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Register a new userEscaping Local Optima in a Class of Multi-Agent Distributed Optimization Problems: A Boosting Function Approach
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We address the problem of multiple local optima commonly arising in optimization problems for multi-agent systems, where objective functions are nonlinear and nonconvex. For the class of coverage control problems, we propose a systematic approach for escaping a local optimum, rather than randomly perturbing controllable variables away from it. We show that the objective function for these problems can be decomposed to facilitate the evaluation of the local partial derivative of each node in the system and to provide insights into its structure. This structure is exploited by defining boosting functions applied to the aforementioned local partial derivative at an equilibrium point where its value is zero so as to transform it in a way that induces nodes to explore poorly covered areas of the mission space until a new equilibrium point is reached. The proposed boosting process ensures that, at its conclusion, the objective function is no worse than its pre-boosting value. However, the global optima cannot be guaranteed. We define three families of boosting functions with different properties and provide simulation results illustrating how this approach improves the solutions obtained for this class of distributed optimization problems.
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This work develops effective distributed strategies for the solution of constrained multi-agent stochastic optimization problems with coupled parameters across the agents. In this formulation, each agent is influenced by only a subset of the entries of a global parameter vector or model, and is subject to convex constraints that are only known locally. Problems of this type arise in several applications, most notably in disease propagation models, minimum-cost flow problems, distributed control formulations, and distributed power system monitoring. This work focuses on stochastic settings, where a stochastic risk function is associated with each agent and the objective is to seek the minimizer of the aggregate sum of all risks subject to a set of constraints. Agents are not aware of the statistical distribution of the data and, therefore, can only rely on stochastic approximations in their learning strategies. We derive an effective distributed learning strategy that is able to track drifts in the underlying parameter model. A detailed performance and stability analysis is carried out showing that the resulting coupled diffusion strategy converges at a linear rate to an $O(mu)-$neighborhood of the true penalized optimizer.
We consider the optimal coverage problem where a multi-agent network is deployed in an environment with obstacles to maximize a joint event detection probability. The objective function of this problem is non-convex and no global optimum is guaranteed by gradient-based algorithms developed to date. We first show that the objective function is monotone submodular, a class of functions for which a simple greedy algorithm is known to be within 0.63 of the optimal solution. We then derive two tighter lower bounds by exploiting the curvature information (total curvature and elemental curvature) of the objective function. We further show that the tightness of these lower bounds is complementary with respect to the sensing capabilities of the agents. The greedy algorithm solution can be subsequently used as an initial point for a gradient-based algorithm to obtain solutions even closer to the global optimum. Simulation results show that this approach leads to significantly better performance relative to previously used algorithms.
This work studies multi-agent sharing optimization problems with the objective function being the sum of smooth local functions plus a convex (possibly non-smooth) function coupling all agents. This scenario arises in many machine learning and engineering applications, such as regression over distributed features and resource allocation. We reformulate this problem into an equivalent saddle-point problem, which is amenable to decentralized solutions. We then propose a proximal primal-dual algorithm and establish its linear convergence to the optimal solution when the local functions are strongly-convex. To our knowledge, this is the first linearly convergent decentralized algorithm for multi-agent sharing problems with a general convex (possibly non-smooth) coupling function.
In this work, we first consider distributed convex constrained optimization problems where the objective function is encoded by multiple local and possibly nonsmooth objectives privately held by a group of agents, and propose a distributed subgradient method with double averaging (abbreviated as ${rm DSA_2}$) that only requires peer-to-peer communication and local computation to solve the global problem. The algorithmic framework builds on dual methods and dynamic average consensus; the sequence of test points is formed by iteratively minimizing a local dual model of the overall objective where the coefficients, i.e., approximated subgradients of the objective, are supplied by the dynamic average consensus scheme. We theoretically show that ${rm DSA_2}$ enjoys non-ergodic convergence properties, i.e., the local minimizing sequence itself is convergent, a distinct feature that cannot be found in existing results. Specifically, we establish a convergence rate of $O(frac{1}{sqrt{t}})$ in terms of objective function error. Then, extensions are made to tackle distributed optimization problems with coupled functional constraints by combining ${rm DSA_2}$ and dual decomposition. This is made possible by Lagrangian relaxation that transforms the coupling in constraints of the primal problem into that in cost functions of the dual, thus allowing us to solve the dual problem via ${rm DSA_2}$. Both the dual objective error and the quadratic penalty for the coupled constraint are proved to converge at a rate of $O(frac{1}{sqrt{t}})$, and the primal objective error asymptotically vanishes. Numerical experiments and comparisons are conducted to illustrate the advantage of the proposed algorithms and validate our theoretical findings.
In this paper, we extend the results from Jiao et al. (2019) on distributed linear quadratic control for leaderless multi-agent systems to the case of distributed linear quadratic tracking control for leader-follower multi-agent systems. Given one autonomous leader and a number of homogeneous followers, we introduce an associated global quadratic cost functional. We assume that the leader shares its state information with at least one of the followers and the communication between the followers is represented by a connected simple undirected graph. Our objective is to design distributed control laws such that the controlled network reaches tracking consensus and, moreover, the associated cost is smaller than a given tolerance for all initial states bounded in norm by a given radius. We establish a centralized design method for computing such suboptimal control laws, involving the solution of a single Riccati inequality of dimension equal to the dimension of the local agent dynamics, and the smallest and the largest eigenvalue of a given positive definite matrix involving the underlying graph. The proposed design method is illustrated by a simulation example.
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