Pranking with Ranking Koby Crammer and Yoram Singer

1. Pranking with RankingKoby Crammer and Yoram Singer Lecture: DuduYanay

2. The Problem • Input:Each instance is associated with a rank or a rating, i.e. an integer from ‘1’ to ‘K’. • Goal:To find a rank-prediction rule which assigns to each instance a rank which is as close as possible to the instance true rank. • Similar problems: • Classifications. • Regression.

3. Natural Setting For… • Information Retrieval. • Collaborative filtering:Predict a user’s rating on new items (books, movies etc) given the user’s past rating of similar items.

4. Possible Solutions • To cast a rating problem as a regression problem. • To reduce a total order into a set of preferences over pairs. • Time consuming since it might require to increase the sample size from to .

5. Lets try another approach… • Online Algorithm (Littlestone 1988): • Each can be computedin polynomial time. • If the problem is separable,after polynomial failures(no) the learner doesn’t makea mistake. Meaning: מורה לומד Animation from Nader Bshouty’s Course.

6. The PERCEPTRON algorithm Animation from Nader Bshouty’s Course.

7. The PERCEPTRON algorithm A slide from Nader Bshouty’s Course.

8. A slide from Nader Bshouty’s Course.

9. PRank algorithm - The model • Input: A sequence • . • Output: A ranking rule where: • . • . • . • Ranking loss after T rounds is: where is the TRUE rank of the instance in round ‘t’ and .

10. PRank algorithm - The update rule • Given an input instance-rank pair , if: • . • . • Lets represent the above inequalities by where The TRUE rank vector

11. PRank algorithm - The update rule • Given an input instance-rank pair , if . • So, let’s “move” the values of and towards each other: • . • , where the sum is only over the indices ‘r’ for which there was a prediction error, i.e., .

12. The update rule - Illustrasion Correct interval Predicted Rank 1 2 3 4 5

13. The PRank algorithm Building the TRUErank vector Checking which thresholdprediction is wrong Updating the hypothesis

14. PRank Analysis – Consistent Hypothesis • First, we need to show that the output hypothesis of Prank is acceptable. Meaning, if and is the final ranking rule then . • Proof – By induction:Since the initialization of the thresholds is such that , then it suffices to show that the claim hold inductively. • Lemma 1 (Order Preservation):Let and be the current ranking rule, where and let be an instance-rank pair fed to Prank on round ‘t’. Denote by and the resulting ranking after the update of Prank, then

15. Lemma 1 – Proof Correct interval Predicted Rank Option 1 1 2 3 4 5 6 Predicted Rank Correct interval Option 2 1 2 3 4 5

16. PRank Analysis – Mistake bound • Theorem 2:Let be an input sequence for PRank where . and . Denote by . Assume that there is a ranking rule with of a unit norm that classifies the entire sequence correctly with margin . . Then, the rank loss of the algorithm , is at the most .

17. Experiments • Comparison between: • Prank. • MultiClassPerceptron – MCP. • Widrow-Hoff (online regression) – WH. • Datasets: • Synthetic. • EachMovie.

18. Synthetic Dataset • Randomly generated points - uniformly at random. • Each point was assign a rank according to: • - noise. • Generated 100 sequences of instance-rank pairs, each of length 7000.

19. EachMovie Dataset • Collaborative filtering dataset. Contains ratings of movies provided by 61,265 people. • 6 possible rating: 0, 0.2, 0.4, 0.6, 0.8, 1. • Only people with atleast 100 rating whereconsidered. • Chose at random oneperson to be the TRUE rank and otherratings where used asfeatures(-0.5,-0.3,-0.1,0.1, 0.3, 0.5).

20. EachMovie Dataset – cont’ • Batch setting • Ran Prank over the training data as an online algorithm and used its last hypothesis to rank the unseen data.

21. Thank You

22. משפט PERCEPTRON הוכחה

23. משפט PERCEPTRON הוכחה

24. משפט PERCEPTRON הוכחה