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3

From [1] we know that we have the following bound between the test and train error for i.i.d samples: $$\mathbb{P}\left(R \leqslant R_{emp} + \sqrt{\frac{d\left(\log{\left(\frac{2m}{d}\right)}+1\right)-\log{\left(\frac{\eta}{4}\right)}}{m}}\right) \geqslant 1-\eta$$ $R$ is the test error, $R_{emp}$ is the training error, $m$ is the size of the training ...

3

Given a hypothesis set $H$, the set of all possible mappings from $X\to Y$ where $X$ is our input space and $Y$ are our binary mappings: $\{-1,1\}$, the growth function, $\Pi_H(m)$, is defined as the maximum number of dichotomies generated by $H$ on $m$ points. Here a dichotomy is the set of $m$ points in $X$ that represent a hypothesis. A hypothesis is just ...

2

We can show that it is not true by a counterexample. For example, $X = \{1,2,3\}$ and $\mathcal H = \{\{\},\{1\},\{2\},\{1,2\}\}$ is the finite set hypothesis class. By definition, in this case, the $\mathcal{VC}$ dimension of $\mathcal H$ over the domain $X$ is $d=2$. Although $A = \{3\} \subset X$, whose size is smaller than the $\mathcal{VC}$ dimension, i....

2

A hypothesis space/class is the set of functions that the learning algorithm considers when picking one function to minimize some risk/loss functional. The capacity of a hypothesis space is a number or bound that quantifies the size (or richness) of the hypothesis space, i.e. the number (and type) of functions that can be represented by the hypothesis space. ...

2

The VC dimension represents the capacity (the same Vapnik, the letter V from VC, calls it the "capacity") of a model (or, in general, hypotheses class), so a model with a higher VC dimension has more capacity (i.e. it can represent more functions) than a model with a lower VC dimension. The VC dimension is typically used to provide theoretical bounds e.g. ...

1

Is there any practical application of knowing whether a concept class is PAC-learnable? If you know that a concept class is PAC-learnable (i.e. its VC dimension is finite), then there's a possibility that you can design an algorithm that can find a function (or concept) that is arbitrarily close to your target (or desired) function. This is not really an ...

1

Consider a target function $f: x \mapsto f(x)$. A hypothesis refers to an approximation of $f$. A hypothesis space refers to the set of possible approximations that an algorithm can create for $f$. The hypothesis space consists of the set of functions the model is limited to learn. For instance, linear regression can be limited to linear functions as its ...

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