy, are covered by our set-up, and the examples have been worked out in detail.