Confusion about the notation of Rademacher Complexity
$begingroup$
The online book:
https://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf
In "Understanding Machine Learning:From Theory to Algorithms" by Shai Shalev-Shwartz and Shai Ben-David, Chapter 26 "Rademacher Complexity",
part 26.1, for reasons of notational brevity, the set $mathcal{F}$ is defined as:
$$
mathcal{F} := mathcal{l} circ mathcal{H} := {z mapsto l(h,z): hin mathcal{H} }
$$
Where $l$ is a loss function, $mathcal{H}$ is the hypothesis space, and $z$ refers to a tuple $(x,y)$.
I'm unsure of how to interpret this.
Does it mean "the set of functions that map $mathcal{Z}$ to $mathbb{R}$ via the loss function $l$ and any $hin mathcal{H}$?" If so, isn't this just the loss function $l$ and thus shouldn't the mapping be from $mathcal{Z} times mathcal{H}$ to $mathbb{R}$ instead?
Alternatively, could this mean "the set of functions $f_h:mathcal{Z} to mathbb{R}$ where $hin mathcal{H}$"? But seeing as the dependency on $h$ is not expressed in subsequent parts of the literature, I have my doubts about this.
I would really appreciate it if someone could clarify this.
machine-learning rademacher-distribution
asked Dec 3 '18 at 0:50
Sean LeeSean Lee
1578
$endgroup$
add a comment |
$begingroup$
The online book:
https://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf
In "Understanding Machine Learning:From Theory to Algorithms" by Shai Shalev-Shwartz and Shai Ben-David, Chapter 26 "Rademacher Complexity",
part 26.1, for reasons of notational brevity, the set $mathcal{F}$ is defined as:
$$
mathcal{F} := mathcal{l} circ mathcal{H} := {z mapsto l(h,z): hin mathcal{H} }
$$
Where $l$ is a loss function, $mathcal{H}$ is the hypothesis space, and $z$ refers to a tuple $(x,y)$.
I'm unsure of how to interpret this.
Does it mean "the set of functions that map $mathcal{Z}$ to $mathbb{R}$ via the loss function $l$ and any $hin mathcal{H}$?" If so, isn't this just the loss function $l$ and thus shouldn't the mapping be from $mathcal{Z} times mathcal{H}$ to $mathbb{R}$ instead?
Alternatively, could this mean "the set of functions $f_h:mathcal{Z} to mathbb{R}$ where $hin mathcal{H}$"? But seeing as the dependency on $h$ is not expressed in subsequent parts of the literature, I have my doubts about this.
I would really appreciate it if someone could clarify this.
machine-learning rademacher-distribution
asked Dec 3 '18 at 0:50
Sean LeeSean Lee
1578
$endgroup$
add a comment |
$begingroup$
The online book:
https://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf
In "Understanding Machine Learning:From Theory to Algorithms" by Shai Shalev-Shwartz and Shai Ben-David, Chapter 26 "Rademacher Complexity",
part 26.1, for reasons of notational brevity, the set $mathcal{F}$ is defined as:
$$
mathcal{F} := mathcal{l} circ mathcal{H} := {z mapsto l(h,z): hin mathcal{H} }
$$
Where $l$ is a loss function, $mathcal{H}$ is the hypothesis space, and $z$ refers to a tuple $(x,y)$.
I'm unsure of how to interpret this.
Does it mean "the set of functions that map $mathcal{Z}$ to $mathbb{R}$ via the loss function $l$ and any $hin mathcal{H}$?" If so, isn't this just the loss function $l$ and thus shouldn't the mapping be from $mathcal{Z} times mathcal{H}$ to $mathbb{R}$ instead?
Alternatively, could this mean "the set of functions $f_h:mathcal{Z} to mathbb{R}$ where $hin mathcal{H}$"? But seeing as the dependency on $h$ is not expressed in subsequent parts of the literature, I have my doubts about this.
I would really appreciate it if someone could clarify this.
machine-learning rademacher-distribution
asked Dec 3 '18 at 0:50
Sean LeeSean Lee
1578
$endgroup$
The online book:
https://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf
In "Understanding Machine Learning:From Theory to Algorithms" by Shai Shalev-Shwartz and Shai Ben-David, Chapter 26 "Rademacher Complexity",
part 26.1, for reasons of notational brevity, the set $mathcal{F}$ is defined as:
$$
mathcal{F} := mathcal{l} circ mathcal{H} := {z mapsto l(h,z): hin mathcal{H} }
$$
Where $l$ is a loss function, $mathcal{H}$ is the hypothesis space, and $z$ refers to a tuple $(x,y)$.
I'm unsure of how to interpret this.
Does it mean "the set of functions that map $mathcal{Z}$ to $mathbb{R}$ via the loss function $l$ and any $hin mathcal{H}$?" If so, isn't this just the loss function $l$ and thus shouldn't the mapping be from $mathcal{Z} times mathcal{H}$ to $mathbb{R}$ instead?
Alternatively, could this mean "the set of functions $f_h:mathcal{Z} to mathbb{R}$ where $hin mathcal{H}$"? But seeing as the dependency on $h$ is not expressed in subsequent parts of the literature, I have my doubts about this.
I would really appreciate it if someone could clarify this.
machine-learning rademacher-distribution
machine-learning rademacher-distribution
asked Dec 3 '18 at 0:50
Sean LeeSean Lee
1578
asked Dec 3 '18 at 0:50
Sean LeeSean Lee
1578
asked Dec 3 '18 at 0:50
Sean LeeSean Lee
1578
asked Dec 3 '18 at 0:50
Sean LeeSean Lee
1578
asked Dec 3 '18 at 0:50
Sean LeeSean Lee
1578
1578
add a comment |
add a comment |
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