Before revision:  For each of the setting , we characterize the oracle complexity of our algorithm to obtain appropriately defined notions of \epsilon-stationary points . Notably, our complexities are independent of the ambient dimension of the Euclidean space in which the manifold is embedded in, and only depend on the intrinsic dimension of the manifold .
After revision:  For each of the setting , we characterize the oracle complexity of our algorithm to obtain appropriately defined notions of \epsilon-stationary point or \epsilon-approximate local minimizer . Notably, our complexities are independent of the ambient dimension of the Euclidean space in which the manifold is embedded in, and only depend on the intrinsic dimension of the manifold .