  Gradient-based solvers risk convergence to local optima, leading to incorrect
researcher inference. Heuristic-based algorithms are able to ``break free" of
these local optima to eventually converge to the true global optimum. However,
given that they do not provide the gradient/Hessian needed to approximate the
covariance matrix and that the significantly longer computational time they
require for convergence likely precludes resampling procedures for inference,
researchers often are unable to quantify uncertainty in the estimates they
derive with these methods. This note presents a simple and relatively fast
two-step procedure to estimate the covariance matrix for parameters estimated
with these algorithms. This procedure relies on automatic differentiation, a
computational means of calculating derivatives that is popular in machine
learning applications. A brief empirical example demonstrates the advantages of
this procedure relative to bootstrapping and shows the similarity in standard
error estimates between this procedure and that which would normally accompany
maximum likelihood estimation with a gradient-based algorithm.
