Study of Optimal Mixing

Hsu, Hung-Wei

 

Prof. Yu, Tian-Li

2017.01.16

Outline

  • Recap : Pasting Problem
  • Again : Pasting Problem

Pasting Problem

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Optimal solution

Target solution

\text{Paste }\,\mathcal{O}{(logn)} \Rightarrow \text{Supply} \,\,2^{c\log{n}}=n^{c}
Paste O(logn)⇒Supply2clogn=nc\text{Paste }\,\mathcal{O}{(logn)} \Rightarrow \text{Supply} \,\,2^{c\log{n}}=n^{c}

Properties of Good Topologies

  • Tree structure / Loop structure

  • Random instances
  • 0-1 loss subfunction

We have discussed on problems with following properties

Counter Example

We have found counter example in 2d spin glass problem

\frac{n}{4}
n4\frac{n}{4}
\sqrt{n}
n\sqrt{n}

An elegant approach

An elegant approach

  • Find the area different to optimal solution
  • Estimate the distribution of these areas size
  • Bound the value by integration

Expected result

We first try exponential distribution : 

\int_{0}^{N}2^{n}\,pdf\left(n\right)\,dn = \mathcal{O}\left(N^{2}\right)
∫0N2npdf(n)dn=O(N2)\int_{0}^{N}2^{n}\,pdf\left(n\right)\,dn = \mathcal{O}\left(N^{2}\right)
pdf \left( n \right) = \lambda \, e^{-\lambda n}
pdf(n)=λe−λnpdf \left( n \right) = \lambda \, e^{-\lambda n}
\lambda \geq \ln{2}
λ≥ln2\lambda \geq \ln{2}

We need : 

pdf \left( n \right) = Cn\,2^{-n}
pdf(n)=Cn2−npdf \left( n \right) = Cn\,2^{-n}

A Smarter Way

  • Ability to cut large areas into small ones

  • Apply a recusive method

  • Probability model

Order Assumption

+1

+3

-2

Order Assumption (recursive)

p_{1}
p1p_{1}
p_{2}
p2p_{2}
p_{3}
p3p_{3}
\int_{0}^{N}2^{n}\,pdf\left(n\right)\,dn
∫0N2npdf(n)dn\int_{0}^{N}2^{n}\,pdf\left(n\right)\,dn

Selection of Cut

Cut path non-overlapping

\sum_{c \in C_{+} }f\left(c\right) = I
∑c∈C+f(c)=I\sum_{c \in C_{+} }f\left(c\right) = I

Conduct Experiments

  1. Square shape

  2. Use half-binomial distribution on boundary

  3. Let the optimal solution fixed (all bit 1's)

  4. Random instance

Fail Rate

Seems to be polynomial decay

Recursive Method

-1

+3

-2

Equal cut

-3

+3

-3

Zero boundary

Fail Rate

Equal cut

Zero boundary

Fail rate grows as area increases

Cut  selection

  • + cut paths
  • Z-shape cut paths
  • General cut paths
\sum_{c \in C_{+} }f\left(c\right) = I
∑c∈C+f(c)=I\sum_{c \in C_{+} }f\left(c\right) = I
\sum_{c \in C_{+} }f\left(c\right) = I
∑c∈C+f(c)=I\sum_{c \in C_{+} }f\left(c\right) = I
\sum_{c \in C_{g} }f\left(c\right) = 2^{N-2}I
∑c∈Cgf(c)=2N−2I\sum_{c \in C_{g} }f\left(c\right) = 2^{N-2}I

Sum

Path

Order

\mathcal{O}(\sqrt{N})
O(N)\mathcal{O}(\sqrt{N})
\mathcal{O}(N)
O(N)\mathcal{O}(N)
\mathcal{O}(2^{N-1})
O(2N−1)\mathcal{O}(2^{N-1})

Path

Overlapping

No

Yes

Yes

Another Problem

In 2d spin glass problem, the complement of optimal solution is also an optimal solution.

So when generating instances, it's not sufficient to use half binomial distribution

But we don't know what is the distribution of optimal solution

By-pass the Problem

  • Find what kind of instance will not have a legal cut path
    • Boundary
    • Inner
  • Conduct brute force experiment with general cut paths to find all these structures
  • Derive some properties from observation
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Properties : 

  • For degree 2 node
    • ...
    • ...
  • For degree 1 node
    • ...
    • ...

Too restricted!!!

Back to Cut Path

We are now discuss on general cut path :

If we flip a inner region, it will never gain!

Means there are two optimal solutions

But we can choose the most similar one as reference

Back to Cut Path

So now we will always have legal cut path:

  • A cut with outer region
  • A legal cut must have positive gain with its outer region part only

Outer region

Inner region

Outer region part

Inner region part

Similar scheme

p_{1}
p1p_{1}
p_{2}
p2p_{2}
p_{3}
p3p_{3}

How to compute p?
What probability model to be used?

Probability Model

  • Random bits for instance
  • Randomly select  region
  • We want the conditional probability :
P\{\nexists \text{ cut}|A\,\,\text{is optimal}\}
P{∄ cut∣Ais optimal}P\{\nexists \text{ cut}|A\,\,\text{is optimal}\}
=\frac{P\{\nexists \text{ cut}\wedge A\,\,\text{is optimal}\}}{P\{A\,\,\text{is optimal}\}}\leq C2^{-|A|}
=P{∄ cut∧Ais optimal}P{Ais optimal}≤C2−∣A∣=\frac{P\{\nexists \text{ cut}\wedge A\,\,\text{is optimal}\}}{P\{A\,\,\text{is optimal}\}}\leq C2^{-|A|}

A Sufficient Condition

  • A region A with flip(Inner region) <= 0
  • Outer region of A is identical to some optimal solution

Finally

Flip from outer region

Scheme

p_{1}
p1p_{1}
p_{2}
p2p_{2}
p_{3}
p3p_{3}

Easier to compute

About p

p^{\prime}+p^{\prime}(1-p^{\prime})+p^{\prime}(1-p^{\prime})^{2}+...
p′+p′(1−p′)+p′(1−p′)2+...p^{\prime}+p^{\prime}(1-p^{\prime})+p^{\prime}(1-p^{\prime})^{2}+...

End

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