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This paper aims to investigate the distributed stochastic optimization problems on compact embedded submanifolds (in the Euclidean space) for multi-agent network systems. To address the manifold structure, we propose a distributed…

Adaptive stochastic gradient algorithms in the Euclidean space have attracted much attention lately. Such explorations on Riemannian manifolds, on the other hand, are relatively new, limited, and challenging. This is because of the…

This paper considers a stochastic optimization problem over the fixed point sets of quasinonexpansive mappings on Riemannian manifolds. The problem enables us to consider Riemannian hierarchical optimization problems over complicated sets,…

In this work, we present a novel approach for solving stochastic shape optimization problems. Our method is the extension of the classical stochastic gradient method to infinite-dimensional shape manifolds. We prove convergence of the…

In this paper, we present a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints.We investigate the…

We propose a stochastic variance-reduced cubic regularized Newton algorithm to optimize the finite-sum problem over a Riemannian submanifold of the Euclidean space. The proposed algorithm requires a full gradient and Hessian update at the…

Riemannian optimization is concerned with problems, where the independent variable lies on a smooth manifold. There is a number of problems from numerical linear algebra that fall into this category, where the manifold is usually specified…

We examine a wide class of stochastic approximation algorithms for solving (stochastic) nonlinear problems on Riemannian manifolds. Such algorithms arise naturally in the study of Riemannian optimization, game theory and optimal transport,…

Given a square complex matrix $A$, we tackle the problem of finding the nearest matrix with multiple eigenvalues or, equivalently when $A$ had distinct eigenvalues, the nearest defective matrix. To this goal, we extend the general framework…

Stochastic optimisation in Riemannian manifolds, especially the Riemannian stochastic gradient method, has attracted much recent attention. The present work applies stochastic optimisation to the task of recursive estimation of a…

Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of…

Gaussian variational approximation is a popular methodology to approximate posterior distributions in Bayesian inference especially in high dimensional and large data settings. To control the computational cost while being able to capture…

We consider the problem of fitting variational posterior approximations using stochastic optimization methods. The performance of these approximations depends on (1) how well the variational family matches the true posterior…

In applications of imprecise probability, analysts must compute lower (or upper) expectations, defined as the infimum of an expectation over a set of parameter values. Monte Carlo methods consistently approximate expectations at fixed…

Motivated by the goal of improving the efficiency of small sample design, we propose a novel Bayesian stochastic approximation method to estimate the root of a regression function. The method features adaptive local modelling and…

We present Zeroth-order Riemannian Averaging Stochastic Approximation (\texttt{Zo-RASA}) algorithms for stochastic optimization on Riemannian manifolds. We show that \texttt{Zo-RASA} achieves optimal sample complexities for generating…

Stochastic minimax optimization on Riemannian manifolds has recently attracted significant attention due to its broad range of applications, such as robust training of neural networks and robust maximum likelihood estimation. Existing…

The performance of standard stochastic approximation implementations can vary significantly based on the choice of the steplength sequence, and in general, little guidance is provided about good choices. Motivated by this gap, in the first…

Projected gradient descent and its Riemannian variant belong to a typical class of methods for low-rank matrix estimation. This paper proposes a new Nesterov's Accelerated Riemannian Gradient algorithm by efficient orthographic retraction…

Approximate computing has shown to provide new ways to improve performance and power consumption of error-resilient applications. While many of these applications can be found in image processing, data classification or machine learning, we…