ABSTRACT
This paper addresses the problem of distributed multi-agent optimization in which each agent i has a local cost function hi(x), and the goal is to optimize a global cost function that aggregates the local cost functions. Such optimization problems are of interest in many contexts, including distributed machine learning, distributed resource allocation, and distributed robotics.
We consider the distributed optimization problem in the presence of faulty agents. We focus primarily on Byzantine failures, but also briey discuss some results for crash failures. For the Byzantine fault-tolerant optimization problem, the ideal goal is to optimize the average of local cost functions of the non-faulty agents. However, this goal also cannot be achieved. Therefore, we consider a relaxed version of the fault-tolerant optimization problem.
The goal for the relaxed problem is to generate an output that is an optimum of a global cost function formed as a convex combination of local cost functions of the non-faulty agents. More precisely, there must exist weights αi for i∈N such that αi ≥ 0 and ∑i≥ Nαi=1, and the output is an optimum of the cost function ∑i≥ N αihi(x). Ideally, we would like αi=1/|N| for all i≥ N, however, this cannot be guaranteed due to the presence of faulty agents. In fact, the maximum number of nonzero weights (αi's) that can be guaranteed is |N|-f, where f is the maximum number of Byzantine faulty agents.
We present an iterative distributed algorithm that achieves optimal fault-tolerance. Specifically, it ensures that at least |N|-f agents have weights that are bounded away from 0 (in particular, lower bounded by 1/2|N|-f}). The proposed distributed algorithm has a simple iterative structure, with each agent maintaining only a small amount of local state. We show that the iterative algorithm ensures two properties as time goes to ∞: consensus (i.e., output of non-faulty agents becomes identical in the time limit), and optimality (in the sense that the output is the optimum of a suitably defined global cost function).
- I. Abraham, Y. Amit, and D. Dolev. Optimal resilience asynchronous approximate agreement. In Principles of Distributed Systems, pages 229--239. Springer, 2005. Google ScholarDigital Library
- D. P. Bertsekas. Convex Optimization Algorithms. Athena Scientific, 2015.Google Scholar
- S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn., 3(1):1--122, Jan. 2011. Google ScholarDigital Library
- S. Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 2004. Google ScholarDigital Library
- S. Chaudhuri. More choices allow more faults: Set consensus problems in totally asynchronous systems. Information and Computation, 105:132--158, 1992. Google ScholarDigital Library
- J. Chen and A. Sayed. Diffusion adaptation strategies for distributed optimization and learning over networks. Signal Processing, IEEE Transactions on, 60(8):4289--4305, August 2012. Google ScholarDigital Library
- D. Dolev, N. A. Lynch, S. S. Pinter, E. W. Stark, and W. E. Weihl. Reaching approximate agreement in the presence of faults. J. ACM, 33(3):499--516, May 1986. Google ScholarDigital Library
- D. Dolev, N. A. Lynch, S. S. Pinter, E. W. Stark, and W. E. Weihl. Reaching approximate agreement in the presence of faults. Journal of the ACM (JACM), 33(3):499--516, 1986. Google ScholarDigital Library
- J. Duchi, A. Agarwal, and M. Wainwright. Dual averaging for distributed optimization: Convergence analysis and network scaling. Automatic Control, IEEE Transactions on, 57(3):592--606, March 2012.Google Scholar
- A. D. Fekete. Asymptotically optimal algorithms for approximate agreement. Distributed Computing, 4(1):9--29, 1990.Google ScholarCross Ref
- R. Friedman, A. Mostefaoui, S. Rajsbaum, and M. Raynal. Asynchronous agreement and its relation with error-correcting codes. Computers, IEEE Transactions on, 56(7):865--875, 2007. Google ScholarDigital Library
- M. Herlihy, S. Rajsbaum, M. Raynal, and J. Stainer. Computing in the presence of concurrent solo executions. In LATIN 2014: Theoretical Informatics, pages 214--225. Springer Berlin Heidelberg, 2014.Google Scholar
- B. Johansson. On distributed optimization in networked systems. 2008.Google Scholar
- H. J. LeBlanc, H. Zhang, S. Sundaram, and X. Koutsoukos. Consensus of multi-agent networks in the presence of adversaries using only local information. In Proceedings of the 1st International Conference on High Confidence Networked Systems, HiCoNS '12, pages 1--10, New York, NY, USA, 2012. ACM. Google ScholarDigital Library
- I. Lobel and A. Ozdaglar. Distributed subgradient methods for convex optimization over random networks. Automatic Control, IEEE Transactions on, 56(6):1291--1306, June 2011.Google Scholar
- N. A. Lynch. Distributed Algorithms. Morgan Kaufmann, 1996. Google ScholarDigital Library
- H. Mendes and M. Herlihy. Multidimensional approximate agreement in byzantine asynchronous systems. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 391--400, New York, NY, USA, 2013. ACM. Google ScholarDigital Library
- A. Mostefaoui, S. Rajsbaum, and M. Raynal. Conditions on input vectors for consensus solvability in asynchronous distributed systems. Journal of the ACM (JACM), 50(6):922--954, 2003. Google ScholarDigital Library
- A. Nedic and A. Ozdaglar. Distributed subgradient methods for multi-agent optimization. Automatic Control, IEEE Transactions on, 54(1):48--61, Jan 2009.Google Scholar
- Y. Nesterov. Introductory lectures on convex optimization, volume 87. Springer Science & Business Media, 2004. Google ScholarDigital Library
- H. Robbins and D. Siegmund. A convergence theorem for non negative almost supermartingales and some applications. In T. Lai and D. Siegmund, editors, Herbert Robbins Selected Papers, pages 111--135. Springer New York, 1985.Google ScholarCross Ref
- L. Su and N. H. Vaidya. Byzantine multi-agent optimization: Part I. arXiv preprint arXiv:1506.04681, 2015.Google Scholar
- L. Su and N. H. Vaidya. Byzantine multi-agent optimization: Part II. CoRR, abs/1507.01845, 2015.Google Scholar
- L. Su and N. H. Vaidya. Byzantine multi-agent optimization: Part III. Technical Report, 2015.Google Scholar
- L. Su and N. H. Vaidya. Fault-tolerant distributed optimization (Part IV): Constrained optimization with arbitrary directed networks. arXiv preprint arXiv:1511.01821, 2015.Google Scholar
- L. Su and N. H. Vaidya. Multi-agent optimization in the presence of byzantine adversaries: Fundamental limits. In Proceedings of IEEE American Control Conference (ACC), July, 2016.Google ScholarCross Ref
- L. Su and N. H. Vaidya. Fault-tolerant multi-agent optimization: Optimal distributed algorithms (full version). In Technical Report, May, 2016.Google Scholar
- S. Sundaram and B. Gharesifard. Consensus-based distributed optimization with malicious nodes. In Proceedings of the 53rd Annual Allerton Conference on Communication, Control and Computing. IEEE, 2015.Google ScholarCross Ref
- L. Tseng and N. H. Vaidya. Iterative approximate byzantine consensus under a generalized fault model. In Distributed Computing and Networking, pages 72--86. Springer, 2013.Google ScholarCross Ref
- L. Tseng and N. H. Vaidya. Iterative approximate consensus in the presence of byzantine link failures. In Networked Systems, pages 84--98. Springer International Publishing, 2014.Google ScholarCross Ref
- K. I. Tsianos, S. Lawlor, and M. G. Rabbat. Push-sum distributed dual averaging for convex optimization. In Decision and Control (CDC), 2012 IEEE 51st Annual Conference on, pages 5453--5458, Dec 2012.Google ScholarCross Ref
- J. Tsitsiklis, D. Bertsekas, and M. Athans. Distributed asynchronous deterministic and stochastic gradient optimization algorithms. Automatic Control, IEEE Transactions on, 31(9):803--812, Sep 1986.Google Scholar
- J. N. Tsitsiklis. Problems in decentralized decision making and computation. Technical report, DTIC Document, 1984.Google Scholar
- N. H. Vaidya. Iterative byzantine vector consensus in incomplete graphs. In Distributed Computing and Networking, pages 14--28. Springer, 2014. Google ScholarDigital Library
- N. H. Vaidya and V. K. Garg. Byzantine vector consensus in complete graphs. In Proceedings of the 2013 ACM symposium on Principles of distributed computing, pages 65--73. ACM, 2013. Google ScholarDigital Library
- N. H. Vaidya, L. Tseng, and G. Liang. Iterative approximate byzantine consensus in arbitrary directed graphs. In Proceedings of the 2012 ACM symposium on Principles of distributed computing, pages 365--374. ACM, 2012. Google ScholarDigital Library
Index Terms
- Fault-Tolerant Multi-Agent Optimization: Optimal Iterative Distributed Algorithms
Recommendations
Approximate Byzantine Fault-Tolerance in Distributed Optimization
PODC'21: Proceedings of the 2021 ACM Symposium on Principles of Distributed ComputingThis paper considers the problem of Byzantine fault-tolerance in distributed multi-agent optimization. In this problem, each agent has a local cost function, and in the fault-free case, the goal is to design a distributed algorithm that allows all the ...
Location functions for self-stabilizing byzantine tolerant swarms
AbstractThis paper proposes a novel framework to realize self-stabilizing Byzantine tolerant swarms. In this framework, non-Byzantine robots execute tasks while satisfying location functions. That is, the robots use a policy for their location choice, ...
Highlights- A novel framework to realize self-stabilizing Byzantine tolerant swarms.
- Non-Byzantine robots execute tasks while satisfying location functions.
- Byzantine-resilient self-stabilizing algorithm based on the location function.
- ...
Comments