Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices (1903.08568v4)

Abstract: We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution $\nu = e^{-f}$ on $\mathbb{R}^n$. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming $\nu$ satisfies a log-Sobolev inequality and the Hessian of $f$ is bounded. Notably, we do not assume convexity or bounds on higher derivatives. We also prove convergence guarantees in R\'enyi divergence of order $q > 1$ assuming the limit of ULA satisfies either the log-Sobolev or Poincar\'e inequality. We also prove a bound on the bias of the limiting distribution of ULA assuming third-order smoothness of $f$, without requiring isoperimetry.

• The paper establishes that ULA converges rapidly under isoperimetric inequalities, eliminating the need for strict convexity assumptions.
• It proves convergence in KL and Rényi divergences with quantified iteration bounds, leveraging log-Sobolev and smoothness conditions.
• It provides bounds on the asymptotic bias of ULA, showing linear scaling with step size while practical observations hint at quadratic behavior.

Overview of the Paper: "Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices"

The paper, "Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices," authored by Santosh S. Vempala and Andre Wibisono, explores the Unadjusted Langevin Algorithm (ULA) for sampling from complex, high-dimensional probability distributions. The primary focus is on establishing rapid convergence results using isoperimetric inequalities rather than relying on strong assumptions like convexity.

Key Contributions and Results

• ULA with Isoperimetric Inequalities: The paper establishes that ULA can converge efficiently under isoperimetric conditions, such as log-Sobolev inequality (LSI) or Poincaré inequality, without assuming convexity or high-order derivative bounds. It effectively extends the applicability of ULA in non-logconcave scenarios, often found in modern applications.
• Convergence in KL and Rényi Divergence: Convergence is established in terms of Kullback-Leibler (KL) divergence, and under certain conditions, in Rényi divergence as well. The paper proves that under LSI and smoothness conditions, ULA can reach a KL divergence of H_\nu(\rho_k) ≤ ε in O(κ^2n/ε) iterations, where κ is a condition number.
• Bound on Asymptotic Bias: The paper provides insights into the bias inherent in ULA when considering the limiting distribution. Specifically, it addresses how these biases can be bounded when using isoperimetric conditions, highlighting how bias scales linearly with step size in the analysis yet should ideally scale quadratically as observed in practical settings.

Implications and Theoretical Insights

From a practical standpoint, this research broadens the scope of ULA for efficient sampling beyond the confines of logconcavity, especially beneficial for non-convex distributions prevalent in various domains. Furthermore, the paper sheds light on the theoretical robustness of ULA when LSI or Poincaré inequalities are leveraged, suggesting that these conditions might suffice for rapid algorithmic convergence.

Future Directions and Open Questions

The research opens several avenues for further exploration:

1. Optimal Analysis of ULA: It remains an open question whether an analysis can be conducted that yields a bias that aligns optimally with empirical observations, particularly for distributions with stronger non-logconcave characteristics.
2. Verification of Isoperimetric Assumptions: The assumptions regarding the biased limit satisfying isoperimetry in the absence of strong convexity need further investigation. There is a potential to relax these assumptions while maintaining rapid convergence.
3. Affine-Invariant Versions of Langevin Dynamics: The exploration of affine-invariant algorithms could present breakthroughs for preserving computational efficiency in high dimensions, circumventing polynomial dependencies on smoothness.

Conclusion

In summary, the paper by Vempala and Wibisono provides a rigorous theoretical framework that extends the utility of the Unadjusted Langevin Algorithm by incorporating isoperimetric inequalities. It paves the way for robust, efficient sampling in complex, high-dimensional spaces, essential for advancing both theoretical and practical applications in sampling and optimization. The inquiry into isoperimetry's sufficiency posits compelling challenges and opportunities for further research, especially within the ambit of AI and machine learning.