Entropy contraction of the Gibbs sampler under log-concavity (2410.00858v1)

Abstract: The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm which iteratively samples from the conditional distributions of a probability measure $\pi$ of interest. Under the assumption that $\pi$ is strongly log-concave, we show that the random scan Gibbs sampler contracts in relative entropy and provide a sharp characterization of the associated contraction rate. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations of $\pi$ needed for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. If $\pi$ is non-strongly log-concave, the convergence rate in entropy degrades from exponential to polynomial. Our techniques are versatile and extend to Metropolis-within-Gibbs schemes and the Hit-and-Run algorithm. A comparison with gradient-based schemes and the connection with the optimization literature are also discussed.

• The paper establishes entropy contraction bounds for the Gibbs Sampler under strong log-concavity, demonstrating a geometric decay in relative entropy.
• It employs variational characterizations and triangular transport maps to derive mixing time estimates scaling as O(κ* M log(1/ε)), independent of the dimensionality.
• The methodology extends to related MCMC algorithms, offering a robust framework for efficient high-dimensional sampling in log-concave models.

Entropy Contraction of the Gibbs Sampler under Log-Concavity

The paper "Entropy Contraction of the Gibbs Sampler under Log-Concavity," authored by Filippo Ascolani, Hugo Lavenant, and Giacomo Zanella, addresses the convergence properties of the Gibbs Sampler (GS), which is a fundamental Markov Chain Monte Carlo (MCMC) algorithm. The paper's focus lies on analyzing and quantifying the contraction of relative entropy for the GS under the assumption of strong log-concavity of the target distribution.

Convergence Analysis of the Gibbs Sampler

Assumptions and Notation

The paper starts by presenting the foundational assumptions for the target distribution, denoted as $\pi$. Specifically, $\pi$ is assumed to be a strongly log-concave distribution, which translates to having a density of the form $\pi(dx) = \exp(-U(x))dx$, where $U$ is a $\lambda$-convex and $L$-smooth function. This setting ensures that $U$ satisfies the conditions for strong convexity and smoothness, characterized by parameters $\lambda$ and $L$, respectively, leading to a condition number $\kappa = L/\lambda$.

Main Result

The primary result of the paper elucidates the entropy contraction property of the random scan Gibbs Sampler. For a target distribution $\pi$ satisfying the strong log-concavity assumption, the paper proves that:

$$KL(\mu P^\mathrm{GS} | \pi) \leq \left(1 - \frac{1}{\kappa^* M} \right) KL(\mu | \pi),$$

where $KL(\cdot | \pi)$ denotes the Kullback-Leibler (KL) divergence relative to $\pi$, and $\kappa^*$ is a "coordinate-wise" condition number. This result implies that the relative entropy decreases geometrically at a rate that depends on the condition number and the number of coordinates $M$.

Implications

The contraction rate of KL divergence directly implies a bound on the mixing time of the Gibbs Sampler. Specifically, for the chain to mix within an $\epsilon$-error in relative entropy, the required number of iterations $n$ scales as $\mathcal{O}(\kappa^* M \log(1/\epsilon))$. Crucially, this rate is independent of the dimensionality $d$, highlighting the efficiency of the Gibbs Sampler in high-dimensional settings, provided that evaluating conditionals is computationally favorable compared to evaluating the joint potential $U$.

Extension to Other Sampling Methods

The techniques developed for the Gibbs Sampler are versatile and extend to other methods such as the Hit-and-Run algorithm and Metropolis-within-Gibbs schemes. For the Hit-and-Run algorithm, the paper demonstrates that the entropy contraction property holds with an analogous contraction rate that scales with the dimensionality of the subspace being sampled. This universality further attests to the robustness of the entropy-based analysis presented.

Comparative Analysis and Computational Considerations

The paper provides a detailed comparative analysis between the Gibbs Sampler and gradient-based MCMC methods such as Langevin and Hamiltonian Monte Carlo. It is shown that GS exhibits favorable scaling properties in terms of computational cost under log-concavity assumptions. Notably, while gradient-based methods typically suffer from a complexity that increases with the dimension $d$, the Gibbs Sampler remains efficient due to its coordinate-wise updates that sidestep the challenges of high-dimensionality.

Analytical Techniques

The proofs hinge on sophisticated variational characterizations of the Gibbs kernel in terms of relative entropy and employ triangular transport maps to decompose the entropy into tractable components. These techniques are instrumental in deriving sharp bounds and could have broader applicability in studying other MCMC algorithms.

Conclusion and Future Directions

The paper establishes a rigorous foundation for understanding the entropy contraction properties of the Gibbs Sampler under log-concavity, providing explicit and sharp bounds on mixing times. These results not only enhance our theoretical understanding but also have practical implications for efficiently sampling from high-dimensional log-concave distributions. Future research could explore further refinements of the contraction rates and extend these techniques to other structured distributions beyond log-concavity, thereby broadening the applicability of these foundational insights in MCMC theory.

By leveraging intricate probabilistic and functional analysis tools, the authors contribute substantially to the landscape of MCMC convergence analysis, promising improvements in both theoretical frameworks and practical algorithms for high-dimensional sampling problems.