11. [CM18] Consider Algorithm Q for the simplest model of interaction based on asking the category of single patterns. The algorithm is based on the computation of Xw ←

worst(X

↓

, nw) and on of ψw

←average-worst(Xw) to decide whether to issue questions.

Why are we using a set of worst patterns instead of simply considering the worst? In other words, why not assume that nw is always set to one?

#!# 12. [HM47] Consider the extended class of developmental functions defined as ζ : T ×

X × X : (t, x, ˙ x) → ζ(t, x, ˙ x). Reformulate the theory of Section 6.5.1 at the light of this extended definition, where ζ(t, x, ˙ x) = ρ(t)φ(x, ˙ x).

#!# 13. [C48] Consider a supervised learning framework in which the learner is forced to keep the risk function R(w, ζ ) =



κ=1 V (yκ , f(w, xκ ))ζκ

small. Here the developmental multipliers ζκ must sum up at least to  over the training set, that is,

must also be close to zero, where s = 
κ=1 ζκ . In so doing, the agent cannot benefit from a lazy behavior by enforcing ζκ = 0. Moreover, the entropy S(ζ ) is defined by S(ζ ) = [s > 0]

−

κ=1 ζκ
s log ζκ
s
is expected to be as larger as possible so as to enforce a weighing as uniform as possible over the training set. Finally, consider the overall cost E(w, ζ ) = R(w, ζ ) − μSS(ζ ) + μGG(ζ).
Formulate a learning algorithm while optimizing in the joint (w, ζ ) space. What if the developmental parameters are interpreted as a truly teaching agent modeled by a neural network by function ζ(w, x)? Suppose we begin learning with the development parameters ζκ = 0.
Discuss the evolution and cognitive links with learning in humans.
(Hint) ζ = 0 prevents from information flooding at the beginning of learning.