Ranking
Some notes on ranking techniques
Basics
Pointwise, Pairwise and Listwise Learning to Rank
Point-wise ranking
In point-wise ranking, you have some scores for you document \(\displaystyle y_i\) so you can train your model \(\displaystyle f\) to predict such scores in a supervised manner.
Pair-wise ranking
If you data is of the form: \(\displaystyle y(x_a) \gt y(x_b)\) then you can train so that your model maximizes \(\displaystyle f(x_a) - f(x_b)\) using a hinge loss: \(\displaystyle L(x_a, x_b) = max(0, 1-(f(x_a) - f(x_b)))\)
Metrics
See https://medium.com/swlh/rank-aware-recsys-evaluation-metrics-5191bba16832
Cumulative Gain
Suppose you have a list of results \(\displaystyle x_1,..., x_n\) with relevency \(\displaystyle r_1,...,r_n\).
Then the cumulative gain at position \(\displaystyle p\) is the sum of the relevency of the first \(\displaystyle p\) results:
\(\displaystyle
\begin{equation}
CG_p = \sum_{i=1}^{p} r_i
\end{equation}
\)
The discounted cumulative gain (DCG) takes the position into account, discounting lower-ranked results: \(\displaystyle \begin{equation} DCG_p = \sum_{i=1}^{p} \frac{r_i}{\log_2 (i+1)} \end{equation} \)
The normalized discounted cumulative gain (NDCG) is 1-normalized by dividing over the best possible ranking: \(\displaystyle \begin{equation} NCDG_p = \frac{DCG_g(\mathbf{r})}{\max_{\mathbf{r}}DCG_p(\mathbf{r})} \end{equation} \)
Mean Reciprocal Rank
If you only have one correct answer which is placed in rank \(\displaystyle i\) then the reciprocal rank is \(\displaystyle 1/i\).
For multiple queries and results, the mean reciprocal rank is simply \(\displaystyle \operatorname{mean}(1/rank)\).