ICML 2025

Riemannian Diffusion Adaptation for Distributed Optimization on Manifolds
Xiuheng Wang 1 Ricardo Borsoi 1 C´edric Richard 2 Ali H. Sayed 3
Abstract
Online distributed optimization is particularly use-
ful for solving optimization problems with stream-
ing data collected by multiple agents over a net-
work. When the solutions lie on a Riemannian
manifold, such problems become challenging to
solve, particularly when efficiency and continuous
adaptation are required. This work tackles these
challenges and devises a diffusion adaptation strat-
egy for decentralized optimization over general
manifolds. A theoretical analysis shows that the
proposed algorithm is able to approach network
agreement after sufficient iterations, which allows
a non-asymptotic convergence result to be derived.
We apply the algorithm to the online decentral-
ized principal component analysis problem and
Gaussian mixture model inference. Experimental
results with both synthetic and real data illustrate
its performance.
1. Introduction
In the decentralized setting, this work deals with the multi-
agent optimization problem seeking consensus on a Rieman-
nian manifold M:
min
w∈M
1
K
K
X
k=1
Jk(w) ,
(1)
where Jk : M →R is a local risk function defined for each
agent by Jk(w) = Exk

Q(w; xk)
	
in terms of the expec-
tation of some loss function Q(w; xk). The computation
of Jk(w) is over the unknown distribution of the data xk,
which makes it necessary to use a stochastic approximation
for the gradient vector based on a set of independent realiza-
tions xk,t, observed sequentially over time. A wide range
Part of this work was done while Xiuheng Wang was a PhD
student at Universit´e Cˆote d’Azur. 1Universit´e de Lorraine, CNRS,
CRAN, France 2Universit´e Cˆote d’Azur, CNRS, OCA, France
3 ´Ecole Polytechnique F´ed´erale de Lausanne, Switzerland. Corre-
spondence to: Xiuheng Wang <dr.xiuheng.wang@gmail.com>.
Proceedings of the 42 nd International Conference on Machine
Learning, Vancouver, Canada. PMLR 26...