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The natural gradient method is widely used in statistical optimization, but its standard formulation assumes a Euclidean parameter space. This paper proposes an inversion-free stochastic natural gradient method for probability distributions…

We consider optimization problems in which the goal is find a $k$-dimensional subspace of $\mathbb{R}^n$, $k<<n$, which minimizes a convex and smooth loss. Such problems generalize the fundamental task of principal component analysis (PCA)…

We introduce two derivative-free projection methods for large-scale systems of nonlinear monotone equations subject to convex constraints. Both methods incorporate an adaptive spectral parameter into established conjugate gradient…

This paper proposes a nonmonotone proximal quasi-Newton algorithm for unconstrained convex multiobjective composite optimization problems. To design the search direction, we minimize the max-scalarization of the variations of the Hessian…

We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang's asymptotic optimal gradient method (Computational Optimization…

Smooth, non-convex optimization problems on Riemannian manifolds occur in machine learning as a result of orthonormality, rank or positivity constraints. First- and second-order necessary optimality conditions state that the Riemannian…

We propose to use stochastic Riemannian coordinate descent on the orthogonal group for recurrent neural network training. The algorithm rotates successively two columns of the recurrent matrix, an operation that can be efficiently…

In the Euclidean setting, the proximal gradient method and its accelerated variants are a class of efficient algorithms for optimization problems with decomposable objective. In this paper, we develop a Riemannian proximal gradient method…

This paper proposes a new steepest gradient descent method for solving nonconvex finite minimax problems using non-monotone adaptive step sizes and providing proof of convergence results in cases of the nonconvex, quasiconvex, and…

We study non-convex subgradient flows for training two-layer ReLU neural networks from a convex geometry and duality perspective. We characterize the implicit bias of unregularized non-convex gradient flow as convex regularization of an…

Gradient descent with momentum has been widely applied in various signal processing and machine learning tasks, demonstrating a notable empirical advantage over standard gradient descent. However, momentum-based distributed Riemannian…

An efficient gradient-based method to solve the volume constrained topology optimization problems is presented. Each iterate of this algorithm is obtained by the projection of a Barzilai-Borwein step onto the feasible set consisting of box…

We propose a unifying algorithm for non-smooth non-convex optimization. The algorithm approximates the objective function by a convex model function and finds an approximate (Bregman) proximal point of the convex model. This approximate…

In this paper, we develop a variant of the well-known Gauss-Newton (GN) method to solve a class of nonconvex optimization problems involving low-rank matrix variables. As opposed to the standard GN method, our algorithm allows one to handle…

We study stochastic projection-free methods for constrained optimization of smooth functions on Riemannian manifolds, i.e., with additional constraints beyond the parameter domain being a manifold. Specifically, we introduce stochastic…

Bilevel optimization has gained prominence in various applications. In this study, we introduce a framework for solving bilevel optimization problems, where the variables in both the lower and upper levels are constrained on Riemannian…

We analyze convergence of gradient-descent methods on Riemannian manifolds. In particular, we study randomization of Riemannian gradient algorithms for minimizing smooth cost functions (of Morse-Bott type). We prove that randomized gradient…

We propose the first global accelerated gradient method for Riemannian manifolds. Toward establishing our result we revisit Nesterov's estimate sequence technique and develop an alternative analysis for it that may also be of independent…

This paper investigates the optimal $H_2$ model order reduction for linear systems with quadratic outputs. In the framework of Galerkin projection, we first formulate the optimal $H_2$ MOR as an unconstrained Riemannian optimization problem…

We study a class of optimization problems on Riemannian manifolds, where the objective function consists of a smooth term and quasi-norm type penalties with exponent $p \in (0, 1]$. The essential difficulty lies in the fact that the…