# Online Selection of Mediated and Domain-Specific Predictions for Improved Recommender Systems

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5 Figure 1: The MAG s % Error over 1000 products consistently is high because of the uniform random weight distribution. X-axis: number of categories and reviewers. Each line: different probability p Figure 2: The DSAG s % Error over 1000 products gets worse as the weight distribution becomes uniform random. X-axis: number of categories and reviewers. Each line: different probability p (p=0 on bottom, p=1 on top) Figures 1 and 2 show the error rates of the MAG and DSAG respectively as we vary 0 p 1 in increments of.1 and the number of categories from 3 to 10. As the number of categories increases, there are more weight distributions that must each converge, increasing the error rate overall until they do. When p = 0, the review matrix is the same as the DSAG example above and the DSAG again has perfect accuracy after converging while the MAG is almost always wrong. As p approaches 1, the domain-specific weights become more uniformly random. However, as we varied p, the mediated weight distribution over the reviewers was consistently uniformly random and the error rate was consistently high. Because it is not holding category information, the MAG doesn t find a difference between truly random (p = 1) and domain-correlated (p = 0) reviews and predicts the same way each time. The DSAG also has a high error rate as the weight distributions become more uniform. When the MAG and DSAG weight distributions become more similar (to uniform or otherwise), the DSAG and MAG predict the same. The DSAG performs better than the MAG until the distributions are more than 50% similar. Very Sparse Review Matrices The previous tests assumed that reviewers gave reviews for all products and all categories. However, this is a strong assumption. Now we relax this assumption and look at much sparser review matrices. For simplicity, we assume that only a single reviewer provides reviews in each category, although this can easily be scaled up to include disjoint sets of reviewers. Here, we will also assume that the single reviewer has similar reviews as the user and both give reviews of 5, although we could easily assume that a few of many reviewers are similar. We define the entries of the review matrix in the following way: { 5 product j in category i R ij = product j not in category i The DSAG in this case only has a single reviewer to assign a weight for each category and gives it full weight. The MAG, however, must combine these reviewers together into a single weight distribution. Because the products are uniformly distributed in the categories, the reviewers are also uniformly weighted. However, unlike the previous examples where the MAG had a high error rate, the MAG performs equally as well as the DSAG here. Because the MAG only has one review in the product column to compare against, it always picks that review to follow and is 100% accurate - the same as the DSAG. We have shown in synthetic, simplified review matrices that the MAG can converge on the weight distribution faster than the DSAG with less data and that the DSAG is more accurate when the categorical weight distributions are different. However, we have also shown that the two advice givers perform equally when the weight distributions are the same across categories or when the review matrix is sparse and there are disjoint sets of reviewers for each category. Next, we analyze the properties of actual recommender systems to understand how the two advice givers perform in these real conditions with real data sparsity and real reviewers. It is unclear whether the advice givers even perform the same for two different users of the same recommender system. We use these results to determine how the tradeoff between data and weight distributions affects the accuracy of predictions for different users. Advice Giver Evaluation - Real Data To understand how often data sparsity affects the accuracy of the DSAG for users, we built two recommender systems using real data sets from popular recommender websites. We compare the accuracy of the MAG to the DSAG across all users and for each user, and evaluate how the weights of the reviewers affect the success of the MAG. First, we describe the recommender systems review matrices R. Experimental Method The reviewers and advice giver provide possible values V = {1, 2, 3, 4, 5} to the user. For each recommender system, the MAG and DSAG provide predictions for the same randomly chosen users. We use the users reviews of products as the truth for accuracy calculations and remove those reviews

7 alyze the number of categories a reviewer must provide reviews in, in order to see a difference in accuracy between the DSAG and MAG advice givers. Then, assuming reviewers provide enough reviews so that the data is not as sparse, we propose two online selection algorithms to autonomously determine which advice giver should make predictions for each user. Very Sparse Data We have shown that in very sparse matrices, there may not be any overlap in reviewers across categories. In other words, reviewers only review products in a single category. The MAG and DSAG produce different weight distributions but the same predictions with the same accuracy. Now, we will show how the MAG is affected as reviewers review products in more categories and as we can collect more data to overcome the extreme sparseness of the review data. We will use the same assumptions as before. The user always gives review 5, so the reviewer that reviews a product with value 5 should have the highest weight. Different Weight Distributions First, instead of completely disjoint sets of reviewers for each category, our reviewers review products in a set of categories. Only a single reviewer is correct in each category but several other reviewers also give incorrect reviews. More concretely, we define the review matrix in the following way: 5 product j in category i R ij = 1 product j in categories (i+1)%m - (i+k)%m otherwise Reviewer i is correct for category i and gives reviews far from the user s preference for categories (i+1)%m to (i+k)%m. The value k is the number of reviewers that provide reviews for each product. By varying k, we can understand how the MAG is affected as the amount of overlap increases in the categories that reviewers review for. We notice that k = 0 means that only a single reviewer reviews products in a single category. Also, k = m means that all reviewers provide reviews for all products. We are interested in how the MAG behaves for values between 0 and m. We know that although it may take the DSAG longer to converge on the categorical weight distributions, it will always have 100% accuracy. Figure 3 shows the error rate (y-axis) for the MAG and DSAG with m = {1,..., 10} (x-axis) and with k = {1,..., 10}. Each line represents a different k. Note that there cannot be more than m reviewers overlapping at a time, so the line for k starts at m. The line for 3 overlapping reviewers starts at 100% error rate at m = 3 and drops a little before converging at a high error rate. Because all reviewers are best for one category, the MAG converges to a uniform distribution. A majority of the reviewers give reviews that conflict with the user, but the MAG follows the majority so it predicts incorrectly nearly all the time depending on Figure 3: The DSAG has almost 0% error no matter how many reviewers review each product. The MAG has perfect accuracy when the reviewers in each category are disjoint (k = 1). However, it is almost always wrong for k > 1 and any amount of overlap with reviewers. the order of products. The DSAG always finds the correct weight distribution and maintains a 0% error over all k. There is no statistical significance between the error rates of the ks. As we increase k, the error is nearly constant. However this error is expected because the weight distributions are so different for each category. Next, we evaluate the MAG performance on categorical weight distributions that are similar but have sparse reviewer data. Similar Weight Distributions We know that the MAG converges faster than the DSAG when the categorical weight distributions are similar and all reviewers always give reviews. Now we create a new review matrix such that there are a few (e.g., 2) best reviewers for a user and the rest give opposing reviews: 5 (p i = 1) 5 ((1 p) i = 2) R ij = 1 (q i 1) otherwise With some probability p, reviewer 1 gives the best response and otherwise reviewer 2 gives the best response. Then the rest of the reviewers each give a response with some other probability q and otherwise give no response. These distributions do not depend on the category, so the DSAG would have to find this same distribution for each category. The MAG finds it just once. If either reviewer 1 or reviewer 2 always provides a review for a product, the MAG can find these best reviewers after getting reviews from each of them only a few times. Because these reviewers are always correct with respect to our user, a MAG that always trusts them will also always be correct. Although these reviewers never appear at the same time, the MAG still has one reviewer that has high weight to

10 (a) The cumulative MSE for User 8 over time. (b) The weighted MSE for User 8 over time. Figure 5: The UdOS used the DSAG for User 8 on products 5-32 when the MAG was providing worse predictions. After product 32, the MAG provided better predictions and the UdOS switched back to using it. Figure 4: For User 19, the UdOS algorithm uses the MAG advice until it has seen enough data to conclude that the DSAG is better and then switches to use its predictions (product 18). from the DSAG than the MAG, because the user s similar reviewers were different for each product domain. The UdOS algorithm recognized that the DSAG had a lower MSE (See Figure 4) and switched to using the DSAG s predictions at product 18. After the switch, the UdOS cumulative MSE converged towards that of the DSAG. The algorithm picked the DSAG for other users like 10, but the switch occurred late enough that the cumulative MSE did not start converging towards the DSAG s MSE. Because the cumulative MSE was the same for both the DSAG and MAG for User 16, it didn t matter which advice giver the UdOS picked (marked with ** in Table 3). As a result, it continually used the MAG that it started with. The UdOS algorithm picked the advice giver with higher overall MSE for three users (marked with * in Table 3). User 8, for example, received better predictions from the UdOS algorithm than either the MAG or DSAG could provide alone because the UdOS switched back and forth be- User MAG DSAG UdOS Selection MAG MAG MAG MAG MAG MAG** MAG* MAG* DSAG* DSAG DSAG DSAG DSAG DSAG DSAG DSAG DSAG DSAG DSAG DSAG Table 3: Six users received better predictions with lower cumulative MSEs (bold) from the MAG while 13 received better predictions from the DSAG. tween the two advice givers before deciding which to use. The MSE of the DSAG is lower than the MAG s because the MAG poorly predicted some products in the middle of the set (specifically products 20-30) (See Figure 5(a)). However, once the MAG s weight distributions identified the most similar reviewers (product 32), its predictions were better than the DSAG s and the UdOS algorithm picked the MAG (See 5(b)). UCdOS Results Overall, nine users received the same predictions as the UdOS, picking either one advice giver or the other for all of the product categories the user requested (See Table 4, no *). Note that this does not mean that the DSAG or MAG

12 tion accuracy in recommender systems. Although the DSAG provides more accurate predictions when there are sufficient reviews, using a MAG was recently suggested when review data is too sparse. We used three real recommender data sets to show that both advice givers provide accurate predictions to some users. The DSAG provides more accurate predictions to users when different reviewers were weighed highly in different domains while the MAG is more accurate when the review matrix and user reviews are sparse. We presented our User-Dependent Online Selection Algorithm and User- and Category-Dependent Online Selection algorithm as two online methods for recommender systems to decide which advice giver is more accurate for each user. The algorithms wait for the reviewers weights to converge before picking the best advice giver. We validated our findings on a fourth recommender data set and demonstrated that the UdOS algorithm successfully picks the better advice giver to provide the most accurate predictions possible for each user. We conclude that a recommender system that uses the UdOS or UCdOS algorithms make better predictions for all users than one that uses only one of the two advice givers. Umyarov, A., and Tuzhilin, A Leveraging aggregate ratings for better recommendations. In Recommender Systems, Webscope, Y. R. 2008a. Movie user ratings of movies and descriptive content information v1.0. Webscope, Y. R. 2008b. Music user ratings of songs with song attributes v1.0. References Adomavicius, G., and Tuzhilin, A Toward the next generation of recommender systems: A survey of the stateof-the-art and possible extensions. IEEE Trans. on Knowledge and Data Engineering Auer, P.; Cesa-Bianchi, N.; Freund, Y.; and Schapire, R Gambling in a rigged casino: The adversarial multiarmed bandit problem. In FOCS, Berkovsky, S.; Kuflik, T.; and Ricci, F. 2007a. Crossdomain mediation in collaborative filtering. In User Modeling, Berkovsky, S.; Kuflik, T.; and Ricci, F. 2007b. Distributed collaborative filtering with domain specialization. In Recommender Systems, Blum, A., and Mansour, Y From external to internal regret. In COLT, Burke, R Hybrid recommender systems: Survey and experiements. User Modeling and User-Adapted Interaction 12(4): Freund, Y.; Schapire, R.; Singer, Y.; and Warmuth, M Using and combining predictors that specialize. In STOC, Leskovec, J.; Adamic, L.; and Huberman, B The dynamics of viral marketing. In ACM Conf. on Electronic Commerce, Littlestone, N., and Warmuth, M The weighted majority algorithm. Information and Computation Nakamura, A., and Abe, N Collaborative filtering using weighted majority prediction algorithms. In International Conference on Machine Learning, Netflix The netflix dataset and prize.

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