Feedback-Based Optimization with Sub-Weibull Gradient Errors and Intermittent Updates

TL;DR

This paper develops a feedback-based projected gradient method for system optimization under sub-Weibull gradient errors and intermittent measurements, providing theoretical error bounds and demonstrating effectiveness in smart grid demand response.

Contribution

It introduces a novel analysis of gradient-based optimization with sub-Weibull errors and missing data, extending convergence guarantees to more realistic noisy and incomplete measurement scenarios.

Findings

Algorithm achieves bounded error from optimal solutions.

Provides expectation and high-probability error bounds.

Numerical results validate approach in smart grid context.

Abstract

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.