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.
Concurrently, @cite_8 looked at a similar problem with different focuses, where the faulty agents are restricted to broadcasting their messages (sending identical messages) to their outgoing neighbors, and the global objective is simply a convex combination of local cost functions at the non-faulty agents. Their algorithm performance is equivalent to simply running iterative Byzantine consensus on the local optima. In addition, their results are based on the assumption that every update matrix has a common left eigenvector corresponding to eigenvalue 1.
@cite_8
1970063032 2122078451 2114791779 1997250627
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 . We present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity. Our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem, where the global objective function is a combination of local agent objective functions. Our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets. To highlight the effects of constraints, we first consider a constrained consensus problem and present a distributed "projected consensus algorithm" in which agents combine their local averaging operation with projection on their individual constraint sets. This algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents. We establish convergence and convergence rate results for the projected consensus algorithm. We next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets. We present a distributed "projected subgradient algorithm" which involves each agent performing a local averaging operation, taking a subgradient step to minimize its own objective function, and projecting on its constraint set. We show that, with an appropriately selected stepsize rule, the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal, and when the weights are time-varying but all agents have the same constraint set. We present a distributed proximal-gradient method for optimizing the average of convex functions, each of which is the private local objective of an agent in a network with time-varying topology. The local objectives have distinct differentiable components, but they share a common nondifferentiable component, which has a favorable structure suitable for effective computation of the proximal operator. In our method, each agent iteratively updates its estimate of the global minimum by optimizing its local objective function, and exchanging estimates with others via communication in the network. Using Nesterov-type acceleration techniques and multiple communication steps per iteration, we show that this method converges at the rate 1 k (where k is the number of communication rounds between the agents), which is faster than the convergence rate of the existing distributed methods for solving this problem. The superior convergence rate of our method is also verified by numerical experiments.