Recommender Systems

Recommender
Systems
João Paulo L. F. Dias da Silva
Oct 2014

Background
(5 min)
Implementation
(5 min)
Demonstration
(5 min)
Agenda

Background 1. Machine Learning Application
• Unsupervised Learning (No
right answers provided)
• Linear Regression
• Gradient Descent Algorithm
2. Content-based Filtering
• Known product features
3. Collaborative Filtering
• Unknown product features
• Features will be “identified”
by the application

Linear Regression
It's a method that allows us to obtain a function that models the
relationship between a scalar dependent variable h and its
explanatory variables X.
Given a dataset {h, x1
, x2
, …, xn
} of statistical units, a linear
regression model assumes that there's a linear relationship
between each variable hi
and its independent variables xi1
, xi2
, …,
xin.
The goal of the linear regression is to obtain a parameter Ɵ so
that the model function h(X) = c + ƟX fits the input dataset as
close as possible.
0 2 4 6 8 10 12 14 16 18
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000

Linear Regression – Model function
hi
= Ɵ1
xi1
+ … + Ɵm
xim
Stacking all examples we can rewrite the above as:
hi
= xi
T
Ɵ
Where X is a nx1 vector with each element being xi
T
and n is the
number of examples of our dataset.
T denotes the transpose operation.
Let h be a function that represents a model for the ith
example of
our dataset:
h = XƟ
Let Ɵ and xi
be mx1 vectors where m is the number of variables
of our model, so hi
becomes:

Linear Regression – Error function
Let h(X) be our hypothesis function (model).
J = (h(X) - Y)2
Let Y be the target values for each example in our dataset.
The squared error function J will be:
Another way of writing the error function is to take into account
the index of each example in our dataset, so that:
m
J = ∑(h(x(i)
) – Y(i)
)2
i=1
The objective of the linear regression is to minimize the error
function J with respect to Ɵ. One way of achieving it is through an
algorithm called Gradient Descent.

Gradient Descent Algorithm
Objective:
Find the minimum values for Ɵ1
,...,Ɵn
that minimize the error
function J(Ɵ1
,...,Ɵn
).
Overview:
• Initialize Ɵ1
,...,Ɵn
with some random values.
• Keep changing Ɵ1
,...,Ɵn
to reduce J(Ɵ1
,...,Ɵn
) until we find a
minimum.
Implementation:
Ɵj
:= Ɵj
– α – ∂ J(Ɵ1
,...,Ɵn
)
∂Ɵj

Gradient Descent
The partial derivative:
m
Ɵj
:= Ɵj
– α 1 – ∑(h(x(i)
) – y(i)
)x(i)
m i=1 j
An example for Ɵ ∈ ℝ3
:
m
Ɵ0
:= Ɵ0
– α 1 – ∑(h(x(i)
) – y(i)
)x(i)
m i=1 0
m
Ɵ1
:= Ɵ1
– α 1 – ∑(h(x(i)
) – y(i)
)x(i)
m i=1 1
m
Ɵ2
:= Ɵ2
– α 1 – ∑(h(x(i)
) – y(i)
)x(i)
m i=2 2

Recommender Systems – Prog. Skills
Skills Ana Beto Carla Daniel
Ruby 5 5 0 0
CSS3 5 ? ? 0
JS ? 4 0 ?
Android 0 0 5 4
iOS 0 0 5 ?
How to predict the values for the unknown skills?

Content-based Filtering
Skills
Ana
Ɵ¹
Beto
Ɵ2 ...
X1
(Web)
X2
(Mobile)
Ruby (X1
) 5 5 ... 0.9 0
CSS3 (X2
) 5 ? ... 1.0 0.01
JS (X3
) ? 4 ... 0.99 0
Android (X4
) 0 0 ... 0.1 1.0
iOS (X5
) 0 0 ... 0 0.9
The skills features are known. Just need to solve one Linear Regression per user.

Content-based Filtering - Predicting
Skills
Ana
Ɵ¹ = [5, 0]
...
X1
(Web)
X2
(Mobile)
Ruby (X1
) 5 ... 0.9 0
CSS3 (X2
) 5 ... 1.0 0.01
JS (X3
) 5 ... 0.99 0
Android (X4
) 0 ... 0.1 1.0
iOS (X5
) 0 ... 0 0.9
Ana(JS) => Ɵ¹ * X3
=> [5, 0] * [0.99, 0] = (5 * 0.99) + (0 * 0) = 5

Collaborative Filtering
Skills
Ana
Ɵ¹
Beto
Ɵ2 ...
X1
(?)
X2
(?)
Ruby (X1
) 5 5 ... ? ?
CSS3 (X2
) 5 ? ... ? ?
JS (X3
) ? 4 ... ? ?
Android (X4
) 0 0 ... ? ?
iOS (X5
) 0 0 ... ? ?
How to predict the values for the unknown skills and features?

Collaborative Filtering – Feature
Learning
We can't find the Ɵ parameters because we don't have the values
for the features vectors.
So we initialize the Ɵ parameters to random values.
Then we can use the Ɵ parameters to apply linear regression in
order to find the features vectors for each skill.
Then we can use the features vectors to apply linear regression to
improve our Ɵ parameters for each user.
We keep doing that until we reach the optimal values for Ɵ and
the features vectors.

Collaborative Filtering – Intuition

Collaborative Filtering - Predicting
Skills
Ana
Ɵ¹ = [5, 0]
... X1 X2
Ruby (X1
) 5 ... 0.9 0
CSS3 (X2
) 5 ... 1.0 0.01
JS (X3
) 5 ... 0.99 0
Android (X4
) 0 ... 0.1 1.0
iOS (X5
) 0 ... 0 0.9
Ana(JS) => Ɵ¹ * X3
=> [5, 0] * [0.99, 0] = (5 * 0.99) + (0 * 0) = 5

Implementa
tion
Python for data scraping
Octave for LR/GD matrix calculations
Missing UI
Hardcoded input

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