Collaborative filtering

Making recommendations

• Not every user rate every movie and it’s important for the system to know which users have rated which movies
• one possible way to approach the problem is to look at the movies that users have not rated

Using per-item feature

Cost function

Notation

• $r(i,j)=1$ if user $j$ has rated movie $i$ (0 otherwise)
• $y^{(i,j)}$ = rating given by user $j$ on movie $i$ (if defined)
• $w^{(j)},b^{(j)}$ = parameters for user $j$
• $x^{(i)}$ = feature vector for movie $i$
• For user $j$ and movie $i$, predict rating: $w^{(j)}\cdot x^{(i)}+b^{(j)}$
• $m^{(j)}$ = number of movies rated by user $j$
• To learn $w^{(j)},b^{(j)}$:
  • $n$ = number of features

$$mi{n}_{w^{(j)}{b}^{(j)}}J(w^{(j)},b^{(j)})=\frac{1}{2m^{(j)}}\sum _{i:r(i,j)=1}(w^{(j)}\cdot x^{(i)}+b^{(j)}-y^{(i,j)}{)}^2+\frac{\mu }{2m^{(j)}}\sum _{k=1}^n(w_k^{(j)}{)}^2$$ $$\frac{1}{2}\sum _{i:r(i,j)=1}(w^{(j)}\cdot x^{(i)}+b^{(j)}-y^{(i,j)}{)}^2+\frac{\mu }{2}\sum _{k=1}^n(w_k^{(j)}{)}^2$$

• To learn parameters $w^{(1)},b^{(1)},...w^{(n_u)},b^{(n_u)}$ for all users:
  • $f(x)=w^{(j)}\cdot x^{(i)}+b^{(j)}$

$$J\left(\genfrac{}{}{0ex}{}{w^{(1)},...,w^{(n_u)}}{b^{(1)},...,b^{(n_u)}}\right)=\frac{1}{2}\sum _{j=1}^{n_u}\sum _{i:r(i,j)=1}(w^{(j)}\cdot x^{(i)}+b^{(j)}-y^{(i,j)}{)}^2+\frac{\mu }{2}\sum _{j=1}^{n_u}\sum _{k=1}n(w_k^{(j)}{)}^2$$

Collaborative filtering algorithm

Gradient Descent