A Distributed Primal-Dual Algorithm for Bandit Online Convex Optimization with Time-Varying Coupled Inequality Constraints

This paper considers distributed bandit online optimization with time-varying coupled inequality constraints. The global cost and the coupled constraint functions are the summations of local convex cost and constraint functions, respectively. The local cost and constraint functions are held privately and only at the end of each period the constraint functions are fully revealed, while only the values of cost functions at queried points are revealed, i.e., in a so called bandit manner. A distributed bandit online primal-dual algorithm with two queries for the cost functions per period is proposed. The performance of the algorithm is evaluated using its expected regret, i.e., the expected difference between the outcome of the algorithm and the optimal choice in hindsight, as well as its constraint violation. We show that O(T-c) expected regret and O(T1-c/2) constraint violation are achieved by the proposed algorithm, where T is the total number of iterations and c is an element of [0.5, 1) is a user-defined trade-off parameter. Assuming Slater's condition, we show that O(root T) expected regret and O(root T) constraint violation are achieved. The theoretical results are illustrated by numerical simulations.