  Two-level stochastic optimization formulations have become instrumental in a
number of machine learning contexts such as continual learning, neural
architecture search, adversarial learning, and hyperparameter tuning. Practical
stochastic bilevel optimization problems become challenging in optimization or
learning scenarios where the number of variables is high or there are
constraints.
  In this paper, we introduce a bilevel stochastic gradient method for bilevel
problems with nonlinear and possibly nonconvex lower-level constraints. We also
present a comprehensive convergence theory that addresses both the lower-level
unconstrained and constrained cases and covers all inexact calculations of the
adjoint gradient (also called hypergradient), such as the inexact solution of
the lower-level problem, inexact computation of the adjoint formula (due to the
inexact solution of the adjoint equation or use of a truncated Neumann series),
and noisy estimates of the gradients, Hessians, and Jacobians involved. To
promote the use of bilevel optimization in large-scale learning, we have
developed new low-rank practical bilevel stochastic gradient methods (BSG-N-FD
and~BSG-1) that do not require second-order derivatives and, in the lower-level
unconstrained case, dismiss any matrix-vector products.
