Skip to content

Commit b6f7697

Browse files
committed
Separate supplementary information
1 parent 864b4e6 commit b6f7697

File tree

2 files changed

+43
-44
lines changed

2 files changed

+43
-44
lines changed

README.md

Lines changed: 1 addition & 44 deletions
Original file line numberDiff line numberDiff line change
@@ -132,47 +132,4 @@ Please cite the following paper:
132132
}
133133
```
134134

135-
This paper has more information as well.
136-
137-
# Supplementary information
138-
## Preliminaries
139-
1. Define a multi-output function as $f: \mathbb{R}^D \rightarrow \mathbb{R}^M$,
140-
2. Assume we run $N$ independent optimization runs and obtain the Pareto sets for each run $\forall i \in \\{1,\dots,M\\}, \mathcal{F}_i \subseteq \mathbb{R}^M$,
141-
3. Define the objective vector $\boldsymbol{f} \in \mathbb{R}^M$ weakly dominates a vector $\boldsymbol{y}$ in the objective space if and only if $\forall m \in \\{1,\dots,M\\}, f_m \leq y_m$ and notate it as $\boldsymbol{f} \preceq \boldsymbol{y}$, and
142-
4. Define a set of objective vectors $F$ weakly dominates a vector $\boldsymbol{y}$ in the objective space if and only if $\exists \boldsymbol{f} \in F, \boldsymbol{f} \leq \boldsymbol{y}$ and notate it as $F \preceq \boldsymbol{y}$
143-
144-
145-
## Attainment surface
146-
147-
As seen in the figure below, the **attainment surface** is the surface in the objective space that we can obtain by splitting the objective space like a step function by the Pareto front solutions yielded during the optimization.
148-
149-
It is simple to obtain the attainment surface if we have only one experiment;
150-
however, it is hard to show the aggregated results from multiple experiments.
151-
To address this issue, we use the $k$% attainment surface.
152-
153-
![Conceptual visualization of the attainment surface](figs/attainment-surface.png)
154-
155-
Credit: Figure 4. in [Indicator-Based Evolutionary Algorithm with Hypervolume Approximation by Achievement Scalarizing Functions](https://dl.acm.org/doi/pdf/10.1145/1830483.1830578?casa_token=wAx-0-6HgLYAAAAA:LTZmyz4H20nnS9aaTJhQA84UejRISpWK_iCkl33LIT2ER6higBIahESB3x9-yZEq8jVkR9BzSjzMPQ).
156-
157-
## $k$% attainment surface
158-
First, we define the following empirical attainment function:
159-
160-
$$
161-
\alpha(\boldsymbol{y}) := \alpha ( \boldsymbol{y} | \mathcal{F}\_{1} , \dots , \mathcal{F}\_{N}) = \frac{1}{N} \sum_{{n=1}}^{N} \mathbb{I} [ \mathcal{F}_{n} \preceq \boldsymbol{y} ] .
162-
$$
163-
164-
The $k$% attainment surface is the attainment surface such that it is achieved by $k$% of independent runs and more formally, it is defined as:
165-
166-
$S = \biggl\\{\boldsymbol{y}\mid\alpha(\boldsymbol{y}) \geq \frac{k}{100}\biggr\\}.$
167-
168-
Note that as we only have $N$ independent runs, we define a control parameter **level** ( $1 \leq L \leq N$ ) and obtain the following set:
169-
170-
$S_L = \biggl\\{\boldsymbol{y}\mid\alpha(\boldsymbol{y}) \geq \frac{L}{N}\biggr\\}.$
171-
172-
The best, median, and worst attainment surfaces could be fetched by $\\{1/N,1/2,1\\}\times 100$% attainment surface, respectively.
173-
174-
Please check the following references for more details:
175-
176-
[2] [An Approach to Visualizing the 3D Empirical Attainment Function](https://dl.acm.org/doi/pdf/10.1145/2464576.2482716?casa_token=b9vWo8MI3i8AAAAA:4UaDmmM1YgQFVo-vEQdNKvk9-12RTT8sO7n16CQIvneP_J33w_eGo2wYhfphwufqY5OcYPYj_Gc3mA)
177-
178-
[3] [On the Computation of the Empirical Attainment Function](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.705.1929&rep=rep1&type=pdf)
135+
This paper has more information as well and we also provide a markdown file [here](supp.md).

supp.md

Lines changed: 42 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,42 @@
1+
# Supplementary information
2+
## Preliminaries
3+
1. Define a multi-output function as $f: \mathbb{R}^D \rightarrow \mathbb{R}^M$,
4+
2. Assume we run $N$ independent optimization runs and obtain the Pareto sets for each run $\forall i \in \\{1,\dots,M\\}, \mathcal{F}_i \subseteq \mathbb{R}^M$,
5+
3. Define the objective vector $\boldsymbol{f} \in \mathbb{R}^M$ weakly dominates a vector $\boldsymbol{y}$ in the objective space if and only if $\forall m \in \\{1,\dots,M\\}, f_m \leq y_m$ and notate it as $\boldsymbol{f} \preceq \boldsymbol{y}$, and
6+
4. Define a set of objective vectors $F$ weakly dominates a vector $\boldsymbol{y}$ in the objective space if and only if $\exists \boldsymbol{f} \in F, \boldsymbol{f} \leq \boldsymbol{y}$ and notate it as $F \preceq \boldsymbol{y}$
7+
8+
9+
## Attainment surface
10+
11+
As seen in the figure below, the **attainment surface** is the surface in the objective space that we can obtain by splitting the objective space like a step function by the Pareto front solutions yielded during the optimization.
12+
13+
It is simple to obtain the attainment surface if we have only one experiment;
14+
however, it is hard to show the aggregated results from multiple experiments.
15+
To address this issue, we use the $k$% attainment surface.
16+
17+
![Conceptual visualization of the attainment surface](figs/attainment-surface.png)
18+
19+
Credit: Figure 4. in [Indicator-Based Evolutionary Algorithm with Hypervolume Approximation by Achievement Scalarizing Functions](https://dl.acm.org/doi/pdf/10.1145/1830483.1830578?casa_token=wAx-0-6HgLYAAAAA:LTZmyz4H20nnS9aaTJhQA84UejRISpWK_iCkl33LIT2ER6higBIahESB3x9-yZEq8jVkR9BzSjzMPQ).
20+
21+
## $k$% attainment surface
22+
First, we define the following empirical attainment function:
23+
24+
$$
25+
\alpha(\boldsymbol{y}) := \alpha ( \boldsymbol{y} | \mathcal{F}\_{1} , \dots , \mathcal{F}\_{N}) = \frac{1}{N} \sum_{{n=1}}^{N} \mathbb{I} [ \mathcal{F}_{n} \preceq \boldsymbol{y} ] .
26+
$$
27+
28+
The $k$% attainment surface is the attainment surface such that it is achieved by $k$% of independent runs and more formally, it is defined as:
29+
30+
$S = \biggl\\{\boldsymbol{y}\mid\alpha(\boldsymbol{y}) \geq \frac{k}{100}\biggr\\}.$
31+
32+
Note that as we only have $N$ independent runs, we define a control parameter **level** ( $1 \leq L \leq N$ ) and obtain the following set:
33+
34+
$S_L = \biggl\\{\boldsymbol{y}\mid\alpha(\boldsymbol{y}) \geq \frac{L}{N}\biggr\\}.$
35+
36+
The best, median, and worst attainment surfaces could be fetched by $\\{1/N,1/2,1\\}\times 100$% attainment surface, respectively.
37+
38+
Please check the following references for more details:
39+
40+
[2] [An Approach to Visualizing the 3D Empirical Attainment Function](https://dl.acm.org/doi/pdf/10.1145/2464576.2482716?casa_token=b9vWo8MI3i8AAAAA:4UaDmmM1YgQFVo-vEQdNKvk9-12RTT8sO7n16CQIvneP_J33w_eGo2wYhfphwufqY5OcYPYj_Gc3mA)
41+
42+
[3] [On the Computation of the Empirical Attainment Function](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.705.1929&rep=rep1&type=pdf)

0 commit comments

Comments
 (0)