A Riemannian Inertial Adaptive Gradient Method with Momentum Restart

Abstract

We propose RIAG-R, a Riemannian Inertial Adaptive Gradient method with gradient-triggered Restart, for optimization on Riemannian manifolds. RIAG-R combines three ingredients: an AdaGrad-type adaptive step-size that requires no knowledge of the Lipschitz constant; inertial extrapolation in the tangent space via the exponential map to accelerate convergence; and a restart rule, costing only one extra inner product per iteration, that detects and corrects the momentum-saturation pathology induced by manifold curvature. We establish a per-iteration descent inequality, an optimal $O(\varepsilon^{-2})$-iteration complexity, a convergence result for cluster points of the generated sequence, and a linear convergence rate under the Riemannian Polyak-\L ojasiewicz inequality. Numerical experiments on four representative matrix manifold optimization problems including sphere, Stiefel, symmetric positive-definite, and Grassmann manifolds demonstrate that RIAG-R consistently outperforms non-inertial and restart-free inertial baselines, confirming that adaptive restart is essential to realizing the benefits of momentum on curved spaces.