In recent years Distributionally Robust Optimization (DRO) has raised to the status of one of the most used tools for robust estimation. This because it shares some nice properties such as good outofsample performances and wellunderstood regularization effects. The estimator we obtain within this framework is computed by minimizing the expected loss under the worstcase distribution among the ones that are close, in a f divergence sense, to the empirical distribution which relies just on historical data. In this thesis we propose a regularized in the distributions’ space approach to compute the Mestimator of an unknown parameter using data coming from linear and noisy measurements. The ultimate goal will be to characterize the estimation error which is in general a challenging task but yet very important. Our analysis is performed under the modern assumption of highdimensional regime in which both the number of measurements and parameters are very large, keeping a fixed proportion while going to infinity which encodes the under/overparametrization of the problem. Our contribution can be summarized as follows. First, we introduce briefly the tools used in the thesis. These are CGMT, which under the assumption of isotropic Gaussian features permits to recover the estimation error by simply solving a deterministic program with few scalar variables and f divergences, which is a family of distances that can be used to quantify the discrepancy between probability measures. Building on these results, we formulate the Distributionally Regularized Estimation problem and we will show a dual reformulation of it. Then, we will discuss what are some of the main challenges encountered when applying CGMT to this problem and we will point out how the choice of the regularization parameter λ is crucial in the highdimension statistics to get a final problem which still encodes robustness parameters. Finally we will show how we can recover the norm of the estimator’s error with a simple deterministic minmax problem.
HighDimensional Analysis of fdivergence Distributionally Regularized Mestimation
CESCON, RICCARDO
2021/2022
Abstract
In recent years Distributionally Robust Optimization (DRO) has raised to the status of one of the most used tools for robust estimation. This because it shares some nice properties such as good outofsample performances and wellunderstood regularization effects. The estimator we obtain within this framework is computed by minimizing the expected loss under the worstcase distribution among the ones that are close, in a f divergence sense, to the empirical distribution which relies just on historical data. In this thesis we propose a regularized in the distributions’ space approach to compute the Mestimator of an unknown parameter using data coming from linear and noisy measurements. The ultimate goal will be to characterize the estimation error which is in general a challenging task but yet very important. Our analysis is performed under the modern assumption of highdimensional regime in which both the number of measurements and parameters are very large, keeping a fixed proportion while going to infinity which encodes the under/overparametrization of the problem. Our contribution can be summarized as follows. First, we introduce briefly the tools used in the thesis. These are CGMT, which under the assumption of isotropic Gaussian features permits to recover the estimation error by simply solving a deterministic program with few scalar variables and f divergences, which is a family of distances that can be used to quantify the discrepancy between probability measures. Building on these results, we formulate the Distributionally Regularized Estimation problem and we will show a dual reformulation of it. Then, we will discuss what are some of the main challenges encountered when applying CGMT to this problem and we will point out how the choice of the regularization parameter λ is crucial in the highdimension statistics to get a final problem which still encodes robustness parameters. Finally we will show how we can recover the norm of the estimator’s error with a simple deterministic minmax problem.File  Dimensione  Formato  

Cescon_Riccardo.pdf
embargo fino al 13/06/2024
Dimensione
606.11 kB
Formato
Adobe PDF

606.11 kB  Adobe PDF 
The text of this website © Università degli studi di Padova. Full Text are published under a nonexclusive license. Metadata are under a CC0 License
https://hdl.handle.net/20.500.12608/40463