# Estimating normalizing constants for log-concave distributions: Algorithms and lower bounds

Journal Article

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=g e dx to within a multiplication factor of 1 ± ϵ for a μ-strongly convex and L-smooth function f, given query access to f(x) and g‡ 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 O(d κ + d κ /ϵ ) queries to g‡ f are sufficient, where κ= L / μ is the condition number. Moreover, we provide an information theoretic lowerbound, showing that at least d /ϵ queries are necessary. This provides a first nontrivial lowerbound for the problem. g., d -f(x) 4/3 7/6 7/6 2 1-o(1) 2-o(1)

### Full Text

### Duke Authors

### Cited Authors

- Ge, R; Lee, H; Lu, J

### Published Date

- June 8, 2020

### Published In

### Start / End Page

- 579 - 586

### International Standard Serial Number (ISSN)

- 0737-8017

### Digital Object Identifier (DOI)

- 10.1145/3357713.3384289

### Citation Source

- Scopus