[Paper Review] Estimating Normalizing Constants for Log-Concave Distributions: Algorithms and Lower Bounds

Rong Ge, Holden Lee+1 more|arXiv (Cornell University)|Nov 8, 2019

Markov Chains and Monte Carlo Methods39 references4 citations

TL;DR

This paper presents a novel algorithm for estimating the normalizing constant of log-concave distributions using multilevel Monte Carlo with underdamped Langevin dynamics, achieving a query complexity of $\widetilde{\mathcal{O}}\left(\frac{d^{4/3}\kappa + d^{7/6}\kappa^{7/6}}{\varepsilon^2}\right)$, where $\kappa = L/\mu$ is the condition number. It also establishes the first nontrivial information-theoretic lower bound of $\frac{d^{1-o(1)}}{\varepsilon^{2-o(1)}}$ queries, closing a key gap in theoretical understanding of this fundamental problem in high-dimensional statistics and machine learning.

ABSTRACT

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 $ abla 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 $ abla 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.

Motivation & Objective

• To develop efficient algorithms for estimating the normalizing constant $Z = \int_{\mathbb{R}^d} e^{-f(x)} dx$ for $\mu$-strongly convex and $L$-smooth functions $f$.
• To close the gap in theoretical understanding by establishing the first nontrivial information-theoretic lower bound for this problem.
• To improve query complexity over standard annealing and Langevin-based methods by leveraging multilevel Monte Carlo techniques.

Proposed method

• Combines annealing with multilevel Monte Carlo estimation using underdamped Langevin dynamics (ULD) to reduce variance and query complexity.
• Employs a hierarchical sampling strategy where multiple levels of discretization are used to estimate the normalizing constant with decreasing error.
• Uses randomized midpoint methods (ULD-RMM) to improve the convergence rate of the underlying Langevin process.
• Applies a novel analysis of the bias and variance trade-offs in multilevel ULD to derive tight bounds on the number of gradient queries.
• Introduces a new annealing strategy that adaptively interpolates between a simple distribution and the target distribution.
• Derives a lower bound via information-theoretic arguments, showing that no algorithm can succeed with fewer than $\frac{d^{1-o(1)}}{\varepsilon^{2-o(1)}}$ queries.

Experimental results

Research questions

• RQ1 What is the optimal query complexity for estimating the normalizing constant of a log-concave distribution defined by a strongly convex and smooth potential?
• RQ2 Can multilevel Monte Carlo techniques significantly reduce the number of gradient queries required in annealing-based normalizing constant estimation?
• RQ3 What is the information-theoretic limit on the number of queries needed to estimate the normalizing constant within $1 \pm \varepsilon$ accuracy?
• RQ4 How do different sampling schemes—MALA, ULD, and ULD-RMM—affect the query complexity in this setting?
• RQ5 Is there a provable gap between the best-known upper bound and the information-theoretic lower bound for this problem?

Key findings

• The proposed multilevel ULD algorithm achieves a query complexity of $\widetilde{\mathcal{O}}\left(\frac{d^{4/3}\kappa + d^{7/6}\kappa^{7/6}}{\varepsilon^2}\right)$, improving over previous methods.
• The algorithm using ULD-RMM with multilevel Monte Carlo achieves the best-known dependence on dimension and condition number.
• An information-theoretic lower bound of $\frac{d^{1-o(1)}}{\varepsilon^{2-o(1)}}$ queries is established, showing that the upper bound is nearly tight in terms of dimension dependence.
• The analysis provides improved convergence rates for estimating expectations under log-concave measures using multilevel ULD, which may be of independent interest.
• The paper resolves a long-standing open problem by providing the first nontrivial lower bound for normalizing constant estimation under strong convexity.
• The results demonstrate that multilevel Monte Carlo is essential to achieving optimal query complexity in this setting, as naive annealing with ULD leads to suboptimal bounds.