Recommender systems are crucial components of most commercial websites to keep users satisfied and to increase revenue. Thus, a lot of effort is made to improve recommendation accuracy. But when is the best possible performance of the recommender reached? The magic barrier, refers to some unknown level of prediction accuracy a recommender system can attain. The magic barrier reveals whether there is still room for improving prediction accuracy or indicates that further improvement is meaningless. In this work, we present a mathematical characterization of the magic barrier based on the assumption that user ratings are afflicted with inconsistencies - noise. In a case study with a commercial movie recommender, we investigate the inconsistencies of the user ratings and estimate the magic barrier in order to assess the actual quality of the recommender system.

1. Users and Noise: The Magic Barrier of Recommender Systems
Alan Said, Brijnesh J. Jain, Sascha Narr, Till Plumbaum
Competence Center Information Retrieval & Machine Learning
@alansaid, @saschanarr, @matip

2. Outline
► The Magic Barrier
► Empirical Risk Minimization
► Deriving the Magic Barrier
► User Study
► Conclusion
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3. The Magic Barrier
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4. The Magic Barrier
► No magic involved....
► Coined by Herlocker et al. in 2004
 “...an algorithm cannot be more accurate than the variance in
a user’s ratings for the same item.”
 The maximum level of prediction that a recommender
algorithm can attain.
► What does this mean?
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5. The Magic Barrier
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6. The Magic Barrier
► Even a “perfect” recommender should not reach RMSE = 0 or
Precision @ N = 1
► Why?
 People are inconsistent and noisy in their ratings
 “perfect” accuracy is not perfect
► So?
 Knowing the highest possible level of accuracy, we can stop
optimizing our algorithms at “perfect” (before overfitting)
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7. The Magic Barrier
So – how do we find the magic barrier?
We employ the Empirical Risk Minimization principle and a
statistical model for user inconsistencies
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8. The Magic Barrier – User Inconsistencies
Assumption:
 If a user were to re-rate all previously rated items, keeping in
mind the inconsistency, the ratings would differ, i.e.
𝑟 𝑢𝑖 = 𝜇 𝑢𝑖 + 𝜀 𝑢𝑖
 where
 𝜇 𝑢𝑖 is the expected rating, and
 𝜀 𝑢𝑖 the rating error (has zero mean)
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9. Empirical Risk Minimization
► … is a principle in statistical learning theory which defines a
family of learning algorithms and is used to give theoretical
bounds on the performance of learning
algorithms.[Wikipedia]
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10. Empirical Risk Minimization
► We formulate our risk function as
 𝑅 𝑓 = 𝑢,𝑖,𝑟 𝑝 𝑢, 𝑖, 𝑟 𝑓 𝑢, 𝑖 − 𝑟 2 The prediction error
The probability of user u rating item i with score r
► Keeping the assumption in mind, we formulate the risk for a
true, unknown, rating function as the sum of the noise
variance, i.e.
 𝑅 𝑓∗ = 𝑢,𝑖 𝑝 𝑢, 𝑖 𝕍 𝜀 𝑢𝑖
 where 𝕍 𝜀 𝑢𝑖 is the noise variance
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11. Deriving the Magic Barrier
► We want to express the risk function in terms of a magic barrier
for RMSE – we take the root of the risk function
 ℬ 𝒰×ℐ = 𝑢,𝑖 𝑝 𝑢, 𝑖 𝕍 𝜀 𝑢𝑖
 RMSE=0 iff 𝜀 𝑢𝑖 = 0 over all ratings users and items
► In terms of RMSE we can express this as
 𝐸 𝑅𝑀𝑆𝐸 𝑓 = ℬ 𝒰×ℐ + 𝐸 𝑓 > ℬ 𝒰×ℐ
 where 𝐸 𝑓 is the error
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12. Estimating the Magic Barrier
1. For each user-item pair in our population
a) Sample ratings on a regular basis, i.e. re-ratings
b) Estimate the expected value of ratings
𝑚
1
𝜇 𝑢𝑖 = 𝑟 𝑡 𝑢𝑖
𝑚
𝑡=1
c. Estimate the rating variance
𝑚
1 2
𝜀 𝑢𝑖 2
=
𝑚
𝜇 𝑢𝑖 − 𝑟𝑡 𝑢𝑖
𝑡=1
2. Estimate the magic barrier by taking the average
1
ℬ= 𝜀 𝑢𝑖 2
𝒳
𝑢𝑖 ∈𝒳
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13. A real-world user study
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14. A User Study
► We teamed up with moviepilot.de
 Germany’s largest online movie recommendation community
 Ratings scale 1-10 stars (Netflix: 1-5 stars)
► Created a re-rating UI
 Users were asked to re-rate at least 20 movies
 1 new rating (so-called opinions) per movie
 Collected data:
 306 users
 6,299 new opinions
 2,329 movies
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15. A User Study
User study moviepilot
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16. A User Study
~4 ratings steps Room for improvement
~1 rating steps
Predictions vs Ratings above Ratings below
Ratings user’s average user’s average
Overall Opinions above Opinions below
Magic Barrier user’s average user’s average
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17. Conclusion
► We created a mathematical characterization of the magic
barrier
► We performed a user study on a commercial movie
recommendation website and estimated its magic barrier
► We concluded the commercial recommender engine still has
room for improvement
► No magic
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18. More?
► Estimating the Magic Barrier of Recommender Systems: A User Study
 SIGIR 2012
► Magic Barrier explained
 http://irml.dailab.de
► Movie rating and explanation user study
 http://j.mp/ratingexplain
► Recommender Systems Wiki
 www.recsyswiki.com
► Recommender Systems Challenge
 www.recsyschallenge.com
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19. Questions?
► Thank You for Listening!
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