L2-Wasserstein Contraction of Modified Euler Schemes for SDEs: Theory and Applications to Non-Asymptotic Bounds and Strong Law of Large Numbers

Core Concepts

This paper proves the weak L2-Wasserstein contraction of a modified Euler scheme for stochastic differential equations (SDEs) with super-linear and locally dissipative drifts under the condition of high diffusivity.

Abstract

• Bibliographic Information: Bao, J., & Hao, J. (2024). L2-Wasserstein contraction of modified Euler schemes for SDEs with high diffusivity and applications. arXiv preprint arXiv:2411.01731.
• Research Objective: To establish the weak L2-Wasserstein contraction for a modified Euler scheme applied to SDEs with super-linear drifts and explore its applications in non-asymptotic L2-Wasserstein bounds and the strong law of large numbers.
• Methodology: The authors utilize a synchronous coupling approach and construct an equivalent quasi-metric to analyze the contraction properties of the modified Euler scheme under the L2-Wasserstein distance. They further analyze specific instances of the modified Euler scheme, including the projected Euler scheme and the tamed Euler scheme.
• Key Findings: The paper proves that the distributions of the modified Euler scheme, starting from different initial distributions, exhibit weak contraction under the L2-Wasserstein distance, given sufficiently high diffusivity. This contraction property is then applied to derive quantitative non-asymptotic L2-Wasserstein bounds between the exact and numerical invariant probability measures. Additionally, the authors establish the strong law of large numbers for additive functionals associated with the modified Euler scheme, demonstrating the convergence of time averages to spatial averages with respect to the equilibrium measure.
• Main Conclusions: The study provides a theoretical framework for analyzing the long-term behavior of a modified Euler scheme for SDEs with super-linear drifts. The established weak L2-Wasserstein contraction has significant implications for understanding the convergence properties of the scheme and its applications in sampling from target distributions and analyzing ergodic properties.
• Significance: This research contributes to the field of numerical analysis for SDEs, particularly in the context of non-convex optimization and sampling algorithms. The results have implications for developing efficient and accurate numerical methods for a wider class of SDEs with super-linear drift terms.
• Limitations and Future Research: The study focuses on the setting of high diffusivity. Further research could explore the contraction properties and convergence behavior of the modified Euler scheme under weaker assumptions on the noise intensity. Additionally, investigating the application of the proposed framework to other variants of Euler schemes and more complex SDE models could be promising research directions.

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$L^2$-Wasserstein contraction of modified Euler schemes for SDEs with high diffusivity and applications

by Jianhai Bao,... at arxiv.org 11-05-2024

https://arxiv.org/pdf/2411.01731.pdf

Deeper Inquiries

How does the choice of the modified drift function b(δ) in the modified Euler scheme impact the contraction rate and the non-asymptotic bounds?

Answer:
The choice of the modified drift function b(δ) significantly impacts both the contraction rate (λ) and the non-asymptotic L2-Wasserstein bounds in the modified Euler scheme. Here's a breakdown:

Contraction Rate (λ):

Dissipativity Outside the Ball: The dissipativity of b(δ) outside the ball of radius R, controlled by the constant K∗R in (H2), directly influences λ. A larger K∗R (stronger dissipativity) leads to a larger λ2 and consequently a faster contraction rate.
Behavior Inside the Ball: While (H2) allows b(δ) to be non-dissipative inside the ball, its behavior there, governed by CR, still affects λ1. A smaller CR (closer to Lipschitz continuity) results in a larger λ1 and thus a faster contraction.
Interplay of Constants: The constants C⋆R, C⋆⋆R, and σ (noise intensity) also play a role in determining λ1 and λ2, highlighting the complex interplay between drift and noise in achieving contraction.

Non-Asymptotic Bounds:

Convergence Rate (β): The convergence rate of the non-asymptotic bound, denoted by β in (H7), is directly determined by how closely b(δ) approximates the original drift b as δ approaches 0. A larger β implies a faster convergence rate.
Constant Factor (C0): The constant factor C0 in the bound is influenced by various factors, including the growth bounds of b(δ) and its derivatives, as well as the constants involved in the contraction rate.

Specific Examples (Tamed vs. Projected Euler):

Tamed Euler: This scheme modifies b to control its growth. The choice of the taming function directly impacts β. Corollary 1.7 shows a rate close to θ (from the taming construction), while Theorem 1.9, with a different taming, achieves β = 1.
Projected Euler: This scheme uses a projection mapping π(δ). The projection's properties, particularly its polynomial growth, dictate the convergence rate, which is β = 1 as shown in Corollary 1.10.
In summary, carefully designing b(δ) to balance approximation accuracy (for small δ) and controlled growth (especially outside the ball) is crucial for optimizing both the contraction rate and the non-asymptotic bounds in the modified Euler scheme.

Could the assumption of high diffusivity be relaxed while still ensuring the contraction property, perhaps by employing alternative analytical techniques or considering different probability metrics?

Answer:
Relaxing the high diffusivity assumption while preserving the L2-Wasserstein contraction is a challenging but important question. Here are some potential avenues for exploration:

Alternative Probability Metrics:

Weighted Wasserstein Distances: Employing weighted Wasserstein distances, where the weight function emphasizes different regions of the state space, could potentially compensate for weaker diffusivity in certain regions. This approach might be particularly relevant when the drift exhibits varying dissipativity properties.
Transportation-Information Inequalities: Exploring connections between L2-Wasserstein contraction and transportation-information inequalities could offer a way to establish contraction with weaker diffusivity assumptions. These inequalities relate transport costs (Wasserstein distances) to relative entropy, potentially providing a more refined analysis.

Analytical Techniques:

Couplings Tailored to Drift Structure: Developing coupling techniques that are specifically designed to exploit the structure of the drift, even in the absence of global dissipativity, could be promising. For instance, if the drift exhibits some form of local contraction or hypoellipticity, tailored couplings might be able to capture these properties and establish contraction.
Lyapunov Function Construction: Constructing more sophisticated Lyapunov functions that capture the system's dynamics more effectively could help relax the diffusivity requirement. This might involve incorporating information about the drift's behavior in different regions or using time-dependent Lyapunov functions.

Trade-offs and Limitations:

Convergence Rate: Relaxing the high diffusivity assumption might lead to slower contraction rates or weaker non-asymptotic bounds.
Applicability: Alternative techniques or metrics might impose additional restrictions on the drift or the noise structure.
In conclusion, while the high diffusivity assumption is convenient for establishing L2-Wasserstein contraction, exploring alternative metrics and analytical techniques tailored to the specific problem structure holds promise for relaxing this assumption in certain scenarios. However, careful consideration of potential trade-offs and limitations is crucial.

What are the implications of these findings for the development of variance reduction techniques or other enhancements to improve the efficiency and accuracy of the modified Euler scheme in practical applications?

Answer:
The findings regarding L2-Wasserstein contraction of the modified Euler scheme have significant implications for enhancing its efficiency and accuracy in practical applications, particularly in the context of sampling and variance reduction:

Variance Reduction Techniques:

Convergence Diagnostics: The contraction rate λ provides a quantitative measure of how quickly the scheme converges to the stationary distribution. This information can be valuable for designing adaptive step-size strategies, where δ is adjusted based on the estimated contraction rate to optimize convergence speed.
Control Variates: The non-asymptotic bounds offer insights into the bias introduced by the discretization. This knowledge can be leveraged to construct control variates that effectively reduce the variance of estimators based on the modified Euler scheme.

Algorithm Design and Analysis:

Optimal Parameter Selection: The explicit dependence of λ and the non-asymptotic bounds on the drift modifications (b(δ) and π(δ)) provides guidance for selecting optimal parameters in the tamed and projected Euler schemes. This can lead to algorithms with improved convergence properties and reduced bias.
Beyond Linearity: The ability to handle super-linear drifts significantly broadens the applicability of these techniques to a wider range of problems, including those with non-convex potentials commonly encountered in machine learning and Bayesian inference.

Practical Considerations:

Computational Cost: While these findings offer valuable theoretical insights, practical implementations need to consider the computational cost associated with different modifications and variance reduction techniques.
High-Dimensional Settings: Extending these results to high-dimensional settings, where the curse of dimensionality can hinder performance, is an important direction for future research.
In summary, the established L2-Wasserstein contraction properties provide a solid theoretical foundation for developing enhanced versions of the modified Euler scheme. By leveraging these insights, we can design more efficient and accurate sampling algorithms, particularly for applications involving complex and high-dimensional distributions.