[Paper Review] Approximating Partition Functions in Constant Time

Vishesh Jain, Frederic Koehler+1 more|arXiv (Cornell University)|Nov 5, 2017

Markov Chains and Monte Carlo Methods20 references3 citations

TL;DR

This paper presents a constant-time algorithm for approximating the log partition function of dense graphical models, including Ising and Potts models, using cut decomposition and the algorithmic regularity lemma. It achieves an $O(\epsilon \|\vec{J}\|_1)$ additive error in time independent of $n$, significantly improving upon prior $n^{O(1/\epsilon^2)}$ methods while extending to low-threshold-rank and $k$-wise interaction models.

ABSTRACT

We study approximations of the partition function of dense graphical models. Partition functions of graphical models play a fundamental role is statistical physics, in statistics and in machine learning. Two of the main methods for approximating the partition function are Markov Chain Monte Carlo and Variational Methods. An impressive body of work in mathematics, physics and theoretical computer science provides conditions under which Markov Chain Monte Carlo methods converge in polynomial time. These methods often lead to polynomial time approximation algorithms for the partition function in cases where the underlying model exhibits correlation decay. There are very few theoretical guarantees for the performance of variational methods. One exception is recent results by Risteski (2016) who considered dense graphical models and showed that using variational methods, it is possible to find an $O(εn)$ additive approximation to the log partition function in time $n^{O(1/ε^2)}$ even in a regime where correlation decay does not hold. We show that under essentially the same conditions, an $O(εn)$ additive approximation of the log partition function can be found in constant time, independent of $n$. In particular, our results cover dense Ising and Potts models as well as dense graphical models with $k$-wise interaction. They also apply for low threshold rank models.

Motivation & Objective

• To develop a constant-time algorithm for approximating the log partition function of dense graphical models, where prior methods required polynomial time in $n$.
• To extend variational approximation methods to achieve sub-polynomial runtime while maintaining theoretical error guarantees.
• To apply the algorithmic regularity lemma to graphical models, particularly for partition function estimation.
• To provide a constant-time approximation of the magnetization per site in Ising and Potts models.
• To generalize results to $k$-wise interaction models and low-threshold-rank systems.

Proposed method

• Utilizes cut decomposition of interaction matrices $J$ to approximate the log partition function via structured, low-complexity representations.
• Employs the algorithmic regularity lemma to decompose $J$ into $O(\epsilon^{-2})$ cut matrices with bounded coefficients and controlled error.
• Applies a randomized sampling procedure to estimate the contribution of each cut matrix to the partition function in constant time.
• Uses the fact that the log partition function can be approximated by optimizing over the decomposed components, leveraging concentration of measure.
• Adapts the weak regularity lemma from Frieze and Kannan to handle higher-order interactions and low-threshold-rank models.
• Employs error bounds via $\ell_1$ and $\ell_2$ norms of $J$, ensuring additive error $O(\epsilon \|\vec{J}\|_1)$.

Experimental results

Research questions

• RQ1 Can the log partition function of dense graphical models be approximated in constant time, independent of $n$?
• RQ2 Does the algorithmic regularity lemma enable constant-time estimation of partition functions in models where correlation decay does not hold?
• RQ3 Can the method be extended to $k$-wise interaction models and low-threshold-rank systems?
• RQ4 Is it possible to achieve a constant-time approximation of the magnetization per site in Ising and Potts models?
• RQ5 How does the error bound scale with $\epsilon$ and $\|\vec{J}\|_1$ in the absence of correlation decay?

Key findings

• The log partition function of dense graphical models can be approximated to within $O(\epsilon \|\vec{J}\|_1)$ additive error in constant time, independent of $n$.
• The algorithm matches the runtime of the fastest known PTAS for Max-Cut on dense graphs, achieving $O(2^{\tilde{O}(1/\epsilon^2)})$ time.
• The method applies to $k$-wise interaction models, with error bounds scaling as $O(\epsilon 2^k \sqrt{N} \|\vec{J}\|_2)$ for $N$-dimensional interaction tensors.
• For low-threshold-rank Ising models, the algorithm achieves $O(\epsilon \|\vec{J}\|_1)$ error using a cut decomposition of width $O(\epsilon^{-2})$.
• The approach enables constant-time approximation of the magnetization per site in Ising and Potts models, with error $O(\epsilon \|\vec{J}\|_1)$.
• The results are the first to use the algorithmic regularity lemma for partition function estimation, providing a new framework for sublinear-time inference.