Quantization Design for Distributed Optimization

• Published in: IEEE transactions on automatic control Vol. 62; no. 5; pp. 2107 - 2120
• Main Authors: Ye Pu, Zeilinger, Melanie N., Jones, Colin N.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-05-2017; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Acceleration; Algorithm design and analysis; Algorithms; Complexity theory; Computer networks; Convergence; Convex functions; Counters; Design optimization; Distributed processing; Inexact proximal- gradient method (inexact PGM); iterative shrinkage-thresholding algorithm (ISTA); Measurement; Optimal control; Optimization; Quantization (signal); Upper bounds
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2016.2600597

We consider the problem of solving a distributed optimization problem using a distributed computing platform, where the communication in the network is limited: each node can only communicate with its neighbors and the channel has a limited data-rate. A common technique to address the latter limitation is to apply quantization to the exchanged information. We propose two distributed optimization algorithms with an iteratively refining quantization design based on the inexact proximal gradient method and its accelerated variant. We show that if the parameters of the quantizers, i.e., the number of bits and the initial quantization intervals, satisfy certain conditions, then the quantization error is bounded by a linearly decreasing function and the convergence of the distributed algorithms is guaranteed. Furthermore, we prove that after imposing the quantization scheme, the distributed algorithms still exhibit a linear convergence rate, and show complexity upper-bounds on the number of iterations to achieve a given accuracy. Finally, we demonstrate the performance of the proposed algorithms and the theoretical findings for solving a distributed optimal control problem.