Database Reference
In-Depth Information
It turns out that the adaptive algorithms for the matrix case carry over rather
smoothly to the higher-order setting.
• Most importantly, our research in Chap.
10
will focus on a combination of
adaptive CF and RL. Specifically, we are heading to apply (tensor) CF to the
transition probability matrices (or tensors) of Markov decision processes as a
means of approximation and regularization to render model-based methods
tractable for large state spaces.
In what follows, the above points will be discussed in more detail.
8.2 Collaborative Filtering
Let us, for the sake of completeness, first address classical collaborative filtering.
The first ever CF system was the
Information Tapestry Project
by
Xerox PARC
in
the early 1990s [GNBT92]. In the further development,
the research group
GroupLens
played an important part.
In classical CF, an identical “decomposition” of the matrix is carried out in (
8.1
),
that is,
X
is taken to be
A
itself and Y to be the identity matrix. Thus, the “low-
dimensional” subspace is given by the data space. Intuitively, there is no decom-
position whatsoever - all sessions are stored as they are.
Having clarified the factorization, we shall briefly address how it is used for
computation. Let
N
(
p
) be the set of all sessions
t
with the considered product
p
(i.e.,
p
has been clicked at least once). Then the predicted reward value
^
a
ps
of a product
p
not observed in the session is computed as follows:
X
s
st
a
pt
b
pt
t
NðÞ
∈
X
a
ps
¼ b
ps
þ
^
,
s
st
t
∈
NðÞ
where
b
ps
is the baseline prediction for
a
ps
(e.g., the mean value of the rewards of all
products in the session) and
s
st
a measure of similarity between the sessions
s
and
t
.
Frequently employed measures of similarity are the
cosine measure
and the
Pearson correlation
; see below.
What does this formula mean? For the prediction of product
a
ps
, we choose all
previously observed sessions that contain product
p
and compute the weighted
mean of their rewards for the product
p
, where the weights are given by the
similarities to the session
s
. That is, if a session
u
is closer to session
s
than another
session
t
, then the reward
a
ps
makes a greater contribution than the reward
a
pt
.
In practice, the described CF approach yields fairly good results, since, in
particular, it takes the entire history of sessions into account. An essential draw-
back, however, is the poor scaling with respect to computing time, as well as
memory space. Indeed, we have to keep the entire matrix
A
in memory, and the
^
