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

Ana M. Ospina, Nicola Bastianello, Emiliano Dall'Anese

TL;DR

This work develops a online feedback-based projected gradient framework for time-varying optimization of systems driven by algebraic maps, where gradient errors follow a sub-Weibull distribution and measurements are intermittently available via Bernoulli sampling. The authors derive iteration-wise expected error bounds and high-probability bounds relative to the time-varying optima ${\mathbf{x}}_{*,t}$, and show the algorithm exhibits input-to-state stability in expectation. They further show how concurrent learning of costs via Gaussian Process regression can be embedded in the online updates, preserving the theoretical guarantees (with GP-induced sub-Weibull gradient errors). A demand-response case in smart grids demonstrates robust tracking under incomplete measurements and data-driven cost learning. Overall, the results provide a principled, data-driven approach for asynchronous online optimization in cyber-physical systems with learning components and communication constraints.

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.

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

TL;DR

This work develops a online feedback-based projected gradient framework for time-varying optimization of systems driven by algebraic maps, where gradient errors follow a sub-Weibull distribution and measurements are intermittently available via Bernoulli sampling. The authors derive iteration-wise expected error bounds and high-probability bounds relative to the time-varying optima , and show the algorithm exhibits input-to-state stability in expectation. They further show how concurrent learning of costs via Gaussian Process regression can be embedded in the online updates, preserving the theoretical guarantees (with GP-induced sub-Weibull gradient errors). A demand-response case in smart grids demonstrates robust tracking under incomplete measurements and data-driven cost learning. Overall, the results provide a principled, data-driven approach for asynchronous online optimization in cyber-physical systems with learning components and communication constraints.

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.

Paper Structure

This paper contains 8 sections, 5 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Online Algorithm
  4. Assumptions
  5. Convergence Analysis
  6. Optimization with Concurrent GP Learning
  7. Numerical Results
  8. Conclusions

Key Result

Proposition 1

(InclusionVladimirova_2020) Let $X \sim \mathrm{subW}(\theta, \nu)$ and let $\theta', \nu'$ s.t. $\theta' \geq \theta$, $\nu' \geq \nu$. Then, $X \sim \mathrm{subW}(\theta', \nu')$. $\Box$

Figures (2)

  • Figure 1: Reference signal ${\mathbf{y}}_{\mathrm{ref}}$ and overall contribution of the non-controllable loads at the point of common coupling.
  • Figure 2: Error $\|{\mathbf{x}}_t - {\mathbf{x}}_{*,t}\|$ for different values of $p$. (a) Learning with GPs. (b) Learning with feedforward NNs. In both cases, the blue line corresponds to the online algorithm with exact knowledge of $\{U_{m}\}$.

Theorems & Definitions (7)

  • Definition 1: Sub-Weibull rv Vladimirova_2020
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • Theorem 1
  • proof