Convergence of Langevin MCMC in KL-divergence (1705.09048v2)

Abstract: Langevin diffusion is a commonly used tool for sampling from a given distribution. In this work, we establish that when the target density $p^*$ is such that $\log p^*$ is $L$ smooth and $m$ strongly convex, discrete Langevin diffusion produces a distribution $p$ with $KL(p||p^*)\leq \epsilon$ in $\tilde{O}(\frac{d}{\epsilon})$ steps, where $d$ is the dimension of the sample space. We also study the convergence rate when the strong-convexity assumption is absent. By considering the Langevin diffusion as a gradient flow in the space of probability distributions, we obtain an elegant analysis that applies to the stronger property of convergence in KL-divergence and gives a conceptually simpler proof of the best-known convergence results in weaker metrics.

• The paper's main contribution is its rigorous proof that Langevin MCMC converges in KL divergence under L-smooth and m-strongly convex conditions.
• It establishes that a distribution within a desired KL threshold is achieved in O(d) steps, unifying convergence metrics including total variation and Wasserstein distances.
• The innovative approach interprets Langevin diffusion as a gradient flow on probability spaces, promising efficiency gains in high-dimensional Bayesian inference and machine learning applications.

Convergence of Langevin MCMC in KL-Divergence

The paper "Convergence of Langevin MCMC in KL-Divergence" by Xiang Cheng and Peter Bartlett addresses the theoretical convergence properties of Langevin Markov Chain Monte Carlo (MCMC) algorithms. Specifically, it establishes the convergence of the Langevin MCMC in Kullback-Leibler (KL) divergence under the conditions that the target density's logarithm is both L-smooth and m-strongly convex. The authors present rigorous mathematical proofs that demonstrate that the Langevin MCMC yields a distribution within a specified KL divergence threshold after $O(d)$ steps, where $d$ represents the dimensionality of the space.

The primary contribution of this work is the demonstration of KL-convergence without depending on the usually considered metrics like total variation or Wasserstein distances. Notably, this convergence in KL divergence implies convergence in the aforementioned metrics, allowing the paper to unify and extend previous results. Furthermore, this is done through an innovative interpretation of the Langevin diffusion process as a gradient flow on the space of probability distributions, leveraging tools from the calculus of variations and probabilistic measure theory.

The authors provide a detailed analysis and comparison to earlier work, particularly the studies by Dalalyan and Durmus & Moulines. They note improvements and extensions in their results compared to established results, especially in high-dimensional settings. Their method circumvents previous approaches by simplifying key aspects of the analysis, leading to an elegant proof structure regarding the convergence behavior in KL divergence, which has crucial implications for practical applications, especially in machine learning and Bayesian inference.

Furthermore, the paper also investigates the scenario when the strong convexity condition is absent. This more challenging analysis necessitates alternative approaches, and the authors present a weaker dependence on KL bounds that still yields practical convergence rates, albeit with a broadened set of assumptions.

Theoretical implications of this research suggest substantial gains in understanding the behavior of Langevin MCMC, contributing to its robustness as an inference tool in large-scale Bayesian problems. Practically, this work supports the design of more efficient sampling methods in machine learning, particularly those involving complex, high-dimensional distributions. Such results are critical as they lead to the potential acceleration of convergence for variational inference and provide enhanced guarantees for sampling in discrete high-dimensional spaces.

Future work extending from this paper could focus on exploring relaxation of the smoothness and convexity conditions further, or on adapting these methods to other complex classes of distributions that arise in real-world data analytics applications. Additionally, examining the scalability and computational trade-offs in various real-world scenarios would be a valuable next step. The intersection of these theoretical insights with advancements in computational power could yield significant benefits in the broad domain of statistical learning and decision-making systems.