Simulation-Driven Parameter Estimation with a Focus on Control and Privacy

QC 20260409

Abstract

Parameter estimation is a pivotal task across various domains such as system identification, statistics, and machine learning. The literature presents numerous estimation procedures, many of which are backed by well-studied asymptotic properties. In the contemporary landscape, highly advanced digital twins (DTs) offer the capability to faithfully replicate real systems through proper tuning. Leveraging these DTs, simulation-driven estimators can alleviate challenges inherent in traditional methods, notably their computational cost and sensitivity to initializations. Furthermore, traditional estimators often rely on sensitive data, necessitating protective measures. In this thesis, we consider simulation-driven and privacy-preserving approaches to parameter estimation that overcome many of these challenges.

The first part of the thesis delves into an exploration of modern simulation-driven estimation techniques, focusing on the two-stage (TS) approach. Operating under the paradigm of inverse supervised learning, the TS approach simulates numerous samples across parameter variations and employs supervised learning methods to predict parameter values. Divided into two stages, the approach involves compressing data into a smaller set of auxiliary statistics, and the second stage utilizes these statistics to predict parameter values. The simplicity of the TS estimator underscores its interpretability, necessitating theoretical justification, which forms a core motivation for this thesis. We establish statistical frameworks for the TS estimator, yielding its Bayes and minimax versions, alongside developing an improved minimax TS variant based on gradient boosting that excels in computational efficiency. We conduct both asymptotic and non-asymptotic analyses of the TS estimator, establishing strong consistency, asymptotic normality, and finite-sample deviation bounds that characterize the estimation error in terms of the number of training samples and observation length. Finally, we address the question of generating diverse and informative training samples by proposing a Metropolis-Adjusted Langevin Algorithm (MALA)-based scheme for sampling training parameters from Jeffreys prior, which reflects the intrinsic geometry of the parameter space via the Fisher Information Matrix.

The second part of the thesis considers the problem of adapting pretrained TS estimators to out-of-distribution scenarios. We introduce two fine-tuning methods: a supervised approach that combines feature-space anomaly detection with Fisher-information-guided retraining, and an unsupervised approach that minimizes the discrepancy between simulated and observed trajectories without requiring labeled data.

The third part of the thesis introduces applications of simulation-driven estimation methods in the design of tuning rules for PI controllers. Leveraging synthetic datasets generated from DTs, we train machine learning algorithms to meta-learn tuning rules, streamlining the calibration process without manual intervention. We explore both TS-based approaches with explicit parameter estimation and direct learning approaches using neural network architectures such as convolutional neural networks and WaveNet.

In the final part of the thesis, we tackle scenarios where estimation procedures must handle sensitive data. Here, we introduce differential privacy constraints into the Bayes point estimation problem to protect sensitive information. By proposing a unified approach, we integrate the estimation problem and differential privacy constraints into a single convex optimization objective, thereby optimizing the accuracy-privacy trade-off. In cases where both observations and parameter spaces are finite, this approach reduces to a tractable linear program which is solvable using off-the-shelf solvers.

In essence, this thesis endeavors to address theoretical foundations, robustness, and privacy concerns within the realm of simulation-driven parameter estimation.

Link to DiVA