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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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 @math , where @math is the set of non-faulty agents, and @math is agent @math 's local cost function, which is initially known only to agent @math . 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 @math for @math such that @math and @math , the output must be an optimum of the cost function @math . Ideally, we would like @math for all @math -- however, this cannot be guaranteed due to the presence of faulty agents. In fact, we show that the maximum achievable number of nonzero weights ( @math 's) is @math , where @math is the upper bound on the number of Byzantine agents. In addition, we present algorithms that ensure that at least @math agents have weights that are bounded away from 0. We also propose a low-complexity suboptimal algorithm, which ensures that at least @math agents have weights that are bounded away from 0, where @math is the total number of agents, and @math ( @math ) is the actual number of Byzantine agents. We study fault-tolerant distributed optimization of a sum of convex (cost) functions with real-valued scalar input output in the presence of crash faults or Byzantine faults. In particular, the goal is to optimize a global cost function @math , where @math is the collection of agents, and @math is agent @math 's local cost function, which is initially known only to agent @math . Since the above global cost function cannot be optimized exactly in presence of crash faults or Byzantine faults, we define two weaker versions of the problem for crash faults and Byzantine faults, respectively. When some agents may crash, the goal for the weaker problem is to generate an output that is an optimum of a function formed as @math where @math is the set of non-faulty agents, @math is the set of faulty agents (crashed agents), @math for each @math and @math is a normalization constant such that @math . We present an iterative algorithm in which each agent only needs to perform local computation, and send one message per iteration. When some agents may be Byzantine, the system cannot take full advantage of the data kept by non-faulty agents. The goal for the associated weaker problem is to generate an output that is an optimum of a function formed as @math such that @math for each @math and @math . We present an iterative algorithm, where only local computation is needed and only one message per agent is sent in each iteration, that ensures that at least @math agents have weights ( @math 's) that are lower bounded by @math .