Loss minimization and parameter estimation with heavy tails (1307.1827v7)

Abstract: This work studies applications and generalizations of a simple estimation technique that provides exponential concentration under heavy-tailed distributions, assuming only bounded low-order moments. We show that the technique can be used for approximate minimization of smooth and strongly convex losses, and specifically for least squares linear regression. For instance, our $d$-dimensional estimator requires just $\tilde{O}(d\log(1/\delta))$ random samples to obtain a constant factor approximation to the optimal least squares loss with probability $1-\delta$, without requiring the covariates or noise to be bounded or subgaussian. We provide further applications to sparse linear regression and low-rank covariance matrix estimation with similar allowances on the noise and covariate distributions. The core technique is a generalization of the median-of-means estimator to arbitrary metric spaces.

• The paper introduces a robust estimator based on a median-of-means generalization that achieves exponential concentration with O(d log(1/δ)) samples under heavy-tailed distributions.
• It demonstrates the estimator's effectiveness in least squares, sparse regression, and low-rank covariance estimation without relying on bounded or subgaussian assumptions.
• The research challenges traditional empirical risk minimization by proving that reliable parameter estimation is possible with only low-order moment conditions.

Overview of "Loss Minimization and Parameter Estimation with Heavy Tails"

The paper by Daniel Hsu and Sivan Sabato addresses the challenge of loss minimization and parameter estimation in scenarios characterized by heavy-tailed distributions. This work introduces and elaborates upon a robust estimation technique that achieves exponential concentration results even under heavy-tailed distributions, provided only that low-order moments are finite. The technique is a generalization of the median-of-means estimator, applied to arbitrary metric spaces, expanding its scope of applicability far beyond traditional settings.

The paper demonstrates the efficacy of this technique in various contexts, emphasizing its utility in minimizing smooth and strongly convex losses, specifically with applications in least squares linear regression. A notable contribution of this work is providing a mathematical guarantee that the proposed $d$-dimensional estimator requires $\tilde{O}(d\log(1/\delta))$ random samples to achieve a constant factor approximation of the optimal least squares loss with high probability $1-\delta$. This is achieved without the need for covariates or noise to be bounded or subgaussian, distinguishing it from many prior approaches that rely on stringent assumptions about distributions.

Key Results and Applications

1. Estimator Efficiency: The core estimator introduced in the paper is shown to require only $\tilde{O}(d\log(1/\delta))$ samples, achieving exponential concentration, which is significant in models dealing with heavy-tailed distributions. This finding is applicable to smooth and strongly convex loss functions, enhancing the practicality of regression tasks in non-ideal data environments.
2. Sparse Linear Regression and Low-Rank Estimation: The research extends beyond least squares linear regression to include sparse linear regression and low-rank covariance matrix estimation. Both these applications benefit from the same relaxed assumptions on noise and covariate distribution, enhancing the robustness of these models under heavy-tailed conditions.
3. Robust Distance Approximation: The introduction of a generalized robust distance approximation method for arbitrary metric spaces marks a pivotal advancement. By minimizing the median of means, the proposed method can effectively manage deviations typical in heavy-tailed data scenarios.
4. Theoretical Implications: The theoretical foundation laid in this paper questions traditional empirical risk minimization, suggesting more robust alternatives that do not rely heavily on bounded or subgaussian assumptions. This opens avenues for re-evaluating algorithms used in heavy-tailed data contexts, potentially extending this methodology to a broader class of loss functions.

Implications and Future Directions

The implications of this research are substantial in both theoretical and practical dimensions, particularly in fields such as econometrics, finance, and environmental data modeling where heavy-tailed distributions frequently appear. Practically, it provides a foundation for developing algorithms that can reliably analyze data prone to extreme values.

The paper also speculates on future research directions, especially the potential for generalizing this technique to even broader classes of learning problems and improving computational efficiency. Further investigations might focus on incorporating this estimator into more complex models, such as generalized linear models or neural networks, ensuring robustness against heavy-tailed variations in data.

The work of Hsu and Sabato encourages a paradigm shift in dealing with non-Gaussian and heavy-tailed data, emphasizing the need for methodologies that balance computational tractability with statistical robustness. This paper is a step toward developing learning systems that effectively operate under realistic, imperfect conditions, fundamentally changing how researchers approach model estimation in machine learning and statistics.