We present an efficient quantum algorithm aiming to find the negative curvature direction for escaping the saddle point, which is a critical subroutine for many second-order non-convex optimization algorithms. We prove that our algorithm could produce the target state corresponding to the negative curvature direction with query complexityÕ(polylog(d)! −1 ), where d is the dimension of the optimization function. The quantum negative curvature finding algorithm is exponentially faster than any known classical method which takes time at least O(d! −1/2 ). Moreover, we propose an efficient algorithm to achieve the classical read-out of the target state. Our classical read-out algorithm runs exponentially faster on the degree of d than existing counterparts.
I am not familiar with the quantum algorithm literature, hence I cannot judge the novelty of the proposed algorithm. Basically, this is a quantum algorithm for computing the smallest negative eigenvalue (and the associated eigenvector) of square matrices. It is worth noting that the algorithm is built on top of the existing SVE model. The authors don't compare this method with any existing work in quantum singular value transformation and decomposition. A throughout discussion and comparison should be made in Section 1.1
Despite the title, I can't find a strong connection between the proposed algorithm and machine learning. Given a loss function f and the parameters x, the paper doesn't specify how to construct the quantum state in the step 2 of Algorithm 2. Is there a cheap way to construct the quantum state from (f, x)? If not, and if one has to compute the Hessian matrix H from (f, x), then the complexity of this pre-processing step will be on the order of O(d^2), which already makes the quantum efficiency meaningless. 
Escaping saddle point is usually an iterative process, because the curvature varies across from place to place. The paper focuses on an individual step of a single iteration (i.e. finding the negative curvature), but it doesn't mention how would the iterative process look like.
Finally, although the quantum complexity is poly-log in dimension d, it has a terrible dependence on the rank r and the Frobenius norm of the Hessian matrix. The potential issues of such dependencies should be discussed.