Analysis of Langevin Monte Carlo via convex optimization (1802.09188v2)

Abstract: In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as a first order optimization algorithm of an objective functional defined on the Wasserstein space of order $2$. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from logconcave smooth target distribution on $\mathbb{R}^d$. Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution, which we analyze as well. Besides, these new algorithms are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm, which is a popular extension of the Unadjusted Langevin Algorithm. Similar to SGLD, they only rely on approximations of the gradient of the target log density and can be used for large-scale Bayesian inference.

• The paper recasts the Unadjusted Langevin Algorithm as a convex optimization problem on the Wasserstein space.
• It derives non-asymptotic error bounds that offer explicit accuracy control for log-concave target distributions.
• The study introduces extensions for non-smooth targets with improved computational complexity in high-dimensional settings.

Analysis of Langevin Monte Carlo via Convex Optimization

The paper "Analysis of Langevin Monte Carlo via Convex Optimization" presents new insights and methodologies for understanding and extending the Unadjusted Langevin Algorithm (ULA), a critical algorithm in the domain of Bayesian inference and machine learning. The authors recast ULA within the framework of convex optimization on the Wasserstein space, a novel approach that allows for a non-asymptotic analysis of its convergence when sampling from log-concave smooth target distributions on $\mathbb{R}^d$.

Key Contributions

1. ULA as a Convex Optimization Problem: The paper uniquely formulates ULA as a first-order optimization algorithm on the Wasserstein space of order 2. This perspective leverages techniques from convex optimization, providing fresh insights into the convergence behavior and error quantification of ULA.
2. Non-Asymptotic Analysis: Through this optimization lens, the authors derive non-asymptotic bounds for ULA's error in approximating a target distribution. These bounds offer explicit control over the accuracy of ULA in practical applications, particularly when the target is log-concave.
3. Extensions to Non-Smooth Distributions: Building upon the standard ULA, the paper introduces two novel algorithms designed to handle non-smooth target distributions. These methods are natural extensions of Stochastic Gradient Langevin Dynamics (SGLD), which is itself a variant of ULA widely used in large-scale Bayesian inference.
4. Computational Complexity: The researchers provide detailed complexity analyses of these new algorithms, comparing them against existing methods. Notably, the paper demonstrates improved dimension dependence in the computational complexity for both the Wasserstein distance and total variation distance under certain conditions.

Numerical Results and Scalability

The numerical experiments underpin the theoretical findings and demonstrate the efficacy of the proposed methods in logistic regression contexts, showcasing improvements over standard techniques in both computational efficiency and result accuracy.

Theoretical Implications

The reinterpretation of ULA as an optimization problem opens pathways to apply more sophisticated optimization techniques to sampling problems in machine learning and Bayesian statistics. This cross-domain approach suggests potential scalability improvements for inference tasks in high-dimensional spaces.

Future Directions

Future work could explore further extensions of ULA and its variants to more general settings, possibly incorporating more advanced stochastic optimization techniques. Considering the paper's comprehensive framework, there is scope for applying this methodology to other stochastic differential equations and related sampling algorithms.

Conclusion

This paper's innovative use of convex optimization principles to analyze and extend ULA marks significant progress in the understanding and application of Monte Carlo methods in complex high-dimensional inference tasks. It provides a robust foundation for improving the practical efficiency of these critical algorithms in machine learning and statistics.