  Here, we consider periodic homogenization for time-fractional
Hamilton--Jacobi equations. By using the perturbed test function method, we
establish the convergence, and give estimates on a rate of convergence. A main
difficulty is the incompatibility between the function used in the doubling
variable method, and the non-locality of the Caputo derivative. Our approach is
to provide a lemma to prove the rate of convergence without the doubling
variable method with respect to the time variable, which is a key ingredient.
