In this paper, we study and analyze zeroth-order stochastic approximation algorithms for solving bilvel 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 used 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.
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