Unified Complexity Bound Established for Logconcave Distribution Sampling
A new preprint on arXiv presents a simple, unified, and nearly tight complexity bound for sampling arbitrary logconcave distributions using the In-and-Out algorithm with exponential lifting. The key technical advance is an improved bound on the Poincaré constant of a lifted distribution, which tightens convergence guarantees across both constrained and well-conditioned settings. The result matters because logconcave sampling underpins a wide range of algorithms in machine learning, statistics, and theoretical computer science, and tighter bounds directly inform practical algorithm design.
A five-page preprint submitted to arXiv on June 10, 2026 by Yunbum Kook and co-authors establishes a unified convergence bound for sampling from arbitrary logconcave distributions starting from a warm initialization. The algorithm analyzed is the In-and-Out sampler combined with exponential lifting, a technique that embeds the target distribution into a higher-dimensional space to improve mixing. The central technical contribution is a sharper bound on the Poincaré constant of the lifted distribution, which controls how quickly the Markov chain converges to stationarity. The resulting rate is shown to be nearly tight in two canonical regimes: constrained sampling (such as a Gaussian restricted to a convex body) and well-conditioned sampling (such as strongly logconcave and smooth densities). By handling both regimes within a single framework, the work simplifies and unifies previously separate lines of analysis. The paper is cross-listed across algorithms, machine learning, probability, and statistics communities, reflecting the broad relevance of logconcave sampling.
What's missing
As a preprint, the work has not yet undergone formal peer review. The paper does not discuss empirical validation or practical runtime comparisons against competing samplers such as Langevin Monte Carlo or hit-and-run, leaving open questions about real-world performance gains.
What different sources said
- arXiv stat.MLCenter
A unified complexity bound for logconcave sampling
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