  We consider the communication complexity of some fundamental convex
optimization problems in the point-to-point (coordinator) and blackboard
communication models. We strengthen known bounds for approximately solving
linear regression, $p$-norm regression (for $1\leq p\leq 2$), linear
programming, minimizing the sum of finitely many convex nonsmooth functions
with varying supports, and low rank approximation; for a number of these
fundamental problems our bounds are nearly optimal, as proven by our lower
bounds.
  Among our techniques, we use the notion of block leverage scores, which have
been relatively unexplored in this context, as well as dropping all but the
``middle" bits in Richardson-style algorithms. We also introduce a new
communication problem for accurately approximating inner products and establish
a lower bound using the spherical Radon transform. Our lower bound can be used
to show the first separation of linear programming and linear systems in the
distributed model when the number of constraints is polynomial, addressing an
open question in prior work.
