具有次威布尔梯度误差与间歇更新的基于反馈的优化
Feedback-Based Optimization with Sub-Weibull Gradient Errors and Intermittent Updates
AI总结：
本文针对梯度受次威布尔误差干扰且输出测量间歇缺失的反馈投影梯度优化问题，结合高斯过程回归等代价学习方法，证明了在线算法解的期望与高概率误差界，并在智能电网需求响应中验证了有效性。
摘要：
本文研究一种基于反馈的投影梯度方法，用于优化建模为代数映射的系统。重点关注如下设置：梯度受到服从次威布尔分布的随机误差干扰，并且系统输出的测量值——在算法更新中替代系统的输入输出映射——可能并非每次迭代都可获得。次威布尔误差模型特别适用于如下框架：问题的代价函数在执行算法的同时通过高斯过程（GP）回归（基于函数评估）进行学习；此外，该模型也自然适用于利用非参数方法和神经网络估计代价函数的设置。利用次威布尔模型，并以伯努利随机变量对系统输出的缺失测量进行建模，我们证明该在线算法生成的点与最优解之间的误差是有界的。特别地，我们给出了期望意义下和高概率意义下的误差界。在智能电网需求响应问题的背景下给出了数值结果。
英文摘要：
This paper considers a feedback-based projected gradient method for optimizing systems modeled as algebraic maps. The focus is on a setup where the gradient is corrupted by random errors that follow a sub-Weibull distribution, and where the measurements of the output -- which replace the input-output map of the system in the algorithmic updates -- may not be available at each iteration. The sub-Weibull error model is particularly well-suited in frameworks where the cost of the problem is learned via Gaussian Process (GP) regression (from functional evaluations) concurrently with the execution of the algorithm; however, it also naturally models setups where nonparametric methods and neural networks are utilized to estimate the cost. Using the sub-Weibull model, and with Bernoulli random variables modeling missing measurements of the system output, we show that the online algorithm generates points that are within a bounded error from the optimal solutions. In particular, we provide error bounds in expectation and in high probability. Numerical results are presented in the context of a demand response problem in smart power grids.
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