We study the problem of constrained distributed optimization in multi-agent networks when some of the computing agents may be faulty. In this problem, the system goal is to have all the non-faulty agents collectively minimize a global objective given by weighted average of local cost functions, each of which is initially known to a non-faulty agent only. In particular, we are interested in the scenario when the computing agents are connected by an arbitrary directed communication network, some of the agents may suffer from crash faults or Byzantine faults, and the estimate of each agent is restricted to lie in a common constraint set. This problem finds its applications in social computing and distributed large-scale machine learning. The fault-tolerant multi-agent optimization problem was first formulated by Su and Vaidya, and is solved when the local functions are defined over the whole real line, and the networks are fully-connected. In this report, we consider arbitrary directed communication networks and focus on the scenario where, local estimates at the non-faulty agents are constrained, and only local communication and minimal memory carried across iterations are allowed. In particular, we generalize our previous results on fully-connected networks and unconstrained optimization to arbitrary directed networks and constrained optimization. As a byproduct, we provide a matrix representation for iterative approximate crash consensus. The matrix representation allows us to characterize the convergence rate for crash iterative consensus.
We considered Byzantine faults and crash faults in @cite_1 @cite_30 @cite_20 . In particular, @cite_1 @cite_30 considered Byzantine faults under synchronous systems, and @cite_20 considered both Byzantine faults and crash faults under synchronous systems, with results partially generalizable to asynchronous systems. It is showed in @cite_1 under Byzantine faults that at most @math non-faulty functions can have non-zero weights. This observation led to the formulation of Problem 2 in ). Six algorithms were proposed in @cite_1 . Algorithms with alternative structure, where only local communication is needed, is proposed in @cite_20 for crash faults and Byzantine faults, respectively. We showed in @cite_30 that when there are sufficient redundancy in the input functions (each input function is not exclusively kept by a single agent), it is possible to solve ), where the summation is taken over all input functions. In addition, a simple low-complexity iterative algorithm was proposed in @cite_30 , and a tight topological condition for the existence of such iterative algorithms is identified.
@cite_30 @cite_1 @cite_20
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This paper proves a necessary and sufficient condition for the existence of iterative, algorithms that achieve approximate Byzantine consensus in arbitrary directed graphs, where each directed edge represents a communication channel between a pair of nodes. The class of iterative algorithms considered in this paper ensures that, after each iteration of the algorithm, the state of each fault-free node remains in the convex hull of the states of the fault-free nodes at the end of the previous iteration. The following convergence requirement is imposed: for any e > 0, after a sufficiently large number of iterations, the states of the fault-free nodes are guaranteed to be within e of each other. To the best of our knowledge, tight necessary and sufficient conditions for the existence of such iterative consensus algorithms in synchronous arbitrary point-to-point networks in presence of Byzantine faults, have not been developed previously. The methodology and results presented in this paper can also be extended to asynchronous systems. The problem of e-approximate agreement in Byzantine asynchronous systems is well-understood when all values lie on the real line. In this paper, we generalize the problem to consider values that lie in Rm, for m ≥ 1, and present an optimal protocol in regard to fault tolerance. Our scenario is the following. Processes start with values in Rm, for m ≥ 1, and communicate via message-passing. The system is asynchronous: there is no upper bound on processes' relative speeds or on message delay. Some faulty processes can display arbitrarily malicious (i.e. Byzantine) behavior. Non-faulty processes must decide on values that are: (1) in Rm; (2) within distance e of each other; and (3) in the convex hull of the non-faulty processes' inputs. We give an algorithm with a matching lower bound on fault tolerance: we require n > t(m+2), where n is the number of processes, t is the number of Byzantine processes, and input and output values reside in Rm. Non-faulty processes send O(n2 d log(m e max δ(d): 1 ≤ d ≤ m )) messages in total, where δ(d) is the range of non-faulty inputs projected at coordinate d. The Byzantine processes do not affect the algorithm's running time. Much of the past work on asynchronous approximate Byzantine consensus has assumed scalar inputs at the nodes [4, 8]. Recent work has yielded approximate Byzantine consensus algorithms for the case when the input at each node is a d-dimensional vector, and the nodes must reach consensus on a vector in the convex hull of the input vectors at the fault-free nodes [9, 13]. The d-dimensional vectors can be equivalently viewed as points in the d-dimensional Euclidean space. Thus, the algorithms in [9, 13] require the fault-free nodes to decide on a point in the d-dimensional space. In our recent work [12], we proposed a generalization of the consensus problem, namely Byzantine convex consensus (BCC), which allows the decision to be a convex polytope in the d-dimensional space, such that the decided polytope is within the convex hull of the input vectors at the fault-free nodes. We also presented an asynchronous approximate BCC algorithm. In this paper, we propose a new BCC algorithm with optimal fault-tolerance that also agrees on a convex polytope that is as large as possible under adversarial conditions. Our prior work [12] does not guarantee the optimality of the output polytope. This paper investigates the task solvability of mobile robot systems subject to Byzantine faults. We first consider the gathering problem, which requires all robots to meet in finite time at a non-predefined location. It is known that the solvability of Byzantine gathering strongly depends on a number of system attributes, such as synchrony, the number of Byzantine robots, scheduling strategy, obliviousness, orientation of local coordinate systems and so on. However, the complete characterization of the attributes making Byzantine gathering solvable still remains open. In this paper, we show strong impossibility results of Byzantine gathering. Namely, we prove that Byzantine gathering is impossible even if we assume one Byzantine fault, an atomic execution system, the n-bounded centralized scheduler, non-oblivious robots, instantaneous movements and a common orientation of local coordinate systems (where n denote the number of correct robots). Those hypotheses are much weaker than used in previous work, inducing a much stronger impossibility result. At the core of our impossibility result is a reduction from the distributed consensus problem in asynchronous shared-memory systems. In more details, we newly construct a generic reduction scheme based on the distributed BG-simulation. Interestingly, because of its versatility, we can easily extend our impossibility result for general pattern formation problems.