Entropy contraction of the Gibbs sampler under log-concavity

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• Giacomo Zanella (Bocconi University)
• Monday 25 November 2024, 14:20-15:10
• Seminar Room 1, Newton Institute.

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SSDW04 - Monte Carlo sampling: beyond the diffusive regime

The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm which iteratively samples from the conditional distributions of a probability measure π of interest. Under the assumption that π is strongly log-concave, we show that the random scan Gibbs sampler contracts in relative entropy and provide a sharp characterization of the associated contraction rate. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations of π needed for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. If π is non-strongly log-concave, the convergence rate in entropy degrades from exponential to polynomial. Our techniques are versatile and extend to Metropolis-within-Gibbs schemes and the Hit-and-Run algorithm. A comparison with gradient-based schemes and the connection with the optimization literature are also discussed. This is joint work with Filippo Ascolani and Hugo Lavenant.

This talk is part of the Isaac Newton Institute Seminar Series series.

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