6 research outputs found
Towards Tight Communication Lower Bounds for Distributed Optimisation
We consider a standard distributed optimisation setting where machines,
each holding a -dimensional function , aim to jointly minimise the sum
of the functions . This problem arises naturally in
large-scale distributed optimisation, where a standard solution is to apply
variants of (stochastic) gradient descent. We focus on the communication
complexity of this problem: our main result provides the first fully
unconditional bounds on total number of bits which need to be sent and received
by the machines to solve this problem under point-to-point communication,
within a given error-tolerance. Specifically, we show that total bits need to be communicated between the machines to find
an additive -approximation to the minimum of . The result holds for both deterministic and randomised algorithms, and,
importantly, requires no assumptions on the algorithm structure. The lower
bound is tight under certain restrictions on parameter values, and is matched
within constant factors for quadratic objectives by a new variant of quantised
gradient descent, which we describe and analyse. Our results bring over tools
from communication complexity to distributed optimisation, which has potential
for further applications
Communication-Efficient Distributed Optimization with Quantized Preconditioners
We investigate fast and communication-efficient algorithms for the classic
problem of minimizing a sum of strongly convex and smooth functions that are
distributed among different nodes, which can communicate using a limited
number of bits. Most previous communication-efficient approaches for this
problem are limited to first-order optimization, and therefore have
\emph{linear} dependence on the condition number in their communication
complexity. We show that this dependence is not inherent:
communication-efficient methods can in fact have sublinear dependence on the
condition number. For this, we design and analyze the first
communication-efficient distributed variants of preconditioned gradient descent
for Generalized Linear Models, and for Newton's method. Our results rely on a
new technique for quantizing both the preconditioner and the descent direction
at each step of the algorithms, while controlling their convergence rate. We
also validate our findings experimentally, showing fast convergence and reduced
communication
Robust Lower Bounds for Graph Problems in the Blackboard Model of Communication
We give lower bounds on the communication complexity of graph problems in the multi-party blackboard model. In this model, the edges of an -vertex input graph are partitioned among parties, who communicate solely by writing messages on a shared blackboard that is visible to every party. We show that any non-trivial graph problem on -vertex graphs has blackboard communication complexity bits, even if the edges of the input graph are randomly assigned to the parties. We say that a graph problem is non-trivial if the output cannot be computed in a model where every party holds at most one edge and no communication is allowed. Our lower bound thus holds for essentially all key graph problems relevant to distributed computing, including Maximal Independent Set (MIS), Maximal Matching, ()-coloring, and Dominating Set. In many cases, e.g., MIS, Maximal Matching, and -coloring, our lower bounds are optimal, up to poly-logarithmic factors