  We study the question of whether parallelization in the exploration of the
feasible set can be used to speed up convex optimization, in the local oracle
model of computation. We show that the answer is negative for both
deterministic and randomized algorithms applied to essentially any of the
interesting geometries and nonsmooth, weakly-smooth, or smooth objective
functions. In particular, we show that it is not possible to obtain a
polylogarithmic (in the sequential complexity of the problem) number of
parallel rounds with a polynomial (in the dimension) number of queries per
round. In the majority of these settings and when the dimension of the space is
polynomial in the inverse target accuracy, our lower bounds match the oracle
complexity of sequential convex optimization, up to at most a logarithmic
factor in the dimension, which makes them (nearly) tight. Prior to our work,
lower bounds for parallel convex optimization algorithms were only known in a
small fraction of the settings considered in this paper, mainly applying to
Euclidean ($\ell_2$) and $\ell_\infty$ spaces. Our work provides a more general
approach for proving lower bounds in the setting of parallel convex
optimization.
