30,266 research outputs found
Distributed Optimization: Convergence Conditions from a Dynamical System Perspective
This paper explores the fundamental properties of distributed minimization of
a sum of functions with each function only known to one node, and a
pre-specified level of node knowledge and computational capacity. We define the
optimization information each node receives from its objective function, the
neighboring information each node receives from its neighbors, and the
computational capacity each node can take advantage of in controlling its
state. It is proven that there exist a neighboring information way and a
control law that guarantee global optimal consensus if and only if the solution
sets of the local objective functions admit a nonempty intersection set for
fixed strongly connected graphs. Then we show that for any tolerated error, we
can find a control law that guarantees global optimal consensus within this
error for fixed, bidirectional, and connected graphs under mild conditions. For
time-varying graphs, we show that optimal consensus can always be achieved as
long as the graph is uniformly jointly strongly connected and the nonempty
intersection condition holds. The results illustrate that nonempty intersection
for the local optimal solution sets is a critical condition for successful
distributed optimization for a large class of algorithms
Nested Distributed Gradient Methods with Adaptive Quantized Communication
In this paper, we consider minimizing a sum of local convex objective
functions in a distributed setting, where communication can be costly. We
propose and analyze a class of nested distributed gradient methods with
adaptive quantized communication (NEAR-DGD+Q). We show the effect of performing
multiple quantized communication steps on the rate of convergence and on the
size of the neighborhood of convergence, and prove R-Linear convergence to the
exact solution with increasing number of consensus steps and adaptive
quantization. We test the performance of the method, as well as some practical
variants, on quadratic functions, and show the effects of multiple quantized
communication steps in terms of iterations/gradient evaluations, communication
and cost.Comment: 9 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1709.0299
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