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Oracle lower bounds for sampling algorithms

Niladri Chatterji
Peter Bartlett
Phil Long
Bernoulli, vol. 28(2) (2022), pp. 1074-1092
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We consider the problem of sampling from a strongly log-concave density in $\Re{d}$, and prove an information theoretic \emph{lower bound} on the number of stochastic gradient queries of the log density needed. Several popular sampling algorithms (including many Markov chain Monte Carlo methods) operate by using stochastic gradients of the log density to generate a sample; our results establish an information theoretic limit for all these algorithms. We show that for every algorithm, there exists a well-conditioned strongly log-concave target density for which the distribution of points generated by the algorithm would be at least $\epsilon$ away from the target in total variation distance if the number of gradient queries is less than $\Omega(\var d/\epsilon^2)$, where $\var d$ is the variance of the stochastic gradient. Our lower bound follows by combining the ideas of Le Cam deficiency routinely used in the comparison of statistical experiments along with standard information theoretic tools used in lower bounding Bayes risk functions. To the best of our knowledge our results provide the first nontrivial dimension-dependent lower bound for this problem.