In this paper, the problem of online distributed optimization with stochastic and nonconvex objective functions is studied by employing a multi-agent system. When making decisions, each agent only has access to a noisy gradient of its own objective function in the previous time and can only communicate with its immediate neighbors via a time-varying digraph. To handle this problem, an online distributed stochastic projection-free algorithm is proposed. Of particular interest is that the dynamic regrets are employed to measure the performance of the online algorithm. Existing works on online distributed algorithms involving stochastic gradients only provide the sublinearity results of regrets in expectation. Different from them, we study the high probability bounds of dynamic regrets, i.e., the sublinear bounds of dynamic regrets are characterized by the natural logarithm of the failure probability’s inverse. Under mild assumptions on the graph and objective functions, we prove that if the variations in both the objective function sequence and its gradient sequence grow within a certain rate, then the high probability bounds of the dynamic regrets grow sublinearly. Finally, a simulation example is carried out to demonstrate the effectiveness of our theoretical results.

中文翻译：

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具有随机目标函数的在线分布式非凸优化：动态遗憾的高概率界限分析


本文利用多智能体系统研究了具有随机非凸目标函数的在线分布式优化问题。在做出决策时，每个智能体只能访问其自身目标函数在前一次的噪声梯度，并且只能通过时变有向图与其直接邻居进行通信。为了解决这个问题，提出了一种在线分布式随机投影免费算法。特别有趣的是，动态遗憾被用来衡量在线算法的性能。现有的涉及随机梯度的在线分布式算法的工作仅提供了期望中遗憾的次线性结果。与它们不同的是，我们研究动态后悔的高概率界限，即动态后悔的次线性界限由失败概率倒数的自然对数来表征。在对图和目标函数的温和假设下，我们证明，如果目标函数序列及其梯度序列的变化在一定速率内增长，则动态遗憾的高概率界限呈次线性增长。最后，通过仿真算例验证了理论结果的有效性。