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I am reading Goodfellow et al Deeplearning Book. I found it difficult to understand the difference between the definition of the hypothesis space and representation capacity of a model.

In Chapter 5, it is written about hypothesis space:

One way to control the capacity of a learning algorithm is by choosing its hypothesis space, the set of functions that the learning algorithm is allowed to select as being the solution.

And about representational capacity:

The model specifies which family of functions the learning algorithm can choose from when varying the parameters in order to reduce a training objective. This is called the representational capacity of the model.

If we take the linear regression model as an example and allow our output $y$ to takes polynomial inputs, I understand the hypothesis space as the ensemble of quadratic functions taking input $x$, i.e $y = a_0 + a_1x + a_2x^2$.

How is it different from the definition of the representational capacity, where parameters are $a_0$, $a_1$ and $a_2$?

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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 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 hypothesis space, or it can be expanded to learn polynomials.

The representational capacity of a model determines the flexibility of a it, its ability to fit a variety of functions (i.e. which functions the model is able to learn), at the same. It specifies the family of functions the learning algorithm can choose from.

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    $\begingroup$ Does it mean that the set of functions described by the representational capacity is strictly included in the hypothesis space ? By definition, is it possible to have functions in the hypothesis space NOT described in the representational capacity ? $\endgroup$ – Qwarzix Aug 23 '18 at 8:43

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