Statistical Learning and Inverse Problems: A Stochastic Gradient Approach

Abstract: Inverse problems are paramount in Science and Engineering. In this paper, we consider the setup of Statistical Inverse Problem (SIP) and demonstrate how Stochastic Gradient Descent (SGD) algorithms can be used in the linear SIP setting. We provide consistency and finite sample bounds for the excess risk. We also propose a modification for the SGD algorithm where we leverage machine learning methods to smooth the stochastic gradients and improve empirical performance. We exemplify the algorithm in a setting of great interest nowadays: the Functional Linear Regression model. In this case we consider a synthetic data example and examples with a real data classification problem.

• The paper introduces a tailored stochastic gradient method for solving statistical inverse problems, integrating measurement uncertainty into the empirical risk framework.
• It demonstrates through both synthetic and real data that the SGD-based approach provides comparable performance to state-of-the-art methods in functional regression and classification tasks.
• The study suggests future exploration into neural network integration and nonlinear inverse problems to further enhance computational accuracy and efficiency.

Statistical Learning and Inverse Problems: A Stochastic Gradient Approach

Introduction

Inverse problems (IPs) encompass the estimation of unknown parameters that satisfy a given equation. These problems are inherently ill-posed, often seen in disciplines like medical imaging, where solutions are sensitive to data perturbations. While classical approaches often involve deterministic models with regularization techniques for noise mitigation, the paper "Statistical Learning and Inverse Problems: A Stochastic Gradient Approach" (2209.14967) opts for a statistical framework that incorporates uncertainties in measurement, defining this as a Statistical Inverse Problem (SIP). Specifically, the paper presents an SGD-based method to tackle SIPs, emphasizing the broader applicability to functional parameters within machine learning contexts.

Problem Formulation and Algorithmic Solution

The paper introduces a new approach for solving SIPs by leveraging stochastic gradients derived from the empirical risk framework associated with functional linear regression models. The central proposition is an adaptation of SGD methods, specifically tailored to manage the ill-posed nature of SIPs by modifying the gradient descent to utilize stochastic gradients. The method incorporates machine learning models to smooth these gradients, contributing to improved empirical performance.

The problem is formally modeled as $y = A[f] + \text{noise}$, where the goal is to estimate the functional parameter $f$ given $y$. Here, $A$ is a known operator, typically linear, mapping between Hilbert spaces. The algorithm proposed involves constructing unbiased estimators for the risk gradient, reformulating the estimation challenge directly within the SGD framework—and importantly, within a probabilistic context.

Main Contributions

1. Numerical Method for SIPs: The paper broadens the application of SGD by integrating it into SIP contexts, traditionally dominated by regularization strategies. This integration extends SGD's efficacy from deterministic inverse problems to statistical settings, providing consistency and finite sample bounds for excess risk.

2. Enhanced Algorithmic Framework: A modification of the SGD algorithm is introduced to substitute stochastic gradients with base learners, akin to boosting, effectively addressing the discretization issues of SIP operators. This adjustment facilitates a smoother estimation process, valuable in machine learning cross-applications where SIP challenges are prevalent (e.g., functional linear regression).

Figure 1

Figure 1: Example of cumulative credits for six different addresses across 501 data points. In red, addresses associated with criminal activity, in blue, addresses associated with noncriminal activities.

Numerical Studies and Results

The paper evaluates the performance of the proposed algorithms using both synthetic and real data. Two main application scenarios are explored: a functional linear regression model employing synthetic data and a classification problem involving bitcoin transaction datasets.

• Synthetic Data Evaluation: Utilizing a functional linear regression setup, the SGD approach demonstrated competencies comparable to existing state-of-the-art methods, even when faced with non-traditional challenges such as high-dimensional functional inputs.
• Real Data Application: The method was applied to a classification task involving bitcoin data, where cumulative credit curves served as input features. The SGD-SIP algorithm, notable for utilizing a single sample at each iteration, successfully rivaled traditional frameworks designed for functional data analysis.

Figure 2

Figure 2: MSE with 2 standard deviations error bars for 10 simulations with $f$ as the sine function. Y-axis in square-root scale.

Figure 3

Figure 3: MSE with 2 standard deviations error bars for 10 simulations with $f$ as a step function and using all samples for the gradient computation.

Implications and Future Directions

The results substantiate the practical efficacy and theoretical soundness of using stochastic gradient methods for SIPs, highlighting significant implications for diverse fields requiring solution estimates for inverse equations. The algorithms, designed for flexible application, exhibit potential for optimizing computational efficiency and accuracy, especially in data-rich environments.

Future developments could explore deeper integrations with neural network architectures or adapt the model to nonlinear SIPs. Additionally, expanding applications into more complex SIP scenarios, such as deconvolution problems or those involving PDEs, could validate the algorithmic robustness across broader scientific inquiries.

Conclusion

This study presents a comprehensive approach to SIPs using machine learning-enhanced SGD, offering a robust solution under an array of statistical settings. By bridging IP methodologies with stochastic optimization, the paper significantly enhances the toolkit available for scientists and engineers tackling SIPs, suggesting a versatile direction for future research explorations.