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In this paper, we propose a variant of Riemannian stochastic recursivegradient method that can achieve second-order convergence guarantee and escapesaddle points using simple perturbation. The idea is to perturb the iterateswhen gradient is small and carry out stochastic recursive gradient updates overtangent space. This avoids the complication of exploiting Riemannian geometry.We show that under finite-sum setting, our algorithm requires$\widetilde{\mathcal{O}}\big( \frac{ \sqrt{n}}{\epsilon^2} + \frac{\sqrt{n}}{\delta^4} + \frac{n}{\delta^3}\big)$ stochastic gradient queries to find a$(\epsilon, \delta)$-second-order critical point. This strictly improves thecomplexity of perturbed Riemannian gradient descent and is superior toperturbed Riemannian accelerated gradient descent under large-sample settings.We also provide a complexity of $\widetilde{\mathcal{O}} \big(\frac{1}{\epsilon^3} + \frac{1}{\delta^3 \epsilon^2} + \frac{1}{\delta^4\epsilon} \big)$ for online optimization, which is novel on Riemannian manifoldin terms of second-order convergence using only first-order information.

Escape saddle points faster on manifolds via perturbed Riemannian stochastic recursive gradient