Projection-Free Methods for Stochastic Simple Bilevel Optimization with
  Convex Lower-level Problem
Jincheng Cao Ruichen Jiang Nazanin Abolfazli Erfan Yazdandoost Hamedani Aryan Mokhtari
In this paper, we study a class of stochastic bilevel optimization problems,
also known as stochastic simple bilevel optimization, where we minimize a
smooth stochastic objective function over the optimal solution set of another
stochastic convex optimization problem. We introduce novel stochastic bilevel
optimization methods that locally approximate the solution set of the
lower-level problem via a stochastic cutting plane, and then run a conditional
gradient update with variance reduction techniques to control the error induced
by using stochastic gradients. For the case that the upper-level function is
convex, our method requires
$\tilde{\mathcal{O}}(\max\{1/\epsilon_f^{2},1/\epsilon_g^{2}\}) $ stochastic
oracle queries to obtain a solution that is $\epsilon_f$-optimal for the
upper-level and $\epsilon_g$-optimal for the lower-level. This guarantee
improves the previous best-known complexity of
$\mathcal{O}(\max\{1/\epsilon_f^{4},1/\epsilon_g^{4}\})$. Moreover, for the
case that the upper-level function is non-convex, our method requires at most
$\tilde{\mathcal{O}}(\max\{1/\epsilon_f^{3},1/\epsilon_g^{3}\}) $ stochastic
oracle queries to find an $(\epsilon_f, \epsilon_g)$-stationary point. In the
finite-sum setting, we show that the number of stochastic oracle calls required
by our method are $\tilde{\mathcal{O}}(\sqrt{n}/\epsilon)$ and
$\tilde{\mathcal{O}}(\sqrt{n}/\epsilon^{2})$ for the convex and non-convex
settings, respectively, where $\epsilon=\min \{\epsilon_f,\epsilon_g\}$.