In this paper, we consider a distributed robust optimization (DRO) problem, where multiple agents in a networked system cooperatively minimize a global convex objective function with respect to a global variable under the global constraints. The objective function can be represented by a sum of local objective functions. The global constraints contain some uncertain parameters which are partially known, and can be characterized by some inequality constraints. After problem transformation, we adopt the Lagrangian primal-dual method to solve this problem. We prove that the primal and dual optimal solutions of the problem are restricted in some specific sets, and we give a method to construct these sets. Then, we propose a DRO algorithm to find the primal-dual optimal solutions of the Lagrangian function, which consists of a subgradient step, a projection step, and a diffusion step, and in the projection step of the algorithm, the optimized variables are projected onto the specific sets to guarantee the boundedness of the subgradients. Convergence analysis and numerical simulations verifying the performance of the proposed algorithm are then provided. Further, for nonconvex DRO problem, the corresponding approach and algorithm framework are also provided.
Ordinal regression methods are widely used to predict the ordered labels of data, among which support vector ordinal regression (SVOR) methods are popular because of their good generalization. In many realistic circumstances, data are collected by a distributed network. In order to protect privacy or due to some practical constraints, data cannot be transmitted to a center for processing. However, as far as we know, existing SVOR methods are all centralized. In the above situations, centralized methods are inapplicable, and distributed methods are more suitable choices. In this paper, we propose a distributed SVOR (dSVOR) algorithm. First, we formulate a constrained optimization problem for SVOR in distributed circumstances. Since there are some difficulties in solving the problem with classical methods, we used the random approximation method and the hinge loss function to transform the problem into a convex optimization problem with constraints. Then, we propose subgradient-based algorithm dSVOR to solve it. To illustrate the effectiveness, we theoretically analyze the consensus and convergence of the proposed method, and conduct experiments on both synthetic data and a real-world example. The experimental results show that the proposed dSVOR could achieve close performance to that of the corresponding centralized method, which needs all the data to be collected together.
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