MRF optimization via dual decomposition

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In dual decomposition a problem is broken into smaller subproblems and a solution to the relaxed problem is found. This method can be employed for MRF optimization.^[1] Dual decomposition is applied to markov logic programs as an inference technique.^[2]

Discrete MRF Optimization (inference) is very important in Machine Learning and Computer vision, which is realized on CUDA graphical processing units.^[3] Consider a graph ${\displaystyle G=(V,E)}$with nodes ${\displaystyle V}$and Edges ${\displaystyle E}$. The goal is to assign a label ${\displaystyle l_{p}}$to each ${\displaystyle p\in V}$so that the MRF Energy is minimized:

(1) ${\displaystyle \min \Sigma _{p\in V}\theta _{p}(l_{p})+\Sigma _{pq\in \varepsilon }\theta _{pq}(l_{p})(l_{q})}$

Major MRF Optimization methods are based on Graph cuts or Message passing. They rely on the following integer linear programming formulation

(2) ${\displaystyle \min _{x}E(\theta ,x)=\theta .x=\sum _{p\in V}\theta _{p}.x_{p}+\sum _{pq\in \varepsilon }\theta _{pq}.x_{pq}}$

In many applications, the MRF-variables are {0,1}-variables that satisfy: ${\displaystyle x_{p}(l)=1}$${\displaystyle \Leftrightarrow }$label ${\displaystyle l}$ is assigned to ${\displaystyle p}$, while ${\displaystyle x_{pq}(l,l^{\prime })=1}$ , labels ${\displaystyle l,l^{\prime }}$ are assigned to ${\displaystyle p,q}$.

The main idea behind decomposition is surprisingly simple:

1. decompose your original complex problem into smaller solvable subproblems,
2. extract a solution by cleverly combining the solutions from these subproblems.

A sample problem to decompose:

${\displaystyle \min _{x}\Sigma _{i}f^{i}(x)}$where ${\displaystyle x\in C}$

In this problem, separately minimizing every single ${\displaystyle f^{i}(x)}$over ${\displaystyle x}$ is easy; but minimizing their sum is a complex problem. So the problem needs to get decomposed using auxiliary variables ${\displaystyle \{x^{i}\}}$and the problem will be as follows:

${\displaystyle \min _{\{x^{i}\},x}\Sigma _{i}f^{i}(x^{i})}$where ${\displaystyle x^{i}\in C,x^{i}=x}$

Now we can relax the constraints by multipliers ${\displaystyle \{\lambda ^{i}\}}$ which gives us the following Lagrangian dual function:

${\displaystyle g(\{\lambda ^{i}\})=\min _{\{x^{i}\in C\},x}\Sigma _{i}f^{i}(x^{i})+\Sigma _{i}\lambda ^{i}.(x^{i}-x)=\min _{\{x^{i}\in C\},x}\Sigma _{i}[f^{i}(x^{i})+\lambda ^{i}.x^{i}]-(\Sigma _{i}\lambda ^{i})x}$

Now we eliminate ${\displaystyle x}$ from the dual function by minimizing over ${\displaystyle x}$ and dual function becomes:

${\displaystyle g(\{\lambda ^{i}\})=\min _{\{x^{i}\in C\}}\Sigma _{i}[f^{i}(x^{i})+\lambda ^{i}.x^{i}]}$

We can set up a Lagrangian dual problem:

(3) ${\displaystyle \max _{\{\lambda ^{i}\}\in \Lambda }g({\lambda ^{i}})=\Sigma _{i}g^{i}(x^{i}),}$ The Master problem

(4) ${\displaystyle g^{i}(x^{i})=min_{x^{i}}f^{i}(x^{i})+\lambda ^{i}.x^{i}}$where ${\displaystyle x^{i}\in C}$The Slave problems

The original MRF optimization problem is NP-hard and we need to transform it into something easier.

${\displaystyle \tau }$is a set of sub-trees of graph ${\displaystyle G}$where its trees cover all nodes and edges of the main graph. And MRFs defined for every tree ${\displaystyle T}$ in ${\displaystyle \tau }$will be smaller. The vector of MRF parameters is ${\displaystyle \theta ^{T}}$and the vector of MRF variables is ${\displaystyle x^{T}}$, these two are just smaller in comparison with original MRF vectors ${\displaystyle \theta ,x}$. For all vectors ${\displaystyle \theta ^{T}}$we'll have the following:

(5) ${\displaystyle \sum _{T\in \tau (p)}\theta _{p}^{T}=\theta _{p},\sum _{T\in \tau (pq)}\theta _{pq}^{T}=\theta _{pq}.}$

Where ${\displaystyle \tau (p)}$and ${\displaystyle \tau (pq)}$denote all trees of ${\displaystyle \tau }$than contain node ${\displaystyle p}$and edge ${\displaystyle pq}$respectively. We simply can write:

(6) ${\displaystyle E(\theta ,x)=\sum _{T\in \tau }E(\theta ^{T},x^{T})}$

And our constraints will be:

(7) ${\displaystyle x^{T}\in \chi ^{T},x^{T}=x_{|T},\forall T\in \tau }$

Our original MRF problem will become:

(8) ${\displaystyle \min _{\{x^{T}\},x}\Sigma _{T\in \tau }E(\theta ^{T},x^{T})}$where ${\displaystyle x^{T}\in \chi ^{T},\forall T\in \tau }$and ${\displaystyle x^{T}\in x_{|T},\forall T\in \tau }$

And we'll have the dual problem we were seeking:

(9) ${\displaystyle \max _{\{\lambda ^{T}\}\in \Lambda }g(\{\lambda ^{T}\})=\sum _{T\in \tau }g^{T}(\lambda ^{T}),}$The Master problem

where each function ${\displaystyle g^{T}(.)}$is defined as:

(10) ${\displaystyle g^{T}(\lambda ^{T})=\min _{x^{T}}E(\theta ^{T}+\lambda ^{T},x^{T})}$where ${\displaystyle x^{T}\in \chi ^{T}}$The Slave problems

Theorem 1. Lagrangian relaxation (9) is equivalent to the LP relaxation of (2).

${\displaystyle \min _{\{x^{T}\},x}\{E(x,\theta )|x_{p}^{T}=s_{p},x^{T}\in {\text{CONVEXHULL}}(\chi ^{T})\}}$

Theorem 2. If the sequence of multipliers ${\displaystyle \{\alpha _{t}\}}$satisfies ${\displaystyle \alpha _{t}\geq 0,\lim _{t\to \infty }\alpha _{t}=0,\sum _{t=0}^{\infty }\alpha _{t}=\infty }$then the algorithm converges to the optimal solution of (9).

Theorem 3. The distance of the current solution ${\displaystyle \{\theta ^{T}\}}$to the optimal solution ${\displaystyle \{{\bar {\theta }}^{T}\}}$, which decreases at every iteration.

Theorem 4. Any solution obtained by the method satisfies the WTA (weak tree agreement) condition.

Theorem 5. For binary MRFs with sub-modular energies, the method computes a globally optimal solution.