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


part of the optimization and statistical learning problems that are widely



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part of the optimization and statistical learning problems that are widely
popular, such as:
Stochastic Convex Optimization in Hilbert Spaces: “Let ‘Z’ be an
arbitrary measurable set, let ‘H’ be a closed, convex, and bounded subset of
a Hilbert space, and let ‘f(h;z)’ be Lipschitz-continuous and convex w.r.t. its
first argument. Here, we want to approximately minimize the objective
function ‘Ez

D [f(h;z)]’, where the distribution over ‘Z’ is unknown, based
on an empirical sample z
1
,...,z
m
”.
Density Estimation: “Let ‘Z’ be a subset of ‘Rn’, let ‘H’ be a set of
bounded probability densities on ‘Z’, and let ‘f (h; z) = − log(h(z))’. Here, ‘f
(·)’ is simply the negative log-likelihood of an instance z according to the
hypothesis density ‘h’. Note that to ensure bounded-ness of ‘f (·)’, we need
to assume that ‘h(z)’ is lower bounded by a positive constant for all ‘

Z’”.
K-Means Clustering in Euclidean Space: “Let Z = R
n
, let ‘H’ be all
subsets of Rn of size k, and let ‘f (h; z) = min
c

h ||c – z||
2’
Here, each h
represents a set of ‘k centroids’, and ‘f (·)’ measures the Euclidean distance
squared between an instance z and its nearest centroid, according to the
hypothesis h”.


Large Margin Classification in a Reproducing Kernel Hilbert Space
(RKHS): “Let ‘Z=X×{0,1}’, where ‘X’ is a bounded subset of an RKHS, let
H’ be another bounded subset of the RKHS, and let ‘f (h; (x, y)) = max{0,
1 − y

x, h

}’. Here, ‘f (·)’ is the popular hinge loss function, and our goal
is to perform margin-based linear classification in the RKHS”.
Regression: “Let ‘Z = X × Y’ where ‘X’ and ‘Y’ are bounded subsets of ‘R
n

and ‘R’ respectively, let ‘H’ be a set of bounded functions ‘h : X

→ R’, and
let ‘f(h;(x,y)) = (h(x)−y)
2
’. Here, ‘f(·)’ is simply the squared loss function”.

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