Byzantine Fault-Tolerant Min-Max Optimization
Shuo Liu, Nitin Vaidya
TL;DR
The paper addresses Byzantine fault-tolerant min-max optimization, where up to $f$ of $n$ cost functions may be arbitrary adversaries, and aims to minimize the worst-case non-faulty cost. It introduces a centralized, rank-based approach using $g_{f+1}(x)=\mathrm{rank}_{f+1}(Q_i(x))$ with bounds relating to non-faulty costs, and also offers a Lipschitz-based approximate method inspired by DIRECT to trade exactness for scalability. To support distributed settings, it develops a DIRECT-based distributed algorithm with convergence guarantees, showing that sampling becomes dense in the feasible region and that the output is bounded between $\min_x h_{f+1}(x)$ and $\min_x h_1(x)$ up to a Lipschitz-determined error that vanishes as the search refines. The work highlights the difficulties of gradient-descent approaches in this context and lays a foundation for further efficient, fault-tolerant distributed min-max optimization methods.
Abstract
In this paper, we consider a min-max optimization problem under adversarial manipulation, where there are $n$ cost functions, up to $f$ of which may be replaced by arbitrary faulty functions by an adversary. The goal is to minimize the maximum cost over $x$ among the $n$ functions despite the faulty functions. The problem formulation could naturally extend to Byzantine fault-tolerant distributed min-max optimization. We present a simple algorithm for Byzantine min-max optimization, and provide bounds on the output of the algorithm. We also present an approximate algorithm for this problem. We then extend the problem to a distributed setting and present a distributed algorithm. To the best of our knowledge, we are the first to consider this problem.