One-Point Sampling for Distributed Bandit Convex Optimization With Time-Varying Constraints

This paper considers the distributed bandit convex optimization problem with time-varying constraints. In this problem, the global loss function is the average of all the local convex loss functions, which are unknown beforehand. Each agent iteratively makes its own decision subject to time-varying inequality constraints which can be violated but are fulfilled in the long run. For a uniformly jointly strongly connected time-varying directed graph, a distributed bandit online primal–dual projection algorithm with one-point sampling is proposed. We show that sublinear dynamic network regret and network cumulative constraint violation (CCV) are achieved if the path–length of the benchmark also increases in a sublinear manner. In addition, an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}({T^{3/4 + g}})$</tex-math></inline-formula> static network regret bound and an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}( {{T^{1 - {g}/2}}})$</tex-math></inline-formula> network CCV bound are established, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> is the total number of iterations and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$g \in ( {0,1/4})$</tex-math></inline-formula> is a trade-off parameter. Moreover, a reduced static network regret bound <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}( {{T^{2/3 + 4g /3}}})$</tex-math></inline-formula> is established for strongly convex local loss functions. Finally, a numerical example is presented to validate the theoretical results.