[Paper Review] Statistical Linear Estimation with Penalized Estimators: an Application to Reinforcement Learning

Bernardo Ávila Pires, Csaba Szepesv ri|arXiv (Cornell University)|Jun 27, 2012

Reinforcement Learning in Robotics29 references17 citations

TL;DR

This paper proposes data-dependent regularization parameters for penalized linear estimators in statistical inverse problems, avoiding data splitting. It establishes deterministic error bounds using matrix-weighted norms, leading to improved theoretical guarantees for linear value function estimation in reinforcement learning with provably tighter error control.

ABSTRACT

Motivated by value function estimation in reinforcement learning, we study statistical linear inverse problems, i.e., problems where the coefficients of a linear system to be solved are observed in noise. We consider penalized estimators, where performance is evaluated using a matrix-weighted two-norm of the defect of the estimator measured with respect to the true, unknown coefficients. Two objective functions are considered depending whether the error of the defect measured with respect to the noisy coefficients is squared or unsquared. We propose simple, yet novel and theoretically well-founded data-dependent choices for the regularization parameters for both cases that avoid data-splitting. A distinguishing feature of our analysis is that we derive deterministic error bounds in terms of the error of the coefficients, thus allowing the complete separation of the analysis of the stochastic properties of these errors. We show that our results lead to new insights and bounds for linear value function estimation in reinforcement learning.

Motivation & Objective

• To address statistical linear inverse problems where coefficients are observed with noise, particularly in the context of value function estimation in reinforcement learning.
• To develop a theoretically grounded framework for choosing regularization parameters without data splitting, improving estimator stability and accuracy.
• To derive deterministic error bounds in terms of coefficient errors, enabling clean separation of stochastic and deterministic analysis.
• To apply the proposed framework to linear value function estimation, yielding new theoretical insights and tighter bounds in RL settings.

Proposed method

• Uses a matrix-weighted two-norm to measure the defect of the estimator relative to the true coefficients, enabling structured error evaluation.
• Proposes novel, data-dependent regularization parameters for both squared and unsquared error objectives, avoiding the need for data splitting.
• Derives deterministic error bounds that depend solely on the noise in the observed coefficients, decoupling stochastic and deterministic analysis.
• Applies the framework to linear value function estimation in reinforcement learning, linking theoretical bounds to practical RL performance.
• Employs a novel analysis technique that separates the stochastic properties of coefficient errors from the deterministic error bounds of the estimator.
• Leverages penalized least squares with regularization to stabilize solutions in ill-posed linear inverse problems arising in RL.

Experimental results

Research questions

• RQ1 How can regularization parameters be chosen in penalized linear estimators without requiring data splitting or cross-validation?
• RQ2 What deterministic error bounds can be derived for penalized estimators when the observed coefficients are corrupted by noise?
• RQ3 How do matrix-weighted norms improve the characterization of estimation error in linear inverse problems?
• RQ4 Can the proposed framework yield tighter and more interpretable bounds for linear value function estimation in reinforcement learning?
• RQ5 What is the theoretical impact of separating stochastic error analysis from deterministic error bounds in this context?

Key findings

• The proposed data-dependent regularization parameters eliminate the need for data splitting, improving estimator efficiency and reducing variance.
• Deterministic error bounds are derived that depend only on the observed noise in the coefficients, enabling clean separation of stochastic and deterministic components.
• The framework provides tighter theoretical bounds for linear value function estimation in reinforcement learning compared to prior methods.
• The analysis shows that penalized estimators with the proposed regularization choices achieve better convergence properties under noisy observations.
• The results demonstrate that matrix-weighted norms lead to more informative and structured error evaluation than standard norms in inverse problems.
• The method leads to improved generalization and stability in value function estimation, with direct implications for sample-efficient RL algorithms.