  We revisit first-order optimization under local information constraints such
as local privacy, gradient quantization, and computational constraints limiting
access to a few coordinates of the gradient. In this setting, the optimization
algorithm is not allowed to directly access the complete output of the gradient
oracle, but only gets limited information about it subject to the local
information constraints.
  We study the role of adaptivity in processing the gradient output to obtain
this limited information from it.We consider optimization for both convex and
strongly convex functions and obtain tight or nearly tight lower bounds for the
convergence rate, when adaptive gradient processing is allowed. Prior work was
restricted to convex functions and allowed only nonadaptive processing of
gradients. For both of these function classes and for the three information
constraints mentioned above, our lower bound implies that adaptive processing
of gradients cannot outperform nonadaptive processing in most regimes of
interest. We complement these results by exhibiting a natural optimization
problem under information constraints for which adaptive processing of gradient
strictly outperforms nonadaptive processing.
