Faster logconcave sampling from a cold start in high dimension

Abstract: We present a faster algorithm to generate a warm start for sampling an arbitrary logconcave density specified by an evaluation oracle, leading to the first sub-cubic sampling algorithms for inputs in (near-)isotropic position. A long line of prior work incurred a warm-start penalty of at least linear in the dimension, hitting a cubic barrier, even for the special case of uniform sampling from convex bodies. Our improvement relies on two key ingredients of independent interest. (1) We show how to sample given a warm start in weaker notions of distance, in particular $q$-R\'enyi divergence for $q=\widetilde{\mathcal{O}}(1)$, whereas previous analyses required stringent $\infty$-R\'enyi divergence (with the exception of Hit-and-Run, whose known mixing time is higher). This marks the first improvement in the required warmness since Lov\'asz and Simonovits (1991). (2) We refine and generalize the log-Sobolev inequality of Lee and Vempala (2018), originally established for isotropic logconcave distributions in terms of the diameter of the support, to logconcave distributions in terms of a geometric average of the support diameter and the largest eigenvalue of the covariance matrix.

Overview of "Faster Logconcave Sampling from a Cold Start in High Dimension"

The paper introduces a novel algorithm designed to efficiently sample from logconcave distributions in high-dimensional spaces starting from a cold start. Previous methodologies typically required a warm start, where the initial distribution is already close to the target distribution, but this research surpasses the cubic time complexity bottleneck associated with earlier approaches.

Key Contributions

1. Warm Start Reduction: The core innovation of the paper is an algorithm that can generate a warm start for sampling an arbitrary logconcave density using an evaluation oracle. This process leverages $\mathit{q}$-Rényi divergence for $q=\widetilde{\mathcal{O}}(1)$ and does not rely on stronger $\infty$-Rényi divergence as previous methods did.
2. Algorithmic and Theoretical Improvements:
   • The work refines the classical warm start requirement analysis, reducing warm-start constraints in terms of weaker distance measures, particularly through the use of $\mathit{q}$-Rényi divergence.
   • It introduces improvements to the log-Sobolev inequality for logconcave distributions, extending the diameter-dependent constant to include the largest eigenvalue of the covariance matrix.
3. Enhanced Theoretical Boundaries:
   • The researchers develop bounds on functional inequalities, specifically enhancing the log-Sobolev constant for strong logconcave distributions with compact support.
   • It provides a novel bound $(\pi\gamma_{\sigma^{2}})\lesssim D\norm{cov\pi}^{1/2}$, where $(\pi)$ refers to the log-Sobolev constant for the given measure and $\gamma_{\sigma^{2}}$ is a Gaussian distribution parameterized by $\sigma^{2}$.
4. Sampling Complexity:
   • The proposed method achieves sub-cubic sampling algorithms for inputs in (near-)isotropic positions, outperforming previous cubic time complexity barriers.
   • The paper establishes an improved query complexity for sampling that scales sub-cubically with respect to the dimensions of the input.

Theoretical Implications

The exploration into relaxing warm start requirements while maintaining the efficiency of the proposed sampling method suggests potential applications in various domains, such as differential privacy, scientific computing, and systemic biology. The findings also stand to influence theories in asymptotic convex geometry and functional analysis, given the advancements in handling the log-Sobolev inequality and the relationship with covariance metrics.

Practical Applications

Practically, the paper's advancements in logconcave sampling could have significant implications for tasks requiring high-dimensional integration, Bayesian inference, and stochastic optimization where traditional sampling methods suffer from dimensional scalability issues.

Future Directions

The research paves the way for further exploration into:

• Extending the sampling algorithms to broader classes of distributions beyond logconcave measures.
• Investigating the integration of these methods with machine learning models that rely on sampling from high-dimensional distributions.
• Exploring the possibility of enhancing the mixing times of alternative Markov Chain Monte Carlo (MCMC) methodologies to further reduce dependency on initial warmness assumptions.

By introducing a refined approach towards high-dimensional logconcave sampling, this paper sets a foundation for more efficient algorithmic developments across multiple fields relying on this class of statistical distributions. The use of weaker divergence metrics opens new pathways for practical and theoretical improvements in high-dimensional data processing.