Modeling Recommendation as a Social Choice Problem ∗ Massimiliano Albanese

Antonio d’Acierno

UMIACS University of Maryland College Park, MD 20742, USA

ISA-CNR Via Roma, 64 83100 Avellino, Italy

[email protected] [email protected] Vincenzo Moscato Fabio Persia Antonio Picariello Dept. of Computer Science University of Naples 80125 Naples, Italy

Dept. of Computer Science University of Naples 80125 Naples, Italy

Dept. of Computer Science University of Naples 80125 Naples, Italy

[email protected]

[email protected]

[email protected]

ABSTRACT In the classical theory of social choice, a set of voters is called to rank a set of alternatives and a social ranking of the alternatives is generated. In this paper, we model recommendation in the context of browsing systems as a social choice problem, where the set of voters and the set of alternatives both coincide with the set of objects in the data collection. We then propose an importance ranking method that strongly resembles the well known PageRank ranking system, and takes into account both the browsing behavior of the users and the intrinsic features of the objects in the collection. We apply the proposed approach in the context of multimedia browsing systems and show that it can generate effective recommendations and can scale well for large data collections.

Categories and Subject Descriptors H.3.7 [Information Storage and Retrieval]: Digital Libraries; H.5.1 [Information Interfaces and Presentation]: Multimedia Information System

General Terms Algorithms, Experimentation, Theory

1. INTRODUCTION AND RELATED WORK Recommendation is becoming the new paradigm of search: interesting items find the user instead of the user explicitly ∗This work has been carried out partially under the financial support of the Programma Faro (Finanziamento per l’Avvio di Ricerche Originali) in the framework of the LATINO Project: Un sistema innovativo di LocAlizzazione e Tracciamento di INdividui per l’Ottimizzazione di percorsi in ambienti indoor.

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searching for them; classical approaches to recommendation can be broadly classified into two major categories: content based recommendation and collaborative filtering. In content based recommender systems [6], the utility of an item o for a user is estimated from the utility assigned by the same user to other items that are similar to o. One of the main drawbacks of similar techniques is that they cannot benefit from the great amount of information that is implicit in the behavior of other users. Instead, the main idea behind collaborative filtering is to associate each user to a set of other users having similar profiles and recommending items based on the similarity between users, rather than on the similarity between data items themselves [9]. These strategies usually suffer from the fact that a new data item will become available for recommendation only when enough users have seen and rated it, and it will not be possible to make reliable recommendations to a user who is not similar enough to any other user. The effectiveness of the recommendation process can be improved by combining content based filtering and collaborative strategies [4] and hybrid systems can be further improved by modeling recommendation as a social choice problem. In the classical theory of social choice, as formulated by Arrow [3], a set of voters is called to rank a set of alternatives. Given the agents’ individual rankings, a social ranking of the alternatives is generated. The theory studies desired properties of the aggregation of agents’ rankings into a social ranking [1]. In particular, Arrow’s impossibility theorem shows that there is no aggregation rule that satisfies some minimal requirements, while by relaxing any of these requirements appropriate social aggregation rules can be defined. The novel feature of ranking systems is that the set of agents and the set of alternatives coincide. Therefore, in this setting one may need to consider the transitive effects of voting, when aggregating individual rankings into a social ranking. For example, if agent a reports on the importance of (i.e., votes for) agent b then this may influence the credibility of a report by b on the importance of agent c. The ranking of agents based on other agents’ input is fundamental to multi-agent systems [7] and it has become a central element of a variety of applications, where the most famous examples are Google’s PageRank [5] and eBay’s reputation system [8]. Axiomatic foundations of trust-based recommendation systems have been studied in [2].

In our work we model recommendation in the context of browsing systems as a social choice problem, where the set of voters and the set of alternatives both coincide with the set of objects in the data collection. Differently from most work on trust-based recommendation, we do not assume that a subset of the agents has explicitly ranked a set of items or that a (partial) trust network (i.e., who trusts who) is known a priori, as in [2]. Instead, we infer links between data items (similar to links between web pages) from the dynamic behavior of the users and the features of objects in the dataset. To this aim, we assume that (i) an item o “links” similar items, and (ii) a user accessing item o2 after item o1 corresponds to a link from o1 to o2 . Additionally, rather than providing the users with absolute recommendations, we aim at making recommendation context-dependent, where the context is determined by the current browsing session. The paper is organized as follows. Section 2 introduces the proposed ranking method. Section 3 reports on preliminary experiments, and concluding remarks are given in Section 4.

where wij is the ∑ fraction of oj ’s importance that is transferred to oi , and oi ∈O wij = 1. Given the iterative nature of this definition, the vector R = [ρ(o1 ) . . . ρ(on )]T can be computed as the solution to the following equation: R=C ·R (2) where C = {wij } is a matrix defining how the importance of each object is transferred to other objects. C must satisfy certain conditions to guarantee that Equation 2 has a solution and such a solution is unique. We will discuss how to build this matrix in Section 2.1, and describe a method to efficiently compute the solution to Equation 2 in Section 2.2.

2.1

Ranking

Given a set U = {u1 , . . . , um } of users, we define a local n × n browsing matrix Al = {alij } for each user ul ∈ U , and a global n × n browsing matrix A = {aij }. The element alij of a local browsing matrix Al is defined as the ratio of the number of times an object oi has been accessed by user ul immediately after oj to the number of times any object in O has been accessed by ul immediately after oj . Similarly, aij is defined as the ratio of the number of times oi has been accessed by any user immediately after oj to the number of times any object in O has been accessed ∑ immediately after oj . It is evident that ∀i, j ∈ [1, n], ul ∈U alij ≤ aij , as anonymous users contribute to A but not to any Al . Additionally, we define the similarity matrix B = {bij } as follows: { fsim (oi ,oj ) , if fsim (oi , oj ) ≥ τ ∧ i ̸= j Γj (3) bij = 0, otherwise where fsim is a similarity function defined over O, τ is a threshold, and Γj∑ , for each j, is a normalization factor which guarantees that i bij = 1. In other words, if the similarity between two given objects is below a given threshold τ , we set the corresponding element of the matrix to 0, which guarantees B to be sparse. The similarity matrix can be efficiently computed using indexes on the image database, but we omit the details for reasons of space. In order to compute customized rankings for each user, we rewrite Equation 2 as Rl = Cl · Rl (4) where Rl = [ρl (o1 ) . . . ρl (on )]T is the vector of recommendation grades, customized for user ul . In order to take into account the browsing behavior of the individual user, the overall behavior of the community of users, and the intrinsic similarity between objects in the collection, we define Cl as a linear combination of Al , A, and B. However, we must first observe that solving Equation 4 corresponds to finding the stationary vector of Cl , i.e., the eigenvector with eigenvalue 1. The Perron-Frobenius theorem ensures that a real square matrix with positive elements has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components. Moreover, the largest eigenvalue of a stochastic matrix is always 1. It is easy to observe that the elements of each column of Al , A, and B either sum up to one or are all 0. To make them into stochastic matrices, we introduce the following auxiliary matrices:

2. THEORETICAL MODEL Building an effective recommender to support intelligent browsing of large data collections implies the capability of identifying the objects that are most likely to satisfy the interests of a user at any given point of her exploration of the collection. We will use the term “browsing session” to denote an ordered sequence of interactions of the user with the system. At each step of a browsing session, the user chooses an object, by either accepting one of the recommendations, or jumping to any object in the collection. In this scenario, we need to address two fundamental questions: (i) How can we select a set of good candidates from the collection? (ii) How can we rank the set of candidates? Similarly to how the Google search engine works, we compute a global ranking of all the objects in the collection, and then use the computed scores to rank candidate objects. In a citation ranking system the importance of a document is influenced by the number and importance of other documents pointing to it. In the case of a generic collection of digital objects, such as a collection of paintings in a virtual museum, usually there are no explicit links between objects. However, the fact that an object oi is chosen immediately after an object oj in the same browsing session can be interpreted as oj “linking” oi . This implies that oi must be relevant to oj . Additionally, if oi is chosen after many different objects, we can conclude it must be important. Similarly, the fact that an object oi is very similar to oj can also be interpreted as oj “linking” oi (and vice-versa). In conclusion, we want to determine an absolute score for each object o in the dataset – we call this score recommendation grade and denote it by ρ(o) – that measures the importance of o and therefore the likelihood that a user will select it. We want to use this score to rank recommendations from a set of candidates and present the user with the top k ranked alternatives. In the following, we focus on presenting the proposed ranking method and a strategy to efficiently compute it. For reason of space, we omit a discussion of the relevant axioms and refer the reader to [1]. Given a set O = {o1 , . . . , on } of objects, we can define the recommendation grade as follows: ∑ ∀oi ∈ O ρ(oi ) = wij · ρ(oj ) (1)

A∗ = A + S(A) ∗

A∗l = Al + S(Al ), ∀l ∈ [1, m] (5)

B = B + S(B) where S is a function that associates with each matrix M = {mij } a matrix S(M ) = {sij } defined as follows: { 0, if ∃k ∈ [1, n] | mkj > 0 sij = (6) 1 , otherwise n

oj ∈O

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In other words, stochastic matrices A∗l , A∗ , and B ∗ are derived from Al , A, and B respectively, by replacing elements in columns of 0’s with 1/n. We can now define Cl as

determined by |λ2 |. In fact, the closer |λ2 | is to 0, the faster λk2 → 0. The fact that the matrix Cl∗ is primitive guarantees that |λ2 | < 1, thus ensuring the convergence of the power method. Additionally, it can be proved that convergence is faster for higher values of σ. However, there is a trade-off in the choice of σ, since higher values also imply a greater weight assigned to the random behavior. We now discuss how to compute Rlk at each step of the power method, by leveraging the specific nature of the matrix Cl∗ . Using Equations 7, 8 and 10, we can write:

Cl = γ · (α · A∗l + (1 − α) · A∗ ) + (1 − γ) · B ∗ (7) Browsing behavior and object similarity both contribute to the overall social ranking, with weights γ and 1−γ respectively. Additionally, the contribution of browsing behavior is a weighted sum of individual and collective behaviors. It is easy to see that this formulation addresses both the cold start and the anonymous user problems. When a user logs into the system for the first time, her browsing matrix has all 0’s, but the system is still able to make recommendations, based on the behavior of the community of users and object similarity. As information about the user is collected over time, Al will start to influence the recommendation process. Similarly, an anonymous user will be considered as a user whose browsing matrix is always all 0’s (alternatively we can assume α = 0 for anonymous users). Cl is guaranteed to be stochastic, as it results from a weighted sum of stochastic matrices. However, Cl is not yet guaranteed to be positive. We then introduce a parameter σ, modeling the probability that the user will choose the next object in a totally random fashion, and we redefine Cl as follows. 1 (8) Cl∗ = (1 − σ) · Cl + σ · · Jn n where Jn is an n × n matrix of ones. Equation 8 guarantees that Cl∗ is stochastic and positive. Therefore, replacing Cl with Cl∗ in Equations 4, we obtain

Rlk = Cl∗ · Rlk−1 = =

(11)

π1 · Al · Rlk−1 + π2 · A · Rlk−1 + π3 · B · Rlk−1 + +(π1 · SAl + π2 · SA + π3 · SB ) · Rlk−1 + π4 · Jn · Rlk−1

where π1 = (1 − σ) · γ · α, π2 = (1 − σ) · γ · (1 − α), π3 = σ are constants. Therefore, Rlk can (1 − σ) · (1 − γ), π4 = n be computed as the sum of several terms, which in turn can be computed efficiently. First, matrices Al , A, and B are sparse, therefore computing the three products Al · Rlk−1 , A · Rlk−1 , and B · Rlk−1 reduces to computing the product of a sparse matrix and a vector, which can be done efficiently. Second, all the rows of matrices SAl , SA , and SB are equal. Therefore, all the elements of the vector (π1 ·SAl +π2 ·SA +π3 · SB )·Rlk−1 are equal and can be computed by multiplying the first row of the matrix S = π1 · SAl + π2 · SA + π3 · SB – which is also sparse – and the vector Rlk−1 . Finally, the elements of matrix π4 · Jn are all equal, therefore all∑the elements k−1 (i), of the vector π4 · Jn · Rlk−1 are equal to π4 · n i=1 Rl which needs to be computed only once. Note that rankings need to be periodically recomputed as new objects can be added to or removed from the collection at any time, and new browsing data is collected on a continuous basis. We can finally describe how to provide a user with a ranked list of recommendations. Assume that a user ul is currently watching object oj . We can define the set of candidate recommendations as

Rl = Cl∗ · Rl (9) which is guaranteed to have a unique solution Rl . In conclusion, the introduction of the factor σ, while contributing to model the behavior of the user, makes the problem solvable.

2.2 Computation There are many ways to find the eigenvectors of a square matrix. However, the challenge here is that the matrix Cl∗ has one column for each object in the collection. This means that Cl∗ might potentially have millions of columns and rows. We want to be able to solve Equation 9 for very large matrices Cl∗ . A method known as the power method can be used for finding the stationary vector of a square matrix. The power method consists in computing a sequence of vectors Rlk as follows:

Cj =

M ∪

{oi ∈ O|Akij > 0} ∪ {oi ∈ O|Bij > 0}

(12)

k=1

In other words, the set of candidates includes the objects that have been accessed by at least one user within k steps from oj , with k between 1 and M , and the objects that are most similar to oj . Note that a positive element akij of Ak indicates that oi was accessed exactly k steps after oj at least once. The ranked list of recommendations is then generated by ranking the objects in Cj according to scores in Rl .

Rlk = Cl∗ · Rlk−1 (10) 0 where Rl is an arbitrary initial vector whose elements sum up to one. The method iterates until convergence is reached or a given number of iterations has been performed. Under certain circumstances the power method converges to the stationary vector of the matrix, as proved below. Assume that the eigenvalues of Cl∗ are λ1 , λ2 , . . . , λn and that 1 = λ1 > |λ2 | ≥ |λ3 | ≥ . . . ≥ |λn |. In addition, assume the existence of a basis vi of eigenvectors for Cl∗ , with corresponding eigenvalues λi . We can write the initial vector Rl0 as Rl0 = c1 · v1 + c2 · v2 + . . . + cn · vn , where c1 , c2 , . . . , cn are arbitrary constants. Then, we compute Rlk = Cl∗ · Rlk−1 until |Rlk − Rlk−1 | ≤ tol ∗ |Rlk−1 | ∨ k ≥ N , where tol is a relative tolerance and N is the maximum number of allowed iterations. Since the eigenvalues λj with j ≥ 2 have absolute value smaller than one, we can conclude that λkj → 0 for all j ≥ 2. Therefore Rlk → Rl = c1 · v1 , an eigenvector corresponding to the eigenvalue 1. The rate at which Rlk → Rl is

3.

EXPERIMENTAL RESULTS

We implemented a prototype of the proposed framework and conducted preliminary experiments, which are briefly described in the following. We applied the proposed recommendation approach to the case of virtual museum, i.e., a museum that offers a web-based access to a multimedia collection of digital reproductions of paintings, educational videos and text documents. Differently from a real world museum, where the order in which artifacts are viewed is mainly dictated by their spatial juxtaposition, a virtual museum enables its users to access artifacts in any order they consider relevant. In particular, we evaluated the impact of the proposed system on enhancing a user’s experience and

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browsing tasks faster and provides a better (less frustrating) user experience. In addition, the fact that browsing tasks can be completed faster using our system with respect to a classical browsing system is an indication that recommendations are effective, as they allow a user to explore interesting and related objects one after another, without the interference of undesired items that would necessarily slow down the process.

Table 1: Comparison with Picasa Task class T1 T2 T3 T4

ta

Picasa (sec.) nc 60.2 15 104.3 26.8 219.8 57.1 402.6 104.2

Our system ta (sec.) nc 52.1 12.6 77.1 20.4 151.7 38.9 240.3 60.3

compared its performance to an existing system for organizing and browsing large photo collection (Picasa Web Albums), which does not take into account browsing behavior of users and intrinsic features of the multimedia objects. The dataset used in the experiments consists of 5,000 paintings of various genres, artists and subjects. In particular, the dataset includes 25 genres (e.g., Cubism, Baroque), about 100 artists (e.g., Caravaggio, Rubens) and about 50 subjects (e.g., Landscapes, Portraits, Mythology). In the first stage of our experiments, we asked a group of 20 undergraduate students to use the system for several days, in order to collect enough browsing data to compute meaningful browsing matrices. Browsing data collected at this stage was also used to learn and tune the parameters – namely α, β, and γ – that we introduced in Section 2.1 to scale the contribution of several factors in the computation of rankings (details are omitted for reasons of space). We then asked a different group of 50 people to browse the collection and complete several browsing tasks (20 tasks per user) of different complexity (5 tasks for each complexity level), using Picasa (taxonomies were implemented as albums, folders and descriptions). After this test, we asked them to browse once again the same collection with the assistance of our recommender system and complete other 20 tasks of the same complexity. We subdivided browsing tasks into four broad categories, as described below:

4.

CONCLUSIONS

In this paper, we have shown that recommendation in the context of browsing systems can be modeled as a social choice problem, where the set of voters and the set of alternatives both coincide with the set of objects in the data collection. We have proposed an importance ranking method that strongly resembles the well known PageRank ranking system, and takes into account both the browsing behavior of the users and the intrinsic features of the objects in the collection. We have then applied the proposed recommendation approach to the case of a virtual museum, and preliminary experiments have shown that the system can generate effective recommendations, improve a user’s experience, and scale well for large data collections.

5.

REFERENCES

[1] A. Altman and M. Tennenholtz. Axiomatic foundations for ranking systems. Journal of Artificial Intelligence Research, 31:473–495, March 2008. [2] R. Andersen, C. Borgs, J. Chayes, U. Feige, A. Flaxman, A. Kalai, V. Mirrokni, and M. Tennenholtz. Trust-based recommendation systems: an axiomatic approach. In Proc. of the 17th Intl. World Wide Web Conf., pages 199–208, April 2008. [3] K. J. Arrow. Social Choice and Individual Values. John Wiley & Sons, second edition edition, 1963. [4] R. Burke. The Adaptive Web: Methods and Strategies of Web Personalization, volume 4321 of Lecture Notes in Computer Science, chapter Hybrid web recommender systems, pages 377–408. Springer-Verlag, 2007. [5] L. Page, S. Brin, R. Motwani, and T. Winograd. The PageRank citation ranking: Bringing order to the web. Tech. Report 1999-66, Stanford InfoLab, Nov. 1999. [6] M. J. Pazzani and D. Billsus. The Adaptive Web: Methods and Strategies of Web Personalization, volume 4321 of Lecture Notes in Computer Science, chapter Content-Based Recommendation Systems, pages 325–342. Springer-Verlag, 2007. [7] P. Resnick, K. Kuwabara, R. Zeckhauser, and E. Friedman. Reputation systems. Communications of the ACM, 43(12):45–48, December 2000. [8] P. Resnick and R. Zeckhauser. The Economics of the Internet and E-Commerce, volume 11 of Advances in Applied Microeconomics, chapter Trust Among Strangers in Internet Transactions: Empirical Analysis of eBay’s Reputation System, pages 127–157. Elsevier Science, November 2002. [9] B. M. Sarwar, J. A. Konstan, A. Borchers, J. Herlocker, B. Miller, and J. Riedl. Using filtering agents to improve prediction quality in the GroupLens research collaborative filtering system. In Proc. of the 1998 ACM Conf. on Computer Supported Cooperative Work, 1998.

1. Low Complexity tasks (T1 ) – e.g. “explore at least 10 paintings of Baroque style painted by Caravaggio and depicting a religious subject”; 2. Medium Complexity tasks (T2 ) – e.g. “explore at least 20 paintings of Baroque artists that have nature as their subject”; 3. High Complexity tasks (T3 ) – e.g. “explore at least 30 paintings of Baroque artists with subject nature and with a predominance of red color ”; 4. Very High Complexity tasks (T4 ) – e.g. “explore at least 50 paintings of Baroque artists with a predominance of red color ”. Note that the complexity of a task depends on several factors: the number of objects to explore, the type of desired features (either low or high-level), and the number of constraints (genre, artist, subject). We used empirical measurements of access complexity in terms of (i) time spent by the users to request and access all the objects for a given class of tasks (ta ), and (ii) number of clicks necessary to collect all the requested objects for a given class of tasks (nc ). Table 1 reports the average values of ta and nc , for both Picasa and our system, for each of the four task complexity levels defined earlier. It is worth noting that the smaller ta and nc are, the easier and faster is for users to browse the collection and achieve their intended goals. Thus, the preliminary results presented here imply that our system allows to complete

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