High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity (1109.3714v4)

Abstract: Although the standard formulations of prediction problems involve fully-observed and noiseless data drawn in an i.i.d. manner, many applications involve noisy and/or missing data, possibly involving dependence, as well. We study these issues in the context of high-dimensional sparse linear regression, and propose novel estimators for the cases of noisy, missing and/or dependent data. Many standard approaches to noisy or missing data, such as those using the EM algorithm, lead to optimization problems that are inherently nonconvex, and it is difficult to establish theoretical guarantees on practical algorithms. While our approach also involves optimizing nonconvex programs, we are able to both analyze the statistical error associated with any global optimum, and more surprisingly, to prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers. On the statistical side, we provide nonasymptotic bounds that hold with high probability for the cases of noisy, missing and/or dependent data. On the computational side, we prove that under the same types of conditions required for statistical consistency, the projected gradient descent algorithm is guaranteed to converge at a geometric rate to a near-global minimizer. We illustrate these theoretical predictions with simulations, showing close agreement with the predicted scalings.

• The paper presents a novel nonconvex framework for high-dimensional regression under noisy and missing data conditions with rigorous error bounds.
• It employs a projected gradient descent algorithm that converges in polynomial time to solutions near the global optimum.
• Simulations validate the theoretical guarantees, demonstrating the method’s robustness for practical applications in complex data environments.

High-Dimensional Regression with Noisy and Missing Data: Analysis and Implications

This paper provides a comprehensive paper of high-dimensional regression in the presence of noisy and missing data, addressing significant challenges associated with nonconvex optimization problems. Authors Loh and Wainwright present a novel framework for handling these issues, with rigorous theoretical guarantees.

Overview

The traditional approach to prediction problems typically assumes fully observed, noiseless data, sampled independently. However, real-world applications often involve scenarios where data is not only noisy and missing but may exhibit dependencies, such as in sensor networks or econometrics. This paper extends sparse linear regression methods to such settings, offering theoretical guarantees for solutions to highly nonconvex problems.

Methodological Contributions

The authors focus on a class of $M$-estimators derived from nonconvex optimization problems, proposing estimators for scenarios where covariates are noisy, missing, and/or dependent. A critical aspect of their methodology is employing a projected gradient descent algorithm, which they prove converges in polynomial time to a neighborhood of the global optima.

Key Results

1. Statistical Guarantees: The paper provides non-asymptotic bounds on the statistical error of the proposed estimators. Particularly, the bounds are shown to hold with high probability even when data is noisy, missing, or dependent.
2. Optimization Guarantees: Despite nonconvexity, a projected gradient descent algorithm converges to a solution that is statistically close to a global optimum. This is a notable result as it extends the applicability of efficiently finding near-global optima in nonconvex settings.
3. Numerical Validation: Through simulations, the authors validate the theoretical predictions, showcasing robustness and practical efficiency across various instances of noisy and missing data.

Implications and Speculation for Future Developments

From a practical standpoint, this research proposes scalable techniques that could potentially transform approaches in fields such as genomics, finance, and environmental science where high-dimensional data with noise and missing entries is commonplace.

Theoretically, the results open avenues for further explorations into nonconvex optimization, particularly in high-dimensional statistics. The projected gradient descent approach may find applications in broader settings, encouraging new algorithmic innovations and theoretical insights.

In future developments, exploring dependencies beyond Gaussian models or extending these techniques to other forms of corruption could offer deeper insights. Moreover, understanding the implications under model misspecification presents an intriguing avenue for research.

Conclusion

The paper makes significant strides in addressing complex problems of high-dimensional regression under realistic data conditions, providing a robust theoretical framework supported by empirical results. It establishes foundational methods that are both theoretically sound and practically viable, offering a substantial contribution to the field of high-dimensional statistics amidst nonconvex challenges.