{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/byzantine-multi-agent-optimization-part-i","title":"Byzantine Multi-Agent Optimization: Part I","arxiv_id":"1506.04681","date":"2015-06-15","proceeding":null,"authors":["Lili Su","Nitin Vaidya"],"abstract":"We study Byzantine fault-tolerant distributed optimization of a sum of convex (cost) functions with real-valued scalar input/ouput. In particular, the goal is to optimize a global cost function $\\frac{1}{|\\mathcal{N}|}\\sum_{i\\in \\mathcal{N}} h_i(x)$, where $\\mathcal{N}$ is the set of non-faulty agents, and $h_i(x)$ is agent $i$'s local cost function, which is initially known only to agent $i$. In general, when some of the agents may be Byzantine faulty, the above goal is unachievable, because the identity of the faulty agents is not necessarily known to the non-faulty agents, and the faulty agents may behave arbitrarily. Since the above global cost function cannot be optimized exactly in presence of Byzantine agents, we define a weaker version of the problem. The goal for the weaker problem is to generate an output that is an optimum of a function formed as a convex combination of local cost functions of the non-faulty agents. More precisely, for some choice of weights $\\alpha_i$ for $i\\in \\mathcal{N}$ such that $\\alpha_i\\geq 0$ and $\\sum_{i\\in \\mathcal{N}}\\alpha_i=1$, the output must be an optimum of the cost function $\\sum_{i\\in \\mathcal{N}} \\alpha_ih_i(x)$. Ideally, we would like $\\alpha_i=\\frac{1}{|\\mathcal{N}|}$ for all $i\\in \\mathcal{N}$ -- however, this cannot be guaranteed due to the presence of faulty agents. In fact, we show that the maximum achievable number of nonzero weights ($\\alpha_i$'s) is $|\\mathcal{N}|-f$, where $f$ is the upper bound on the number of Byzantine agents. In addition, we present algorithms that ensure that at least $|\\mathcal{N}|-f$ agents have weights that are bounded away from 0. We also propose a low-complexity suboptimal algorithm, which ensures that at least $\\lceil \\frac{n}{2}\\rceil-\\phi$ agents have weights that are bounded away from 0, where $n$ is the total number of agents, and $\\phi$ ($\\phi\\le f$) is the actual number of Byzantine agents.","url_abs":"http://arxiv.org/abs/1506.04681v2","url_pdf":"http://arxiv.org/pdf/1506.04681v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"byzantine-multi-agent-optimization-part-i","repo_url":"https://github.com/kkuwaran/resilient-distributed-optimization","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}