<p>In this paper, we study and analyze zeroth-order stochastic approximation algorithms for solving bilevel problems when neither the upper/lower objective values nor their unbiased gradient estimates are available. In particular, exploiting Stein’s identity, we first use Gaussian smoothing to estimate first- and second-order partial derivatives of functions with two independent block of variables. We then use these estimates in the framework of a stochastic approximation algorithm for solving bilevel optimization problems and establish its non-asymptotic convergence analysis. To the best of our knowledge, this is the first time that sample complexity bounds are established for a fully stochastic zeroth-order bilevel optimization algorithm.</p>