Relative Density-Ratio Estimation for Robust Distribution Comparison (1106.4729v1)

Abstract: Divergence estimators based on direct approximation of density-ratios without going through separate approximation of numerator and denominator densities have been successfully applied to machine learning tasks that involve distribution comparison such as outlier detection, transfer learning, and two-sample homogeneity test. However, since density-ratio functions often possess high fluctuation, divergence estimation is still a challenging task in practice. In this paper, we propose to use relative divergences for distribution comparison, which involves approximation of relative density-ratios. Since relative density-ratios are always smoother than corresponding ordinary density-ratios, our proposed method is favorable in terms of the non-parametric convergence speed. Furthermore, we show that the proposed divergence estimator has asymptotic variance independent of the model complexity under a parametric setup, implying that the proposed estimator hardly overfits even with complex models. Through experiments, we demonstrate the usefulness of the proposed approach.

• The paper proposes relative density-ratio estimation to enhance stability and non-parametric convergence in distribution comparisons.
• It demonstrates that the estimator’s variance is independent of model complexity, reducing overfitting risks.
• Experiments validate improved accuracy in tasks like two-sample tests and outlier detection even with limited samples.

Relative Density-Ratio Estimation for Robust Distribution Comparison

The paper introduces a novel method for comparing distributions through relative density-ratio estimation, addressing inadequacies in traditional density-ratio approaches. Conventional divergence estimators often encounter issues when estimating the density ratio directly due to the high fluctuation of these ratios, especially in cases with small denominator values. This high variability can result in poor convergence speeds and reliability issues, particularly when outlier detection, transfer learning, or two-sample homogeneity tests are involved.

Key Contributions

1. Relative Density-Ratio Estimation: The authors propose approximating relative density-ratios instead of ordinary density-ratios. The benefit of relative ratios is their inherent smoothness, which provides more stable non-parametric convergence speeds. The relative density-ratio for two probability distributions $p$ and $p'$ is defined relative to a mixture density, smoothing out fluctuations that occur with traditional density-ratios.
2. Non-Parametric and Parametric Analysis: The authors establish that their method offers non-parametric convergence rates that are less sensitive to fluctuations in the density ratio. Furthermore, under a parametric model, the variance of the proposed estimator does not depend on the complexity of the model, indicating a reduced risk of overfitting even when complex models are employed.
3. Asymptotic Behavior: The paper reveals that the asymptotic variance of the proposed relative Pearson divergence estimator is independent of model complexity, a significant advantage over traditional methods.
4. Experimental Validation: The paper reports extensive experiments demonstrating the superiority of the proposed method in several tasks, including two-sample homogeneity tests and outlier detection. In these experiments, the relative divergence estimator showed robustness and higher accuracy in detecting distribution discrepancies, even when the sample size was limited or the distributions were complex.

Implications and Future Directions

The introduction of relative density-ratio estimation has broad implications for machine learning domains where robust distribution comparison is crucial. Its application in tasks such as outlier detection showcases its practical utility. Notably, the method provides a scalable and reliable alternative to classic density ratio estimation, which suffers from instability and high variance.

Theoretically, the relative approach aligns with improving the robustness of estimation procedures in statistical machine learning. By mitigating the problem of overfitting through bounded variance, the paper advances the current state of divergence estimation methods, providing a more stable framework for handling high-dimensional data in practical applications.

Future research directions may explore expanding this methodology to accommodate other divergence measures beyond the Pearson divergence, enhancing model selection criteria under relative density-ratio frameworks, and applying this method to broader contexts such as anomaly detection and adaptive learning systems.

This paper represents a significant advancement in the field, offering a reliable solution for the enduring challenges associated with density ratio estimation, especially in non-parametric contexts. By facilitating more accurate distribution comparisons, this approach has the potential to significantly improve the performance of various machine learning algorithms in real-world scenarios.