Measuring Sample Quality with Kernels (1703.01717v9)

Abstract: Approximate Markov chain Monte Carlo (MCMC) offers the promise of more rapid sampling at the cost of more biased inference. Since standard MCMC diagnostics fail to detect these biases, researchers have developed computable Stein discrepancy measures that provably determine the convergence of a sample to its target distribution. This approach was recently combined with the theory of reproducing kernels to define a closed-form kernel Stein discrepancy (KSD) computable by summing kernel evaluations across pairs of sample points. We develop a theory of weak convergence for KSDs based on Stein's method, demonstrate that commonly used KSDs fail to detect non-convergence even for Gaussian targets, and show that kernels with slowly decaying tails provably determine convergence for a large class of target distributions. The resulting convergence-determining KSDs are suitable for comparing biased, exact, and deterministic sample sequences and simpler to compute and parallelize than alternative Stein discrepancies. We use our tools to compare biased samplers, select sampler hyperparameters, and improve upon existing KSD approaches to one-sample hypothesis testing and sample quality improvement.

• The paper introduces Kernel Stein Discrepancies (KSDs) that unify Stein’s method with reproducing kernel Hilbert spaces to measure sample convergence.
• It shows that kernel choice critically affects the detection of non-convergence, with heavier-tailed kernels outperforming light-tailed options in high dimensions.
• Empirical tests confirm KSDs offer algorithmic efficiency and practical utility in hyperparameter tuning and one-sample hypothesis testing.

Insight into "Measuring Sample Quality with Kernels"

This paper addresses the significant challenge of ensuring reliable inference in approximate Markov Chain Monte Carlo (MCMC) methodologies, which trade off accuracy in favor of computational efficiency. The classical MCMC diagnostics tend to fall short in capturing biases introduced by approximate methods; hence, novel measures such as the Kernel Stein Discrepancies (KSDs) are introduced. By unifying Stein's method with reproducing kernel Hilbert spaces, the authors propose a tractable and theoretically sound approach for assessing sample quality.

Objective and Motivation

The primary objective of the paper is to develop a standardized measure that can effectively determine the convergence quality of samples — whether they are biased, exact, or derivatively generated. As biased MCMC methods can produce rapid enough samples to allow real-world applications but lack classical asymptotic reliability, developing a measure that captures their convergence characteristics is paramount.

Methodology

Here, the paper introduces KSDs, which leverage Stein’s operator to yield a discrepancy measure devoid of integration under complex target distributions. By demonstrating how common KSDs can fail even under basic targets like Gaussian distributions, the authors highlight the critical role kernel choice plays in effectively capturing non-convergence traits.

KSDs are defined using reproducing kernels, capable of elucidating the convergence properties of samples toward their target distributions via kernel evaluations. This approach benefits from a closed-form solution, considerably simplifying implementations and making it conducive to parallelization — a critical advantage in large-scale computations involving high-dimensional data.

Key Results and Analysis

Several significant theoretical findings emerge from the paper:

1. Theoretical Boundaries: The paper rigorously establishes conditions under which KSDs are indicative of sample convergence. For univariate distributions, they prove that KSDs dominate weak convergence, setting a compelling argument for their adoption.
2. Kernel Impact: Not all kernels are equal — with higher-dimensional settings introducing failures for kernels with light tails (e.g., Gaussian kernels). Instead, using kernels with heavier tails (e.g., inverse multiquadric kernels) ensures that undesirable, non-convergent behaviors in sample sequences are more reliably detected.
3. Algorithmic Efficiency: The paper demonstrates that KSDs are not only theoretically appealing but also computationally viable. Empirical tests show that computations can be performed with lower time complexities compared to graph-based Stein discrepancies, especially as dimensionality increases.
4. Practical Implications: Beyond theoretical contributions, the authors provide use cases where KSDs underscore their utility, including hyperparameter tuning for MCMC methods and one-sample hypothesis testing. Both tasks benefit from KSD’s nuanced ability to discriminate sample quality under challenging conditions.

Implications and Future Work

The introduction of KSDs suggests a wide array of implications for probabilistic inference methodologies. Practically, KSDs can become essential diagnostic tools to guide the development and tuning of modern MCMC applications. Theoretical implications are equally significant, as they inspire a reevaluation of kernel choice in both probabilistic modeling and function approximation contexts.

The authors suggest exploring stochastic, low-rank, and sparse approximations of the kernel matrices to enhance scalability with increasing data points, a pivotal consideration given the growing size of modern datasets.

Concluding Thoughts

This paper provides a comprehensive framework for diagnosing and ensuring the quality and reliability of samples generated by approximate inference methods. Through robust theoretical examinations paired with practical applications, it forms a solid foundation for future explorations in Monte Carlo methods and their utilization in expansive data-centric domains. The concession that kernel choice has profound impacts on inference tasks underscores a nuanced understanding of statistical modeling that resonates well with ongoing advancements in machine learning and computational statistics.