Oracle lower bounds for stochastic gradient sampling algorithms

We consider the problem of sampling from a strongly log-concave density in Rd, and prove an information theoretic lower bound on number stochastic gradient queries log needed. Several popular algorithms (including many Markov chain Monte Carlo methods) operate by using gradients to generate sample; our results establish limit for all these algorithms. show that every algorithm, there exists well-conditioned target which distribution points generated algorithm would be at least ε away total variation distance if is less than Ω(σ2d/ε2), where σ2d variance gradient. Our follows combining ideas Le Cam deficiency routinely used comparison statistical experiments along with standard tools bounding Bayes risk functions. To best knowledge provide first nontrivial dimension-dependent this problem.