Inversion-Free Natural Gradient Descent on Riemannian Manifolds

Mirror Descent (MD) is a scalable first-order method widely used in large-scale optimization, with applications in image processing, policy optimization, and neural network training. This paper generalizes MD to optimization on Riemannian manifolds. In particular, we develop a Riemannian Mirror Descent (RMD) framework via reparameterization and further propose a stochastic variant of RMD. We also establish non-asymptotic convergence guarantees for both RMD and stochastic RMD. As an application to the Stiefel manifold, our RMD framework reduces to the Curvilinear Gradient Descent (CGD) method proposed in [26]. Moreover, when specializing the stochastic RMD framework to the Stiefel setting, we obtain a stochastic extension of CGD, which effectively addresses large-scale manifold optimization problems.

We consider the problem of approximating a function by an element of a nonlinear manifold which admits a differentiable parametrization, typical examples being neural networks with differentiable activation functions or tensor networks. Natural gradient descent (NGD) for the optimization of a loss function can be seen as a preconditioned gradient descent where updates in the parameter space are driven by a functional perspective. In a spirit similar to Newton's method, a NGD step uses, instead of the Hessian, the Gram matrix of the generating system of the tangent space to the approximation manifold at the current iterate, with respect to a suitable metric. This corresponds to a locally optimal update in function space, following a projected gradient onto the tangent space to the manifold. Still, both gradient and natural gradient descent methods get stuck in local minima. Furthermore, when the model class is a nonlinear manifold or the loss function is not ideally conditioned (e.g., the KL-divergence for density estimation, or a norm of the residual of a partial differential equation in physics informed learning), even the natural gradient might yield non-optimal directions at each step. This work introduces a natural version of classical inertial dynamic methods like Heavy-Ball or Nesterov and show how it can improve the learning process when working with nonlinear model classes.

Natural gradient descent (NGD) underlies common methods in ML. For dually flat families, idealized NGD on the forward Kullback--Leibler objective follows the mixture geodesic which is often longer than the shortest Fisher--Rao path. We quantify this overhead by the inefficiency ratio $R \ge 1$, the Fisher length of the mixture geodesic divided by the Fisher--Rao distance, and bound its supremum over endpoint pairs as a function of the parameter dimension $d$. A tensor criterion identifies the regime (I) families, with $R=1$ everywhere: exactly those with quadratic potential or dimension one, such as fixed-covariance Gaussians. For non-quadratic families, we prove two further regimes: (II) bounded third-order skewness plus finite Fisher--Rao diameter yields a dimension-independent bound; and (III) for products of scale families---including Gaussian covariances and Gamma rates---$R$ grows as $Θ(\sqrt{\log d})$, unbounded in $d$. Under a per-step Fisher-chord budget, $R$ translates to a practical computational cost: NGD requires asymptotically at least $R$ times as many steps as an optimizer following the Fisher--Rao geodesic. Experiments confirm all three regimes: $R=1$ to machine precision for quadratic-potential families (I), the categorical bound $π/(2\sqrt{2})$ is approached but not attained (II), and sampled scale-product $R$ grows with $d$, reaching $R \approx 1.5$ for long, high-dimensional moves (III).