Empirical Bernstein Bounds and Sample Variance Penalization (0907.3740v1)

Abstract: We give improved constants for data dependent and variance sensitive confidence bounds, called empirical Bernstein bounds, and extend these inequalities to hold uniformly over classes of functionswhose growth function is polynomial in the sample size n. The bounds lead us to consider sample variance penalization, a novel learning method which takes into account the empirical variance of the loss function. We give conditions under which sample variance penalization is effective. In particular, we present a bound on the excess risk incurred by the method. Using this, we argue that there are situations in which the excess risk of our method is of order 1/n, while the excess risk of empirical risk minimization is of order 1/sqrt/{n}. We show some experimental results, which confirm the theory. Finally, we discuss the potential application of our results to sample compression schemes.

• The paper introduces empirical Bernstein bounds that provide variance-sensitive confidence intervals for enhanced risk estimation.
• The paper proposes Sample Variance Penalization as an alternative to ERM, achieving a faster risk decay rate of 1/n under low variance conditions.
• The paper leverages classical inequalities to establish a robust theoretical foundation and outlines future directions for practical algorithm improvements.

Empirical Bernstein Bounds and Sample Variance Penalization: An Overview

Introduction

The paper by Maurer and Pontil addresses the domain of machine learning with a focus on empirical Bernstein bounds and the novel learning method of Sample Variance Penalization (SVP). The research aims to improve confidence bounds and propose a variance-sensitive approach, extending these insights to potentially enhance learning methodologies beyond traditional Empirical Risk Minimization (ERM).

Key Contributions

The authors introduce empirical Bernstein bounds, a mechanism to establish more nuanced confidence intervals that are sensitive to variance. This generalization allows for bounds that are not only data-dependent but also provide uniform applicability across function classes with polynomial growth relative to sample size $n$.

Sample Variance Penalization (SVP)

SVP is presented as an alternative to ERM, motivated by these tailored confidence bounds. Unlike ERM, which treats all hypotheses uniformly, SVP incorporates the empirical variance of the loss function into its selection process. This additional step has been shown under certain conditions to produce an excess risk that decays at a rate $1/n$ compared to the $1/\sqrt{n}$ decay typical of ERM. This improved bound is particularly significant when hypotheses have low variance.

Theoretical Foundations

The paper grounds its approach in well-established inequalities such as Hoeffding’s and Bennett’s, using them to derive new bounds that are not only rigorous but practically relevant. Empirical Bernstein bounds particularly shine due to their ability to provide tighter estimates for hypotheses with small variance, offering a potentially clearer view of the hypothesis space.

Practical Implications

The theoretical improvements have tangible implications in scenarios where noise arises from many independent sources that are not disproportionately impactful. In such scenarios, an optimal hypothesis can be effectively screened as SVP seeks to minimize both empirical risk and sample variance.

Experimental Evidence

The authors validate their theoretical proposals with experiments that demonstrate SVP's advantages over ERM in terms of risk minimization. This is particularly evident in synthetic settings where risks are clearly delineated, and composite variances can skew empirical risk estimates in real-world applications.

Future Research Directions

The research opens various avenues for future exploration. Efficient algorithms that can incorporate SVP's non-convex nature are needed, alongside practical implementations in real-world datasets. Furthermore, the combination of empirical Bernstein bounds with sample compression techniques presents a productive research front for creating compact, efficient, and accurate learning models.

Conclusion

This research offers significant advancements in variance-sensitive learning frameworks. By restructuring the confidence bounds and penalization mechanisms within learning algorithms, Maurer and Pontil provide both a robust theoretical framework and compelling initial evidence for the potential of SVP as an effective alternative to traditional ERM approaches. Future work exploring real-world applications and computational optimizations will be crucial in assessing the full impact and utility of these findings.