Fisher information lower bounds for sampling

10/05/2022

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by   Sinho Chewi, et al.

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We prove two lower bounds for the complexity of non-log-concave sampling within the framework of Balasubramanian et al. (2022), who introduced the use of Fisher information (FI) bounds as a notion of approximate first-order stationarity in sampling. Our first lower bound shows that averaged LMC is optimal for the regime of large FI by reducing the problem of finding stationary points in non-convex optimization to sampling. Our second lower bound shows that in the regime of small FI, obtaining a FI of at most ε^2 from the target distribution requires poly(1/ε) queries, which is surprising as it rules out the existence of high-accuracy algorithms (e.g., algorithms using Metropolis-Hastings filters) in this context.