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The paper introduces a distributed continuous-time gradient flow method, called Dist-AGM, that aims to minimize the sum of smooth convex functions. Dist-AGM achieves an unprecedented convergence rate of O(1/t^(2-β)), where β > 0 can be arbitrarily small.
The authors establish an energy conservation perspective on optimization algorithms, where the associated energy functional remains conserved within a dilated coordinate system. This generalized framework can be used to analyze the convergence rates of a wide range of distributed optimization algorithms.
The authors provide a consistent rate-matching discretization of Dist-AGM using the Symplectic Euler method, ensuring that the discretized algorithm achieves a convergence rate of O(1/k^(2-β)), where k represents the number of iterations.
Experimental results demonstrate the accelerated convergence behavior of the proposed distributed optimization algorithm, particularly on problems with poor condition numbers.
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