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 ...


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 ...


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 ...


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