We consider optimization problems for (networked) systems, where we minimize a cost that includes a known time-varying function associated with the systems outputs and an unknown function of the inputs. We focus on a data-based online projected gradient algorithm where: i) the input-output map of the system is replaced by measurements of the output whenever available (thus leading to a closed-loop setup); and ii) the unknown function is learned based on functional evaluations that may occur infrequently. Accordingly, the feedback-based online algorithm operates in a regime with inexact gradient knowledge and with random updates. We show that the online algorithm generates points that are within a bounded error from the optimal solution of the problem; in particular, we provide error bounds in expectation and in high-probability, where the latter is given when the gradient error follows a sub-Weibull distribution and when missing measurements are modeled as Bernoulli random variables. We also provide results in terms of input-to-state stability in expectation and in probability. Numerical results are presented in the context of a demand response task in power systems.
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