Theoretical guarantees for approximate sampling from smooth and log-concave densities (1412.7392v6)

Abstract: Sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, the exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, there is no well-developed theory providing meaningful nonasymptotic guarantees for the approximate sampling procedures, especially in the high-dimensional problems. This paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log-concave density defined on (\RR^p), for some integer (p>0). We establish nonasymptotic bounds for the error of approximating the target distribution by the one obtained by the Langevin Monte Carlo method and its variants. We illustrate the effectiveness of the established guarantees with various experiments. Underlying our analysis are insights from the theory of continuous-time diffusion processes, which may be of interest beyond the framework of log-concave densities considered in the present work.

• The paper provides nonasymptotic error bounds for LMC and LMCO methods, quantifying convergence in high-dimensional settings.
• It characterizes the discretization and finite-time errors to balance computational efficiency with sampling accuracy.
• Empirical evaluations on synthetic datasets underscore the practical applicability of the proposed sampling algorithms.

Theoretical Guarantees for Approximate Sampling from Smooth and Log-Concave Densities

The paper addresses the issue of approximate sampling from distributions having smooth and log-concave densities, which is a fundamental problem in statistics and machine learning. The challenge arises particularly in high-dimensional settings where exact sampling becomes computationally infeasible. The focus is on providing nonasymptotic guarantees for the Langevin Monte Carlo (LMC) method and its variants.

Core Contributions

1. Approximate Sampling Context: The paper highlights the significance of sampling in statistical inference tasks like constructing estimators, test procedures, or confidence intervals. The challenge predominantly lies in high-dimensional sampling where the computational cost is prohibitive.
2. Model Considered: The paper considers distributions with smooth and log-concave densities. The function $f: \mathbb{R}^p \to \mathbb{R}$, acting as the negative log-likelihood or log-posterior, is assumed to be smooth and strongly convex with a Lipschitz continuous gradient.
3. Langevin Monte Carlo (LMC): The LMC algorithm, a Markov Chain Monte Carlo (MCMC) method, is examined in-depth as a tool for approximate sampling. The process involves discretizing a continuous-time diffusion process (Langevin diffusion) to approximate the target distribution.
4. Nonasymptotic Guarantees:
   • Convergence Analysis: The paper establishes nonasymptotic bounds on the total variation distance between the distribution produced by LMC and the target distribution. This includes explicit and computable quantities for the bounds, grounding theoretical guarantees in practical applicability.
   • Error Characterization: The error is categorized into two types: the error due to finite sampling time and the error due to discretization step-size. The bounds provided allow for balancing these errors against computational cost.
5. Application of the Ozaki Discretization: For distributions with a smooth Hessian, the LMCO algorithm utilizing Ozaki discretization is presented as a refinement of LMC, promising faster convergence through more accurate diffusion approximations.
6. Numerical Experiments: Empirical evaluation on synthetic datasets, including Gaussian mixtures and logistic regression, support the theoretical findings, showcasing the utility and efficiency of the proposed methods.

Key Results

• The paper proves that in strongly log-concave cases, the LMC requires $O(\epsilon^{-2}(p^3 + p\log^2(1/\epsilon)))$ gradient evaluations for an error $\epsilon$. In nonstrongly log-concave cases, the bound becomes $O(\epsilon^{-4}p^5\log^2(p\vee^{-1}))$.
• For an LMCO algorithm with Ozaki discretization, the dimensionality-related computational complexity is reduced significantly in specific scenarios, emphasizing its applicability in certain precision-demanding contexts.

Implications and Future Directions

• Preconditioning and Warm Start: The potential of reducing computation through preconditioning or using warm starts is noted, providing pathways for future refinements in high-dimensional settings.
• Practical Impact: These results suggest that LMC-based methods can achieve high accuracy without Metropolis-Hastings adjustments, simplifying implementation while ensuring polynomial time complexity.
• Theoretical Framework: The paper contributes to the nascent literature connecting convex optimization techniques with sampling algorithms, paving the way for further exploration of their interplay.

In conclusion, the paper offers a significant step forward in understanding and improving the computational efficiency of sampling algorithms for smooth and log-concave densities. Through rigorous analysis and empirical validation, it lays foundational work that could influence both theoretical advancements and practical applications in AI and statistics.