[Paper Review] Byzantine Multi-Agent Optimization: Part I

Lili Su, Nitin H. Vaidya|arXiv (Cornell University)|Jun 15, 2015

Islamic Finance and Banking Studies19 citations

TL;DR

This paper addresses Byzantine fault-tolerant distributed optimization in multi-agent systems, where up to f agents may behave arbitrarily. It proposes a suboptimal algorithm that ensures at least ⌈n/2⌉−ϕ non-faulty agents have weights bounded away from zero in a convex combination of local cost functions, achieving a consensus optimum despite Byzantine faults. The key contribution is a tight bound on the number of non-faulty agents that can influence the solution, with a practical algorithm matching this bound under realistic assumptions.

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 $α_i$ for $i\in \mathcal{N}$ such that $α_i\geq 0$ and $\sum_{i\in \mathcal{N}}α_i=1$, the output must be an optimum of the cost function $\sum_{i\in \mathcal{N}} α_ih_i(x)$. Ideally, we would like $α_i=\frac{1}{|\mathcal{N}|}$ 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 ($α_i$'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 $\lceil \frac{n}{2} ceil-ϕ$ agents have weights that are bounded away from 0, where $n$ is the total number of agents, and $ϕ$ ($ϕ\le f$) is the actual number of Byzantine agents.

Motivation & Objective

• To address the challenge of distributed optimization in multi-agent systems where up to f agents may be Byzantine faulty and behave arbitrarily.
• To define a weaker optimization problem that remains solvable under Byzantine faults, since exact average optimization of non-faulty agents' cost functions is impossible.
• To maximize the number of non-faulty agents whose local cost functions non-trivially influence the global solution in a convex combination.
• To design a low-complexity algorithm that guarantees a minimum number of non-faulty agents have weights bounded away from zero in the solution.
• To establish theoretical bounds on the maximum number of non-faulty agents that can be meaningfully included in the optimization, showing it is at most |N|−f.

Proposed method

• The problem is reformulated as Problem 2, where the output must be an optimum of a convex combination ∑ᵢ∈N αᵢhᵢ(x), with αᵢ ≥ 0 and ∑ᵢ∈N αᵢ = 1.
• A novel algorithm is proposed that uses a median-based approach to estimate the solution, leveraging the fact that non-faulty agents' values are bounded and consistent.
• The algorithm ensures that at least ⌈n/2⌉−ϕ non-faulty agents have weights bounded away from zero, by analyzing the subgradient signs at the candidate solution.
• The solution is derived by partitioning agents into sets based on their value estimates: A(x̃) for those with positive subgradient sum, B(x̃) for negative, and C(x̃) for zero.
• A convex combination is formed using weights proportional to the size of these sets and a balancing parameter ζ ∈ [0,1], ensuring the subgradient condition is satisfied.
• The analysis proves that the resulting solution is an optimum of a weighted sum, with at least ⌈n/2⌉−ϕ non-faulty agents contributing with weights bounded below by 1/(2|N|).

Experimental results

Research questions

• RQ1 Can exact optimization of the average of non-faulty agents' cost functions be achieved in the presence of Byzantine faults?
• RQ2 What is the maximum number of non-faulty agents that can non-trivially influence the solution in a Byzantine-tolerant distributed optimization setting?
• RQ3 Is it possible to design a low-complexity algorithm that ensures a significant number of non-faulty agents contribute meaningfully to the solution despite Byzantine behavior?
• RQ4 Can the solution be guaranteed to be an optimum of a convex combination of non-faulty agents' cost functions, even when the identities of faulty agents are unknown?
• RQ5 What is the theoretical lower bound on the number of non-faulty agents whose cost functions can be meaningfully included in the solution?

Key findings

• The maximum number of non-faulty agents that can have non-zero weights in the convex combination is bounded by |N|−f, where f is the upper bound on Byzantine agents.
• The proposed algorithm ensures that at least ⌈n/2⌉−ϕ non-faulty agents have weights bounded away from zero, where ϕ is the actual number of Byzantine agents.
• The solution is guaranteed to be an optimum of a convex combination ∑ᵢ∈N αᵢhᵢ(x), with ∑ᵢ∈N αᵢ = 1 and αᵢ ≥ 0.
• The algorithm achieves this result with low computational complexity by using median-based estimation and subgradient sign analysis.
• The theoretical analysis proves that the solution is optimal for a weighted sum, even when the faulty agents' behavior is arbitrary and adversarial.
• The result holds under the condition that n > 3f, ensuring sufficient non-faulty agents to dominate the consensus process.