Optimal rates for non-log-concave sampling and log-partition estimation are sometimes equal to or faster than optimization rates, but polynomial-time algorithms fall short of near-optimal performance.
Since ˜S cannot distinguish the zero function f ≡ 0 and f2, we must have max{|Lf − ˜S(f)|, |Lf2 − ˜S(f2)|} ≥ Ωm,d(Bn−m/d)
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Convergence Rates for Non-Log-Concave Sampling and Log-Partition Estimation
Optimal rates for non-log-concave sampling and log-partition estimation are sometimes equal to or faster than optimization rates, but polynomial-time algorithms fall short of near-optimal performance.