Unified Complexity Bound Established for Logconcave Distribution Sampling
Researchers have derived a unified and nearly tight complexity bound for sampling arbitrary logconcave distributions using the In-and-Out algorithm with exponential lifting. The analysis introduces an improved bound on the Poincaré constant of lifted distributions, a key technical contribution. This result is significant because it provides nearly optimal convergence rates across both constrained settings (like Gaussians restricted to convex bodies) and well-conditioned settings (like strongly logconcave smooth densities).
A new theoretical result in computational mathematics provides a simplified and unified approach to bounding the complexity of sampling from logconcave distributions, a fundamental problem in machine learning and statistics. The work leverages the In-and-Out algorithm combined with exponential lifting, with the main technical innovation being an improved bound on the Poincaré constant of lifted distributions. The resulting convergence rate is nearly tight across diverse problem settings, including both constrained optimization scenarios and well-conditioned density estimation problems. This unified framework eliminates the need for separate analyses of different problem classes, potentially streamlining both theoretical understanding and practical algorithm design. The paper represents a contribution to the intersection of algorithms, machine learning, and probability theory.
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- arXiv stat.MLCenter
A unified complexity bound for logconcave sampling
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