Log-concave sampling: Metropolis-Hastings algorithms are fast (1801.02309v4)

Abstract: We consider the problem of sampling from a strongly log-concave density in $\mathbb{R}^d$, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin diffusion, combined with an accept-reject step. Relative to known guarantees for the unadjusted Langevin algorithm (ULA), our bounds show that the use of an accept-reject step in MALA leads to an exponentially improved dependence on the error-tolerance. Concretely, in order to obtain samples with TV error at most $\delta$ for a density with condition number $\kappa$, we show that MALA requires $\mathcal{O} \big(\kappa d \log(1/\delta) \big)$ steps, as compared to the $\mathcal{O} \big(\kappa^2 d/\delta^2 \big)$ steps established in past work on ULA. We also demonstrate the gains of MALA over ULA for weakly log-concave densities. Furthermore, we derive mixing time bounds for the Metropolized random walk (MRW) and obtain $\mathcal{O}(\kappa)$ mixing time slower than MALA. We provide numerical examples that support our theoretical findings, and demonstrate the benefits of Metropolis-Hastings adjustment for Langevin-type sampling algorithms.

• The paper derives non-asymptotic upper bounds on MALA’s mixing time for strongly log-concave distributions, demonstrating improved error tolerance compared to ULA.
• It shows that MALA converges exponentially faster from a warm start with linear dimension dependence, outperforming zeroth-order methods like MRW.
• Numerical experiments validate MALA’s efficiency gains, highlighting its practical benefits for high-dimensional Bayesian and machine learning applications.

Log-concave Sampling: Metropolis-Hastings Algorithms Are Fast

This essay provides a detailed examination of the paper "Log-concave sampling: Metropolis-Hastings algorithms are fast" by Dwivedi, Chen, Wainwright, and Yu. This paper presents a comprehensive paper on sampling from strongly log-concave distributions using the Metropolis-adjusted Langevin Algorithm (MALA).

Main Contributions

The paper's primary contribution is the derivation of a non-asymptotic upper bound on the mixing time of MALA when sampling from a strongly log-concave density in $\mathbb{R}^d$. MALA improves upon the Unadjusted Langevin Algorithm (ULA) by integrating an accept-reject step, leading to enhanced mixing properties:

• Improved Dependence on Error Tolerance: MALA requires $\mathcal{O}(\kappa d \log(1/\delta))$ steps to achieve samples with a total variation (TV) error of $\delta$, compared to the $\mathcal{O}(\kappa^2 d/\delta^2)$ steps needed by ULA. Here, $\kappa$ denotes the condition number of the target density.
• Warm Start Analysis: The paper demonstrates that MALA converges exponentially faster than ULA from a warm start, with linear dimension dependence and exponentially better tolerance dependence.
• Zeroth vs. First-Order Methods: The paper provides a detailed comparison, showing that first-order methods (MALA) outperform zeroth-order methods (like Metropolized Random Walk - MRW) in terms of mixing times. It derives $\mathcal{O}(d^2 \log(1/\delta))$ steps for MRW, which is slower compared to MALA for the same task.
• Weakly Log-concave Distributions: A modified version of MALA is proposed for distributions that are weakly log-concave, achieving a mixing time of $$\tilde{\mathcal{O}}(\kappa^{3/2}L^{1.5}d^{1.5}/\delta^{1.5})$$ in contrast to $\tilde{\mathcal{O}}(\kappa^3L^2d^{2.5}/\delta^4)$ for ULA in similar settings.

Numerical Results and Practical Implications

The authors support their theoretical findings with numerical experiments demonstrating the superior efficacy of MALA over ULA and MRW. The practical implications of these results are significant for fields that require efficient sampling methods for stochastic models, such as Bayesian statistics and machine learning. MALA's improved sampling efficiency allows practitioners to achieve equivalent or higher accuracy with fewer computational resources, broadening the scope of feasible applications involving high-dimensional data.

Theoretical and Practical Relevance

The theoretical insights presented in this paper extend the foundational understanding of MCMC algorithms, particularly in analyzing the intricate relationship between algorithmic modifications (e.g., the accept-reject step) and performance enhancements. Practically, these advances pave the way for more efficient algorithms in real-world applications where log-concave distributions are prevalent.

Future Directions

The results invite several avenues for future research, including:

• Investigating whether the observed performance differences between first-order and zeroth-order methods can be formally established as fundamental algorithmic limitations.
• Exploring further improvements or variations of the Metropolis-Hastings framework that might close the gap between theoretical guarantees and empirical performance.
• Extending the analysis to scenarios with non-log-concave distributions, potentially expanding the applicability of MALA beyond the current scope.

This paper stands out for its detailed theoretical analysis, strong numerical validation, and clear implications for both the academic and practical aspects of MCMC sampling methods.