#### Presentation Title

Improved randomized approximation schemes for partition functions

November 2016

November 2016

HUB 302-16

Poster

#### Abstract

An important family of distributions in statistics and statistical mechanics is known as Gibbs distribution. The normalizing constant of a Gibbs distribution is called a partition function. In many applications of Gibbs distributions, such as finding the maximum likelihood estimator for a parameter, or approximating the size of a set of combinatorial objects, depend on the ability to compute the partition function for various parameter values. It is known that finding the partition function exactly is a #P complete problem. Therefore, approximation algorithms are needed to estimate the value of a partition function. The specific problem addressed in this work is: given the ability to draw samples from Gibbs distribution, what is the best way to estimate the partition function. An algorithm is called (ε, δ)-randomized approximation algorithm if with probability at least 1-δ, the relative error between the approximation and the exact value is less than ε. Recently, an (ε,1/4)-approximation algorithm was presented by Mark Huber. In this work, we applied a novel M-estimator to this (ε,1/4)-approximation algorithm, and improved it to an (ε, δ)-randomized approximation algorithm. More importantly, by making modifications to the cooling schedule, we reduced the number of samples needed to obtain the same (ε, δ) level.

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Improved randomized approximation schemes for partition functions

HUB 302-16

An important family of distributions in statistics and statistical mechanics is known as Gibbs distribution. The normalizing constant of a Gibbs distribution is called a partition function. In many applications of Gibbs distributions, such as finding the maximum likelihood estimator for a parameter, or approximating the size of a set of combinatorial objects, depend on the ability to compute the partition function for various parameter values. It is known that finding the partition function exactly is a #P complete problem. Therefore, approximation algorithms are needed to estimate the value of a partition function. The specific problem addressed in this work is: given the ability to draw samples from Gibbs distribution, what is the best way to estimate the partition function. An algorithm is called (ε, δ)-randomized approximation algorithm if with probability at least 1-δ, the relative error between the approximation and the exact value is less than ε. Recently, an (ε,1/4)-approximation algorithm was presented by Mark Huber. In this work, we applied a novel M-estimator to this (ε,1/4)-approximation algorithm, and improved it to an (ε, δ)-randomized approximation algorithm. More importantly, by making modifications to the cooling schedule, we reduced the number of samples needed to obtain the same (ε, δ) level.