Content-Based Recommender Systems III Profile Learner

Introduction

As my previous posts, I described several aspects of Recommender System. In this post, I will add more Machine Learning falvor into this topic. I will introduce the mathmatical problem fomulation and some algorithm to learn user profile.

Problem Formulation

Notation:

$r(i,j)=1$ if user $j$ has rated object $i$ (0 otherwise)

$y^{(i,j)}=$ rating by user $j$ on object $i$ (if defined)

${\theta }^{\left(j\right)}=$ parameter vector for user $j$

$x^{\left(i\right)}=$ feature vector for object $i$ (this is to describe the similarity among objects. eg. For movies, we can have a vector to denote how much percent it looks like a romance or action)

$m^{\left(j\right)}=$ number of objects rated by user $j$

The user’s profile is defined by ${\theta }^{\left(j\right)}$. The recommendation is based on the prediction rating of $\left({\theta }^{\left(j\right)}{)}^T\right(x^{\left(i\right)})$

Learning

It’s actually a regression problem.

Goal: To learn ${\theta }^{\left(j\right)}$ for user $j$

$mi{n}_{{\theta }^{\left(j\right)}}$ $\frac{1}{2}{\sum }_{i,r(i,j)=1}$ $({\theta }^{\left(j\right)}{)}^T$ $\left(x^{\left(i\right)}\right)$ $-{y^{(i,j)}}^2$ $+$ $\frac{\lambda }{2}{\sum }_{k=1}^n({\theta }_k^{\left(j\right)}{)}^2$

To learn all users’ profiles, just need to sum up through all users

We can use Gradient