  Optimization problems with access to only zeroth-order information of the
objective function on Riemannian manifolds arise in various applications,
spanning from statistical learning to robot learning. While various
zeroth-order algorithms have been proposed in Euclidean space, they are not
inherently designed to handle the challenging constraints imposed by Riemannian
manifolds. The proper adaptation of zeroth-order techniques to Riemannian
manifolds remained unknown until the pioneering work of
\cite{li2023stochastic}. However, zeroth-order algorithms are widely observed
to converge slowly and be unstable in practice. To alleviate these issues, we
propose a Riemannian accelerated zeroth-order algorithm with improved
robustness. Regarding efficiency, our accelerated algorithm has the function
query complexity of $\mathcal{O}(\epsilon^{-7/4}d)$ for finding an
$\epsilon$-approximate first-order stationary point. By introducing a small
perturbation, it exhibits a function query complexity of
$\tilde{\mathcal{O}}(\epsilon^{-7/4}d)$ for seeking a second-order stationary
point with a high probability, matching state-of-the-art result in Euclidean
space. Moreover, we further establish the almost sure convergence in the
asymptotic sense through the Stable Manifold Theorem. Regarding robustness, our
algorithm requires larger smoothing parameters in the order of
$\tilde{\mathcal{O}}(\epsilon^{7/8}d^{-1/2})$, improving the existing result by
a factor of $\tilde{\mathcal{O}}(\epsilon^{3/4})$.
