Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm
This addresses efficient sampling for heavy-tailed distributions in statistics and machine learning, representing an incremental improvement with specific theoretical bounds.
The paper tackles sampling from heavy-tailed densities by applying the Unadjusted Langevin Algorithm to transformed versions of the target, achieving polynomial-order oracle complexities in dimension and accuracy for specific classes of densities.
We analyze the oracle complexity of sampling from polynomially decaying heavy-tailed target densities based on running the Unadjusted Langevin Algorithm on certain transformed versions of the target density. The specific class of closed-form transformation maps that we construct are shown to be diffeomorphisms, and are particularly suited for developing efficient diffusion-based samplers. We characterize the precise class of heavy-tailed densities for which polynomial-order oracle complexities (in dimension and inverse target accuracy) could be obtained, and provide illustrative examples. We highlight the relationship between our assumptions and functional inequalities (super and weak Poincaré inequalities) based on non-local Dirichlet forms defined via fractional Laplacian operators, used to characterize the heavy-tailed equilibrium densities of certain stable-driven stochastic differential equations.