Machine Learning: 2 Books in 1: Machine Learning for Beginners, Machine Learning Mathematics. An Introduction Guide to Understand Data Science Through the Business Application

h
 F
S
(h)”, and ensures that the expected value of “F(hˆ
S
)” converges to the
optimal value “F
∗
= inf 
h
 F(h)”.
Although dependency on "h" is nonlinear, uniform convergence can still be
established with the use of "covering number arguments" provided "H" is
a finite dimension. Regrettably, uniform convergence may not take place if
we go to infinite-dimensional hypothesis and empirical minimization may

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not make impart the ability to learn to the algorithm. Remarkably, this does
not mean that the problem can be deemed "unlearnable". It can
be shown that with regularization mechanisms, even when uniform
convergence doesn't exist, a learning algorithm can be developed to solve
any "stochastic convex optimization" issue. This mechanism directly relates
to the principle of stability. For example, let’s look at the “convex stochastic
optimization” problem given by the equation in the picture below, where for
this example “H” will be the “d-dimensional unit sphere H = h
∈
Rd : | h|
≤1”, “z=(x,α) with α 
∈
[0, 1]d” and “x 
∈
H” , and “u 
∗
 v” can be defined
as an element-wise product. 
Now, considering a series of learning problems, where “d = 2
m
” for any
sample size “m”, and establishing that a “convergence rate independent of
the dimensionality ‘d’ cannot be expected”. This case can be formalized
into infinite dimensions. The learning problem in the equation above can be
considered as “that of finding the center of an unknown distribution over x
∈
 Rd, where stochastic per-coordinate confidence measures “α[i]” are also
available”. For now, we will be focusing on the scenario wherein certain
coordinates are missing, meaning “α[i] = 0”.
By taking into consideration the distribution given below over “(x, α): x =
0” with probability as 1, and “α” is uniform over “{0, 1}
d
”. That is, “α[i]”
are independent and identically distributed uniform “Bernoulli”. For a
random sample “(x