High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm (1605.01559v4)

Abstract: We consider in this paper the problem of sampling a high-dimensional probability distribution $\pi$ having a density with respect to the Lebesgue measure on $\mathbb{R}^d$, known up to a normalization constant $x \mapsto \pi(x)= \mathrm{e}^{-U(x)}/\int_{\mathbb{R}^d} \mathrm{e}^{-U(y)} \mathrm{d} y$. Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that $U$ is continuously differentiable, $\nabla U$ is globally Lipschitz and $U$ is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order $2$ and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims.

• The paper establishes non-asymptotic convergence bounds for ULA using Wasserstein and total variation metrics under strong convexity and Lipschitz conditions.
• It quantifies convergence rates with both fixed and decreasing step sizes, delineating performance scaling in high-dimensional settings.
• Empirical results on binary regression underline ULA's efficiency, comparing favorably against methods like the Polya-Gamma Gibbs sampler.

High-Dimensional Bayesian Inference via the Unadjusted Langevin Algorithm

The paper by Alain Durmus and Eric Moulines explores the utilization of the Unadjusted Langevin Algorithm (ULA) for sampling from high-dimensional probability distributions, which is a critical task in Bayesian inference and machine learning. The main focus is on deriving non-asymptotic convergence bounds in Wasserstein distance and total variation distance for the ULA when applied to certain classes of probability distributions. By investigating both constant and decreasing step sizes in the Euler discretization of the Langevin stochastic differential equation (SDE), the authors provide insights into the theoretical performance of the algorithm in high-dimensional settings.

Core Contributions

1. Non-Asymptotic Bounds: The paper presents non-asymptotic bounds for the convergence of ULA in terms of Wasserstein distance of order 2 and total variation distance. This is done under assumptions that the potential function $U$ is continuously differentiable, its gradient $\nabla U$ is Lipschitz continuous, and $U$ is strongly convex. The dependence of these bounds on the dimension $d$ is explicit.
2. Convergence Rates: The authors quantify the rates of convergence for the measure generated by the ULA towards the target distribution $\pi$. They consider both fixed and decreasing step sizes, providing bounds on the number of iterations needed for the algorithm to reach a desired level of accuracy. They emphasize that these bounds are dependent on the smoothness of $U$.
3. Empirical Measures: Another significant aspect addressed is the convergence of empirically weighted measures from ULA. The authors report mean square error bounds and explore exponential deviation inequalities for functions that are either measurable and bounded or Lipschitz continuous.
4. Practical Implications: The paper includes an application to Bayesian inference for binary regression, demonstrating the practicality and effectiveness of the ULA in a real-world scenario.

Numerical Results and Implications

• Efficiency and Performance: The numerical experiments conducted on a binary regression model illustrate the practical performance of ULA. The algorithm's capability to approximate the posterior distribution's marginals was vividly compared with the Polya-Gamma Gibbs sampler, showcasing the potential of ULA as a robust and efficient method for Bayesian computations.
• High-Dimensional Scalability: The results suggest that ULA is particularly useful in high-dimensional settings where conventional methods may struggle due to extensive computational requirements.

Theoretical and Practical Implications

The theoretical implications of this work are substantial for the design and analysis of Markov Chain Monte Carlo (MCMC) algorithms in high-dimensional statistics. Specifically, the results refine existing knowledge about the performance of ULA and similar Langevin-based methods without requiring Metropolis-Hastings adjustments.

On the practical side, the research underscores ULA's applicability to machine learning problems that involve sampling from large-dimensional spaces, such as those found in Bayesian neural networks and other advanced probabilistic modeling frameworks.

Future Directions

The findings open several avenues for future research, particularly in extending the results to more general classes of non-convex potentials, where ULA and its variants could find even broader applications. Moreover, investigating the integration of adaptive step size strategies or preconditioning methods to further improve the scalability and efficiency of ULA could be beneficial.

Overall, this paper provides a rigorous and comprehensive assessment of ULA's potential for high-dimensional Bayesian inference, offering both theoretical foundations and practical guidelines for its deployment in complex statistical and machine learning tasks.