Zeroth-Order Federated Methods for Stochastic MPECs and Nondifferentiable Nonconvex Hierarchical Optimization

Motivated by the emergence of federated learning (FL), we design and analyze federated methods for addressing: (i) Nondifferentiable nonconvex optimization; (ii) Bilevel optimization; (iii) Minimax problems; and (iv) Two-stage stochastic mathematical programs with equilibrium constraints (2s-SMPEC). Research on these problems has been limited and afflicted by reliance on strong assumptions, including the need for differentiability of the implicit function and the absence of constraints in the lower-level problem, among others. We make the following contributions. In (i), by leveraging convolution-based smoothing and Clarke's subdifferential calculus, we devise a randomized smoothing-enabled zeroth-order FL method and derive communication and iteration complexity guarantees for computing an approximate Clarke stationary point. To contend with (ii) and (iii), we devise a unifying randomized implicit zeroth-order FL framework, equipped with explicit communication and iteration complexities. Importantly, our method utilizes delays during local steps to skip calls to the inexact lower-level FL oracle. This results in significant reduction in communication overhead. In (iv), we devise an inexact implicit variant of the method in (i). Remarkably, this method achieves a total communication complexity matching that of single-level nonsmooth nonconvex optimization in FL. We empirically validate the theoretical findings on instances of federated nonsmooth and hierarchical problems.