Estimating divergence functionals and the likelihood ratio by convex risk minimization (0809.0853v2)

Abstract: We develop and analyze $M$-estimation methods for divergence functionals and the likelihood ratios of two probability distributions. Our method is based on a non-asymptotic variational characterization of $f$-divergences, which allows the problem of estimating divergences to be tackled via convex empirical risk optimization. The resulting estimators are simple to implement, requiring only the solution of standard convex programs. We present an analysis of consistency and convergence for these estimators. Given conditions only on the ratios of densities, we show that our estimators can achieve optimal minimax rates for the likelihood ratio and the divergence functionals in certain regimes. We derive an efficient optimization algorithm for computing our estimates, and illustrate their convergence behavior and practical viability by simulations.

• The paper presents a variational characterization of f-divergences that transforms their estimation into a convex empirical risk minimization problem.
• It employs M-estimation procedures and kernel-based methods to develop efficient estimators for the KL divergence and likelihood ratios.
• Simulation results and convergence analyses demonstrate the practical viability and theoretical robustness of the proposed approach in high-dimensional settings.

Estimating Divergence Functionals and the Likelihood Ratio by Convex Risk Minimization

The paper "Estimating divergence functionals and the likelihood ratio by convex risk minimization" by XuanLong Nguyen, Martin J. Wainwright, and Michael I. Jordan tackles the problem of estimating divergence functionals and the likelihood ratios between two probability distributions using $M$-estimation methods. This approach leverages a non-asymptotic variational characterization of $f$-divergences to transform the estimation problem into a convex empirical risk optimization challenge.

Main Contributions

1. Variational Characterization of Divergences: The authors establish a variational representation of $f$-divergences that connects it to a risk minimization problem. This transforms the problem of estimating divergences into solving convex optimization problems, which can be tackled using $M$-estimation techniques.
2. $M$-Estimation Procedures: By placing the problem into the framework of convex risk minimization, the paper develops simple estimators for both the Kullback-Leibler (KL) divergence and the likelihood ratios. These estimators are computationally efficient since they reduce to standard convex programs, readily solvable by existing optimization techniques.
3. Consistency and Convergence Analysis: The authors provide a detailed analysis of the consistency and convergence properties of these estimators. They show that under certain conditions on the density ratios, the estimators can achieve optimal minimax rates for the likelihood ratio and divergence functionals.
4. Practical Implementation Using Kernel Methods: The practical viability of the proposed methods is demonstrated through an efficient implementation using reproducing kernel Hilbert spaces (RKHS). The computational algorithms for these kernel-based methods are derived, ensuring efficient scalability to high-dimensional problems.
5. Simulation Results: Extensive simulations validate the theoretical claims, illustrating the performance and convergence behavior of the proposed estimators. The results indicate that these methods perform well compared to existing techniques, particularly in higher dimensions.

Theoretical Implications

The variational characterization of divergence functionals introduced in this paper offers a profound theoretical insight. Specifically, the relationship between $f$-divergences and Bayes decision problems opens new avenues for analyzing and estimating divergences. This correspondence implies that estimating divergences can fundamentally be viewed as solving a Bayes decision problem under a convex risk minimization framework.

The convergence analysis presented is particularly robust. Under conditions involving density ratios and continuity of empirical processes, the authors prove that the proposed estimators achieve nearly optimal rates. Notably, in the context of high-dimensional statistics, achieving rates proportional to $n^{-\alpha/(d + 2\alpha)}$ for smooth function classes in a Sobolev space marks a significant theoretical advancement.

Practical Implications

On the practical side, the paper’s methods have substantial applications in fields like information theory, statistical machine learning, and signal processing. For instance, accurately estimating KL divergence is crucial in applications like hypothesis testing, channel coding, data compression, and independent component analysis.

The use of kernel-based function approximations standardizes the approach, facilitating its application in a wide range of practical problems involving multivariate distributions. This broad applicability, coupled with the robust theoretical foundation, makes these methods highly valuable for empirical researchers and practitioners.

Future Directions

Several interesting directions for future research emerge from this work:

• Extensions to Other Divergence Functionals: While the paper primarily focuses on KL and $f$-divergences, further research could explore extensions to other divergence measures like Renyi divergence or Tsallis entropy.
• Adaptive Function Classes: Investigating whether adaptive selection of function classes based on the sample size and properties of the data could yield improvements in convergence rates and practical performance.
• Alternative Estimators: The exploration of different $M$-estimators or penalization schemes may provide further refinements in both theoretical properties and practical utility.
• High-dimensional Settings: Extending the theoretical results to high-dimensional data settings where $d$ can be large relative to $n$, possibly leveraging advanced techniques in high-dimensional statistics and machine learning.

In summary, the paper "Estimating divergence functionals and the likelihood ratio by convex risk minimization" provides a comprehensive and effective framework for estimating divergence functionals through convex empirical risk minimization. This work bridges a critical gap in both theoretical understanding and practical application, offering robust and efficient methodologies aligned with the needs of modern statistical analysis and machine learning.