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Controlling Moments with Kernel Stein Discrepancies: Establishing Equivalence to q-Wasserstein Convergence


Core Concepts
The kernel Stein discrepancy (KSD) can be used to control the convergence of expectations of polynomially growing continuous functions (q-Wasserstein convergence) under appropriate conditions on the reproducing kernel.
Abstract

The paper analyzes the convergence control properties of kernel Stein discrepancies (KSDs). It first shows that standard KSDs used for weak convergence control fail to control moment convergence. To address this limitation, the authors provide sufficient conditions under which alternative diffusion KSDs control both moment and weak convergence.

The key insights are:

  1. Standard KSDs with kernels previously recommended for weak convergence control cannot enforce q-Wasserstein convergence, which is a stronger mode of convergence that controls expectations of polynomially growing continuous functions.

  2. The authors establish sufficient conditions on the kernel to guarantee that the KSD controls both q-Wasserstein convergence and weak convergence. Specifically, they show that using a kernel of a particular form, the KSD is equivalent to q-Wasserstein convergence.

  3. Under additional assumptions, the authors obtain an explicit upper bound on the q-Wasserstein distance in terms of the KSD, demonstrating the rate of q-Wasserstein convergence relative to KSD convergence.

The results provide a theoretical foundation for using the KSD as a quality measure for distribution approximation, especially when dealing with unbounded functions of polynomial growth.

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Key Insights Distilled From

by Heishiro Kan... at arxiv.org 10-01-2024

https://arxiv.org/pdf/2211.05408.pdf
Controlling Moments with Kernel Stein Discrepancies

Deeper Inquiries

Practical Considerations in Choosing the Reproducing Kernel for the KSD

When selecting a reproducing kernel for the Kernel Stein Discrepancy (KSD), several practical considerations must be taken into account to ensure the desired convergence control properties, especially in high-dimensional settings: Growth Properties: The kernel should exhibit appropriate growth properties to control the convergence of expectations for functions of polynomial growth. Specifically, the kernel must satisfy conditions that allow it to approximate q-growth functions effectively. This is crucial for ensuring that the KSD can enforce uniform integrability and weak convergence. Smoothness and Regularity: The chosen kernel should belong to a class of smooth functions, ideally being continuously differentiable. This ensures that the Stein operator can be applied effectively, and the resulting Stein kernel maintains the necessary properties for convergence control. Universality: Utilizing universal kernels, which can approximate any continuous function in a given space, can enhance the flexibility of the KSD. For instance, kernels that are universal to (C^1_0(\mathbb{R}^d)) can provide a robust framework for approximating a wide range of functions, thus improving the KSD's applicability. Computational Efficiency: In high-dimensional settings, computational efficiency becomes paramount. The kernel should allow for efficient computation of the KSD, ideally through closed-form expressions or manageable numerical approximations. This is particularly important when dealing with large datasets or complex models. Tail Behavior: The kernel should be chosen to accommodate the tail behavior of the target distribution. For instance, if the target distribution has heavy tails, the kernel must be capable of handling such characteristics to ensure accurate approximation and convergence. Parameterization: The kernel may require tuning of hyperparameters (e.g., bandwidth, scaling factors) to optimize performance in specific applications. A careful selection of these parameters can significantly impact the KSD's ability to control convergence. By considering these factors, practitioners can select a reproducing kernel that not only meets the theoretical requirements for KSD but also performs well in practical applications, particularly in high-dimensional spaces.

Extending Theoretical Results to General Function Classes

The theoretical results concerning the KSD can be extended to handle more general function classes beyond polynomial growth, such as exponential or sub-exponential growth, through the following approaches: Generalized Growth Conditions: By redefining the growth conditions to accommodate exponential or sub-exponential functions, one can establish new classes of functions that the KSD can control. For instance, one might consider functions that grow at rates defined by (e^{|x|^p}) for some (p > 0) or functions that decay faster than any polynomial. Modified Stein Operators: The Stein operators can be adapted to account for the specific characteristics of these broader function classes. This may involve altering the diffusion matrix or the form of the Stein operator to ensure that the resulting KSD can still provide meaningful convergence control. Robustness of KSD: The KSD's inherent properties can be leveraged to show that it retains its convergence control capabilities even when applied to more complex function classes. This may involve demonstrating that the KSD can still enforce uniform integrability and weak convergence under the new growth conditions. Empirical Validation: Extending the theoretical framework should be complemented by empirical studies that validate the KSD's performance on various function classes. This can help identify practical limitations and guide further theoretical developments. Integration with Other Metrics: By integrating the KSD with other metrics that are known to handle exponential or sub-exponential growth, one can create a hybrid approach that retains the strengths of both methodologies, thus broadening the applicability of the KSD. Through these strategies, the KSD can be effectively adapted to handle a wider range of function classes, enhancing its utility in various statistical and machine learning applications.

Potential Applications of the KSD-Wasserstein Equivalence Result

The KSD-Wasserstein equivalence result has significant implications across various fields, including Bayesian inference, generative modeling, and reinforcement learning. Here are some potential applications: Bayesian Inference: In Bayesian statistics, the KSD can be employed to assess the quality of posterior approximations. The KSD-Wasserstein equivalence allows practitioners to quantify how well approximate posterior distributions converge to the true posterior, facilitating model diagnostics and improving the reliability of inference. Generative Modeling: In generative models, such as Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs), the KSD can serve as a loss function to guide the training process. By leveraging the KSD-Wasserstein equivalence, one can ensure that the generated samples closely match the target distribution, leading to improved sample quality and diversity. Reinforcement Learning: In reinforcement learning, the KSD can be used to evaluate the convergence of policy distributions. The KSD-Wasserstein equivalence provides a framework for measuring how closely the learned policy approximates the optimal policy, enabling better convergence guarantees and more stable training processes. Model Selection and Comparison: The KSD can be utilized as a criterion for model selection, allowing researchers to compare different statistical models based on their KSD values. The equivalence with Wasserstein distance provides a robust metric for assessing model fit and guiding the selection of the best-performing model. Goodness-of-Fit Testing: The KSD can be applied in goodness-of-fit tests for statistical models, particularly in scenarios where the normalizing constant is intractable. The KSD-Wasserstein equivalence allows for a principled approach to testing how well a model fits the observed data, enhancing the robustness of statistical inference. By leveraging the KSD-Wasserstein equivalence, researchers and practitioners can enhance the performance and reliability of various statistical and machine learning methodologies, leading to more accurate and interpretable models.
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