Estimating Normalizing Constants for Log-Concave Distributions: Algorithms and Lower Bounds
This addresses a fundamental computational challenge in fields like machine learning and statistics, providing both algorithmic improvements and the first nontrivial lower bounds for this problem.
The paper tackles the problem of estimating the normalizing constant for log-concave distributions, achieving an algorithm with query complexity of Õ(d^{4/3}κ + d^{7/6}κ^{7/6}/ε²) and proving a lower bound of d^{1-o(1)}/ε^{2-o(1)} queries.
Estimating the normalizing constant of an unnormalized probability distribution has important applications in computer science, statistical physics, machine learning, and statistics. In this work, we consider the problem of estimating the normalizing constant $Z=\int_{\mathbb{R}^d} e^{-f(x)}\,\mathrm{d}x$ to within a multiplication factor of $1 \pm \varepsilon$ for a $μ$-strongly convex and $L$-smooth function $f$, given query access to $f(x)$ and $\nabla f(x)$. We give both algorithms and lowerbounds for this problem. Using an annealing algorithm combined with a multilevel Monte Carlo method based on underdamped Langevin dynamics, we show that $\widetilde{\mathcal{O}}\Bigl(\frac{d^{4/3}κ+ d^{7/6}κ^{7/6}}{\varepsilon^2}\Bigr)$ queries to $\nabla f$ are sufficient, where $κ= L / μ$ is the condition number. Moreover, we provide an information theoretic lowerbound, showing that at least $\frac{d^{1-o(1)}}{\varepsilon^{2-o(1)}}$ queries are necessary. This provides a first nontrivial lowerbound for the problem.