Recommendation systems are ubiquitous and impact many domains. One common application of these systems is online platforms for video, music, and product purchases through service providers such as Netflix, Pandora, and Amazon. Recommendation systems have the potential to influence how users perceive the world by filtering access to news, media, and books. Even more gravely, these systems impact crucial decision-making processes, such as loan approvals, criminal profiling, and medical interventions.
In the real world, these systems are updated regularly to incorporate new data and deployed systems are retrained based on observed data that is influenced by the recommendation system itself, forming a feedback loop (fig:cartoon). It seems undesirable to ignore new data entirely, but the effects of algorithmic confounding should be understood. In this paper, we expose unintended consequences of algorithmic confounding in recommendation systems so that these effects may be countered.
The findings of this paper are relevant to a variety of individuals. Recommendation system researchers need to ensure that their models are evaluated convincingly to prove the efficacy of their systems for broader use and adoption; to do so, they need to consider these confounding factors in the generation of the data and account for them in both training and testing. Social science researchers look to online platforms as a rich source of data about human behavior; they should similarly account for algorithmic confounding when possible. Practitioners or platform developers who wish to increase user satisfaction with recommendations (either as an end, or as a means to an end, e.g., to increase platform engagement or sales) should account for algorithmic confounding when they update their recommendation algorithms. Platform users and policy makers may be concerned with the impact of recommendation systems on themselves as individuals or on society more broadly. Those interested in understanding the risks of these systems will be able to use the insights we provide to develop and suggest methods for greater transparency and accountability in these platforms.
We begin with a summary of our claims (sec:claims). To provide evidence for these claims, we introduce a model for users interacting with recommendations (sec:interaction); this allows us to analyze the impact of algorithmic confounding on simulated communities (sec:simulations). We find that algorithmic confounding disadvantages some algorithms in training (sec:utility), biases held-out evaluation (sec:eval), and amplifies the homogenization of user behavior without gains in utility (sec:homogenization). We briefly discuss weighting approaches to account for these effects (sec:confounded_eval) and situate this work among related lines of inquiry (sec:related) before we conclude (sec:conclusion). In an appendix, we outline a general framework for recommendation systems and frame the algorithms we study in this context (sec:framework). This general recommendation system framework highlights the commonalities between seemingly distinct recommendation approaches, a contribution in itself.
2. Consequences of the Feedback Loop
Real-world recommendation systems are often part of a feedback loop (fig:cartoon): the underlying recommendation model is trained using data that are confounded by algorithmic recommendations from a previously deployed system. We attempt to characterize the impact of this feedback loop; we have three core findings.
Utility of Confounded Data (sec:utility).
Algorithmically confounded data may be used to train recommendation models; some (but not all) algorithms perform well with confounded data, indicating these data have utility for training, but should be used with care.
Evaluation Using Confounded Data (sec:eval).
When a recommendation system is evaluated using confounded held-out data, results are biased toward recommendation systems similar to the confounding algorithm. This means that the choice of data can considerably impact held-out evaluation and subsequent conclusions.
Homogenization Effects (sec:homogenization).
The recommendation feedback loop causes homogenization of user behavior, which is amplified with more cycles through the loop. Homogenization occurs at both a population level (all users behave more similarly) and at an individual level (each user behaves more like its nearest neighbors). Users with lower relative utility have higher homogenization.
3. Interaction Model
In order to reason about the feedback dynamics of algorithmic recommendations and user behavior, we need a model of how users engage with recommended items on a platform; we model engagement and not ratings, which is justified in sec:related. We draw on a model recently proposed by Schmit and Riquelme (schmit2017human, ) that captures the interaction between recommended items and users, and we modify this model to allow for personalized recommendations and multiple interactions for a given user.111Unlike the Schmit and Riquelme model (schmit2017human, ), we include no additional noise term because we model utility (or “quality” in the Schmit and Riquelme model) probabilistically instead of holding it fixed.
Definition 3.1 ().
The utility of user consuming item at time is
where and are the utilities that are known and unknown to the user, respectively. Neither utilities are known to the platform.
When a user considers whether or not they wish to engage with items, they have some notion of their own preferences; these preferences come from any information displayed and external knowledge. The quantity captures these preferences that are known to the user but unknown to the recommendation system platform. Users rely on in combination with the ordering of recommended content to select items with which to engage. Users must also have some unknown utility
, or else they would be omniscient, and recommendation systems would be of no use. The underlying objective of these systems is to estimate the total utility.
The utility of a user interacting with an item is approximately static over time, or .
In the real world, utility fluctuates due to contextual factors such as user mood. However, the variance around utility is likely small and inversely related to the importance of the choice. Moving forward, we will omit the time notation for simplicity.
The total utility
is beta-distributed,222We use an atypical parameterization of the beta distribution with mean and fixed variance and distinguish this parameterization as . For our simulations in sec:simulations, we set . To convert to the standard parameterization, and .
and is parameterized by the dot product of user general preferences for user and item attributes for item .
This assumption will constrain utility values to be in the range
; this representation is flexible because any utility with finite support can be rescaled to fall within this range. The use of the dot product to parameterize the utility is likewise a flexible representation; when the underlying vectorsand have a dimensionality of and either preferences or attributes use a one-hot representation, then all possible utility values can be captured.
General preferences and attributes are fixed but unknown to the user or the recommendation system. They are drawn from Dirichlet distributions, or
for all users and all items , respectively. Individual preferences are parameterized by a vector of global popularity of preferences over all users. Individual item attributes are similarly parameterized by a global popularity of attributes over all items.
A draw from the Dirichlet distribution produces a vector that sums to one, allowing for easy interpretation; this distribution is also flexible enough to capture a range of possible preferences and attributes, including the one-hot representation discussed previously. Further, as:utility requires and this guarantees that and will satisfy this constraint. Most importantly, when aggregated by user (a proxy for activity) or item (popularity), this construction produces a distribution of utility values with a long tail, as seen in real platforms (celma2010long, ).
The proportion of the utility known to user is ;333Mean proportion in our simulations (sec:simulations). this results in
This assumption implies that each user is approximately consistent at assessing utility, but introduces some uncertainty so that the known utility is a noisy approximation of the true utility . While each user could theoretically have a different mean proportion , in practice this is not important because the known utilities are not compared across users. This representation is simple; however, a more realistic model might vary the known proportion of utility as a function of item attributes or include a time dependency. These additional complexities would likely amplify the effects our findings or have no impact.
At every discrete time step , each user will interact with exactly one item, .
Users in the real world have varying levels of activity; we argue that the long tail of utility (see the justification for as:prefs) captures the essence of this behavior and that we could adopt different levels of user activity without substantially altering our results.
Definition 3.2 ().
To select an item at time , user relies on her own preferences and a function of the rank of the items provided by recommendation system .444For our simulations (sec:simulations), we used , which approximates observed effects of rank on click-through rate (jansen2013effect, ); our results held for other choices of . The chosen item is
where such that , according to recommender system ’s ordering of items, as described in sec:framework.
Users are more likely to click on items presented earlier in a ranked list (jansen2013effect, ); the function captures this effect of rank on the rate of interaction. In keeping with our earlier discussion, to allow for various levels of user activity, one need only add some threshold such that if the function of rank and preference inside eq:choice are less than this threshold, then no interaction occurs.
Each user interacts with item at most once.
When a user engages with an item repeatedly, utility decreases with each interaction. This is the simplest assumption that captures the notion of decreasing utility; without it, a generally poor recommendation system might ostensibly perform well due to a single item. The interaction model could alternatively decrease utility with multiple interactions, but this would not alter results significantly.
New and recommended items are interleaved.
As in the real world, new items are introduced with each time interval. When no recommendation algorithm is in place (early “start-up” iterations), the system randomly recommends the newest items. Once a recommendation algorithm is active, we interleave new items with recommended items; this interleaving procedure is a proxy for users engaging with items outside of the recommendation system, or elsewhere on the platform. Since this procedure is identical for all systems, it does not impact the comparison across systems.
4. Simulated Communities
In this section, we explore the performance of various recommendation systems on simulated communities of users and items. We first describe the simulation procedure, and then discuss three claims.
4.1. Simulation Procedure
We consider six recommendation algorithms: popularity, content filtering (“content”), matrix factorization (“MF”), social filtering (“social”), random, and ideal. sec:framework provides further details the first four approaches and describes our general recommendation framework. The core idea of this framework is that each recommendation system provides some underlying score of how much a user will enjoy item at time . These scores are constructed using user preferences and item attributes :
and each recommendation approach has a different way of constructing or modeling these preferences and attributes.
For our simulations, all of the six approaches recommend from the set of items that exist in the system at the time of training; random recommends these items in random order. Ideal recommends items for each user based on the user’s true utility for those items. Comparison with these two approaches minimizes the impact of the interaction model assumptions (sec:interaction) on our results.
In all of our simulations, a community consists of users and is run for time intervals with ten new items being introduced at each interval; each simulation is repeated with ten random seeds and all our results are averages over these ten “worlds.”
We generate the distributions of user preference and item attribute popularity, as used in eq:pref, in dimensions; we generate uneven user preferences, but approximately even item attributes. The user preference parameter is generated as follows:
This mirrors the real world where preferences are unevenly distributed, which allows us to expose properties of the recommendation algorithms. Item attribute popularity is encouraged to be more even in aggregate, but still be sparse for individual items; we draw:
While item attributes are not evenly distributed in the real world, this ensures that all users will be able to find items that match their preferences. With these settings for user preferences and item attributes, the resulting matrix of true utility is sparse (e.g., fig:world), which matches commonly accepted intuitions about user behavior.
We generate social networks using the covariance matrix of user preferences; we impose that each user must have at least one network connection and binarize the covariance matrix using this criteria. This procedure enforces that the network is homophilous, which is generally (but not always) true in the real world.
We consider two cases of observing user interactions with items: a simple case where each recommendation algorithm is trained once, and a more complicated case of repeated training; this allows us to compare a single cycle of the feedback loop (fig:cartoon) to multiple cycles. In the simple paradigm, we run 50 iterations of “start-up” (new items only each iteration), train the algorithms, and then observe 50 iterations of confounded behavior. In the second paradigm, we have ten iterations of “start-up,” then train the algorithms every iteration for the remaining 90 iterations using all previous data.
4.2. Utility of Confounded Data
When data is collected from a platform where users are recommended content, the resulting algorithmically confounded user behavior data may be used to train a new recommendation system (or update an old one). We wish to identify which algorithms can use confounded data to increase utility for users.
Because our interaction model (sec:interaction) includes an explicit notion of utility, we can simply observed the utility in our simulations and compare recommendation algorithms to randomly recommended content. If an algorithm trained with confounded data results in higher utility than random recommendations, then confounded data are “useful” in training that algorithm.
We found that content and social filtering made best use of confounded data (fig:utility); this is likely because the content information and social network ground the algorithms in more static representations of the world. While matrix factorization cumulatively outperformed random recommendations in the single training case (fig:cumulative_utility), both MF and popularity under-performed random recommendations in the repeated training case, indicating that naively training on confounded data can be disadvantageous in practice.
We conclude that using confounded data for training may still increase the utility of recommendation platforms, but it is not universally beneficial. This means that practitioners can still train models on confounded data and see improvements in user satisfaction (and therefore other metrics like click-through-rate and revenue), but that in some cases performance may be less optimal.
4.3. Evaluation Using Confounded Data
Recommendation system are often evaluated using confounded held-out data. This form of offline evaluation is used by both researchers and practitioners alike, but the choice is not as innocuous as one might expect. When a recommendation system is evaluated using confounded held-out data, results are biased toward recommendation systems similar to the confounding algorithm. Researchers can, intentionally or unintentionally, select data sets that highlight their proposed model, thus overstating its performance.
To illustrate this point, we simulated identical communities interacting with social filtering and matrix factorization recommendations. After generating confounded data, we compared the held-out evaluation of the confounding algorithms using 80% of the data for training and 20% for testing.We used normalized discounted cumulative gain (nDCG), a typical rank-based evaluation metric (see sec:related). The unnormalized variant of this metric is
where is the set of held-out items for user and is the th recommended item for user , as defined in sec:framework, eq:rank. This is then normalized
We found that held-out evaluation favored the algorithm used to confound the data: social factorization outperformed matrix factorization (MF) when evaluated on data confounded by social recommendations and MF had superior held-out performance when evaluated on data confounded by MF (fig:ndcg). This held for both the simple single-training case and for the case of repeated cycles through the feedback loop. The underlying models of utility in the simulations were identical, so one might naively assume that the training data would be approximately equivalent no matter the confounding algorithm; instead, confounded data yielded conflicting results. These results demonstrate that held-out evaluation using confounded data can be unreliable and that the choice of data matters.
There are specific instances in recommendation system literature where this may be troubling. For example, MovieLens is a research website that uses collaborative filtering to make movie recommendations (miller2003movielens, ); the researchers behind it have released several public datasets555https://grouplens.org/datasets/movielens/ that are used prolifically in evaluation recommendation systems (harper2016movielens, ). Recommendation systems based on collaborative filtering will likely perform better on this data.
Douban is a social networking service that recommends popular content to users; there is a publicly available data set from this platform that is used to evaluate many recommendation systems (hao:sr, ). Because of the popularity features on this website, algorithms that include a notion of popularity will perform better on this data.
Facebook and Etsy are among websites that use social networks to recommend content. Many algorithms use social network information to improve recommendations (Chaney:2015, ; Jamali:2010, ; Yang2013, ; guo2015trustsvd, ; Ma:2009, ; SoRec, ) but when training and evaluating these algorithms on data from social platforms, it is not clear if the models capture the true preferences of users, or if they capture the biasing effects of platform features.
These biases are problematic for researchers and platform developers who attempt to evaluate models offline or rely on academic publications that do.
4.4. Homogenization Effects
Recommendation systems may not change the underlying preferences of user (especially not when used on short time scales), but they do impact user behavior, or the collection of items with which users interact. Recommendation algorithms encourage similar users to interact with the same set of items, therefore homogenizing their behavior, relative to the same platform without recommended content. For example, Popularity-based systems represent all users in the same way; this homogenizes all users, as seen in previous work (celma2008hits, ; treviranus2009value, ). Social recommendation systems homogenize connected users or within cliques, and matrix factorization homogenizes users along learned latent factors; in this latter case, the homogeneity of user interactions will increase as the cosine similarity in their latent representations increases.
Homogenizing effects are not inherently bad as they indicate that the models are learning patterns from the data, as intended. However, homogenization of user behavior does not correspond directly with an increase in utility. There is an optimum amount of homogenization for a given user representation, or we can observe an increase in homogenization without a corresponding increase in utility. This is related to the explore/exploit paradigm, where we wish to exploit the user representation to maximize utility, but not to homogenize users more than necessary. When a representation of users is over-exploited, users are being pushed to be have similar behaviors when, based on their true preferences, they would enjoy a broader range of items. This would indicate that the “tyranny of majority” and niche “echo chamber” effects may both be manifestations of the same problem: over-exploitation of recommendation models.
We measure homogenization of behavior by first pairing each user with their most similar user according to the recommendation system, or user is partnered with user that maximizes the cosine similarity of and . Next, we compute the Jaccard index of the sets of observed items for these users. If at time , user has interacted with a set if items the Jaccard index of the two users’ interactions can be written as
We compared the Jaccard index for paired users against the Jaccard index of that same set of users exposed to ideal recommendations; this difference captures how much the behavior has homogenized relative to ideal. fig:jaccard_time shows these results for both the single training and the repeated training cases. We found that in the single training case, users became slightly homogenized after training, but returned to the ideal homogenization with time. For the repeated training, all recommendation systems (except random), homogenized user behavior beyond what was needed to achieve ideal utility. As the number of cycles in the feedback loop (fig:cartoon) increases, we observe homogenization effects continue to increase, without corresponding increases in utility (fig:utility).
We can also consider global homogenization to reveal the impact of the feedback loop at the population level; instead of comparing to paired users based on , we can compare users matched randomly (fig:jaccard_global). In this setting, we find that all recommendation systems (except, again, random) increased global homogeneity of user behavior. The popularity system increased homogeneity the most; after that, matrix factorized and social filtering homogenized users comparably, and content filtering homogenized randomly pair users least of all, but still more than ideal.
We have shown that when practitioners update their models without considering the feedback loop of recommendation and interaction, they encourage users to consume a more narrow range of items, both in terms of local niche behavior and global behavior.
Changes in utility due to these effects are not necessarily born equally across all users. For example, users whose true preferences are not captured well by the low dimensional representation of user preferences may be disproportionately impacted. These minority users may see lesser improvements or even diminishes in utility when homogenization occurs. fig:jaccard_util breaks down the relationship between homogenization and utility by user; for all recommendation algorithms, we find that users who experience lower utility generally have higher homogenization with their nearest neighbor.
Note that we have assumed that each user/item pair has fixed utility (as:static_utility). In reality, a collection of similar items is probably less useful than a collection of diverse items(drosou2017diversity, ). With a more nuanced representation of utility that includes the collection of interactions as a whole, these effects would likely increase in magnitude.
5. Accounting for Confounding
Researchers and practitioners alike would benefit from methods to address these concerns. Weighting techniques, such as those proposed by Schnabel, et al. (schnabel2016recommendations, ) and De Myttenaere, et al. (de2014reducing, ) for offline evaluation seem promising. We performed a preliminary exploration of weighting techniques and found that in the repeated training case, weighting can simultaneously increase utility and decrease homogenization. These weighting techniques could also be of use when attempting to answer social science questions using algorithmically confounded user behavior data. We leave a full exploration of these methods for future work.
As proposed by Bottou, et al. (bottou2013counterfactual, ), formal causal inference techniques can assist in the design of deployed learning systems to avoid confounding. This would likely reduce the effects we have described (sec:simulations), but needs to be studied in greater depth. Regardless, practitioners would do well to incorporate measures to avoid confounding, such as these. At the very least, they should log information about the recommendation system in deployment along with observations of behavior; this would be useful in disentangling recommendation system influence from true preference signal as weighting and other techniques are refined.
6. Related Work
Bias, confounding, and estimands
Schnabel, et al. (schnabel2016recommendations, ) note that users introduce selection bias; this occurs during the interaction component of the feedback loop shown in fig:cartoon. They consider a mechanism for interaction in which users first select an item and then rate it. Other work also considers similar notions of missingness in rating data (marlin2009collaborative, ; ying2006leveraging, ). However, many platforms exist where users express their preferences implicitly by viewing or reading content, as opposed to explicitly rating it. In the implicit setting, the observations of user behavior are the selections themselves. The quantity of interest (estimand) is no longer the rating of an item, but the probability of the user selecting an item. Thus, we no longer wish to correct for the user preference aspect of selection bias; instead, we wish to predict it.
Recommendation systems introduce confounding factors in this setting; it is difficult to tell which user interactions stem from users’ true preferences and which are influenced by recommendations. The core problem is that recommendation algorithms are attempting to model the underlying user preferences, making it difficult to make claims about user behavior (or use the behavior data) without accounting for algorithmic confounding. In this paper, we describe various problems that arise from using data confounded by recommendation systems. Among these problems is offline evaluation using confounded data; De Myttenaere, et al. (de2014reducing, ) propose addressing this with weighting techniques and Li, et al. (li2011unbiased, ) propose a method specifically for reducing this bias in contextual bandit algorithms.
Previous work has investigated how many algorithms rely on data that is imbued with societal biases and explored how to address this issue (baeza2016data, ; Sweeney:2013, ; chander2016racist, ; sandvig2014auditing, ). This work is complementary to these efforts as the described feedback effects may amplify societal biases. Due to these and other concerns, regulations are emerging to restrict automated individual decision-making, such as recommendation systems (goodman2016european, ); this work will aid in making such efforts effective.
Evaluating recommendation systems
Rating prediction is a common focus for recommendation algorithms, owing its popularity at least in part to the Netflix challenge (bennett2007netflix, ; bell2007lessons, ), which evaluated systems using RMSE on a set of held-out data. In practice, however, recommendation systems are deployed to rank items. Even Netflix has moved away from its original 5-star system in favor of a ranking-friendly thumbs-up/down interaction (youtubeThumbs, ) and now advocates for ranking items as the primary recommendation task, as it considers all items in a collection during evaluation instead of only held-out observed items (steck2013evaluation, ).
Simply put, top- accuracy metrics such as precision, recall, and nDCG are better for evaluating the real-world performance of these systems (cremonesi2010performance, ). Accuracy metrics are popular because they are straightforward to understand and easy to implement, but still they do not necessarily capture the usefulness of recommendations; there are a variety of system characteristics that are thought to be important, along with methods for evaluating them (herlocker2004evaluating, ).
Among these characteristics, diversity is often framed as a counterpart to accuracy (zhou2010solving, ; shi2017long, ; liu2012solving, ; javari2015probabilistic, ). Diversity can be considered at multiple levels: in aggregate, within groups of users, and individually. Many efforts have been made to understand whether or not various recommender systems impact diversity by reinforcing the popularity of already-popular products or by increasing interactions with niche items (fleder2009blockbuster, ; treviranus2009value, ; anderson2006long, ; celma2008hits, ; mooney2000content, ; fleder2007recommender, ; park2008long, ; massa2007trust, ; zeng2012reinforcing, ; zeng2015modeling, ; dan2013long, ). These systems also have the potential to create “echo-chambers,” which result in polarized behavior (dandekar2013biased, ).
Causality in recommendation systems
Formal causal inference techniques have only recently been applied to recommendation systems. Liang, et al. (liang2016modeling, ) draw on the language of causal analysis in describing a model of user exposure to items; this is related to distinguishing between user preference and our confidence in an observation (Hu08collaborativefiltering, ). Some work has also been done to understand the causal impact of these systems on behavior by finding natural experiments in observational data (Sharma:2015, ; su2016effect, ) (approximating expensive controlled experiments (kohavi2009controlled, )), but it is unclear how well these results generalize. As previously mentioned, Schnabel, et al. (schnabel2016recommendations, ) use propensity weighting techniques to remove users’ selection bias for explicit ratings. Bottou, et al. (bottou2013counterfactual, ) use ad placement as an example to motivate the use of causal inference techniques in the design of deployed learning systems to avoid confounding; this potentially seminal work does not, however, address the use of already confounded data (e.g., to train and evaluate systems or ask questions about user behavior), which is abundant.
Connections with the explore/exploit trade-off
In considering the impact of recommendation systems, some investigations model temporal dynamics (koren2010collaborative, ) or frame the system in an explore/exploit paradigm (Vanchinathan:2014, ; li2010contextual, )
. Recommendation systems have natural connections with the explore/exploit trade-off; for example, should a system recommend items that have high probability of being consumed under the current model, or should the system recommend low probability items in order to learn more about a user’s preferences? Reinforcement learning models already build in some notion of a feedback loop to maximize reward. One major challenge with this setup, however, is knowing how to construct the reward functions. Usually the reward is based on click-through rate or revenue for companies; we, however, focus on utility for the users of a platform. Our analysis and simulations may be informative for the construction of reward functions in reinforcement-style recommendation systems.
We have explored the impact of algorithmic confounding on a range of simulated recommendation systems. We found that algorithmic confounding disadvantages some algorithms in training (sec:utility), biases held-out evaluation (sec:eval), and amplifies the homogenization of user behavior without corresponding gains in utility (sec:homogenization). These findings have implications for any live recommendation platform; those who design these systems need to consider how a system influences its users and how to account for this algorithmic confounding. Researchers who use confounded data to test and evaluate their algorithms should also be aware of these effects, as should researchers who wish to use confounded data to make claims about user behavior from a social science perspective. Platform users and policy makers should take these effects into consideration as they make individual choices or propose policies to guide or govern the use of these algorithms.
Appendix A Recommendation Framework
In this appendix, we cast ostensibly disparate recommendation methods into a general mathematical framework that encompasses many standard algorithmic recommendation techniques.
A recommendation system provides some underlying score of how much a user will enjoy item at time . This score is generated from two components: the system’s representations at time of both user preferences for user and item attributes for item . The dot product of these vectors produces the score:
This construction is reminiscent of the notation typically used for the matrix factorization approach discussed in sec:collaborative_filtering, but it also provides us a more general framework.
The scores are not necessarily comparable across recommendation techniques. Recommendation systems that focus on rating prediction will provide scores comparable to other rating prediction models. For these systems, the goal is to predict how users will explicitly rate items, most commonly on a five-star scale, and are typically evaluated with prediction error on held-out ratings. Low errors for predicting ratings, however, do not always correspond to high accuracy in rank-based evaluation (cremonesi2010performance, ; steck2013evaluation, ) (e.g., “what should I watch next?”), which is the ultimate goal in many recommendation applications. Additionally, limiting our scope to rating prediction systems would omit models that focus on learning rankings (karatzoglou2013learning, ) or that otherwise produce rankings directly, such as ordering items by popularity.
Given a collection of scores , a recommendation system then produces, for each user, an ordered list (or sequence) of items sorted according to these scores. Formally, we represent these recommendations as a set of sequences
where is the set of all items and is the set of all users. For each user , the system provides a sequence, or ranked list of items, where is the position in the ranked list and is the recommended item for user at rank . This sequence of items for a given user is defined as all items sorted descendingly by their respective score , or
Our simulated experiments (sec:simulations) revealed that it is important to break ties randomly when performing this sort; if not, the random item recommender baseline receives a performance advantage on early iterations by exposing users to a wider variety of items.
We now cast a collection of standard recommendation systems in this framework by defining the user preferences and item attributes for each system; this emphasizes the commonalities between the various approaches.
Intuitively, the popularity recommendation system ranks items based on the overall item consumption patterns of a community of users. All users are represented identically, or for all , and thus every user receives the same recommendations at a given time . Item attributes are based on the interactions of users with items up to time ; these interactions can be structured as a collection of triplets , where each triplet indicates that user interacted with item at time .
There are many permutations of the popularity recommendation technique, including windowing or decaying interactions to prioritize recent behavior; this prevents recommendations from stagnating. For our analysis (sec:simulations), we employ the simplest popularity recommendation system; it considers all interactions up to time , or
a.2. Content Filtering
Content-based recommender systems match attributes in a user’s profile with attribute tags associated with an item (RSH, , ch. 3). In the simplest variant, the set of possible attribute tags are identical for both users and items. Then, user preferences is a vector of length , and the attribute tags for a given user are in the set ; this gives us
The item attribute tags can similarly be represented as a vector of length with values
where is the set of attributes for an item .
Attributes for both users and items can be input manually (e.g., movie genre), or they can be learned independent of time with, for example, a topic model (Blei:2012, ) for text or from the acoustic structure of a song (tingle2010exploring, ) for music; when learned, the attribute tags can be real-valued instead of binary. For our simulations (sec:simulations), we use binary item attributes and learn real-valued user preferences.777Item attributes are determined by applying a binarizing threshold to in eq:pref such that every item has at least one attribute. User representations are then learned using scipy.optimize.nnls (scipy, ).
a.3. Social Filtering
Social filtering recommendation systems rely on a user’s social network to determine what content to suggest. In the simplest variant, the user preferences are a representation of the social network, or a matrix; for each user connected to another user ,
where is the set of people follows (directed network) or with which they are connected as “friends” (undirected network) as of time . Alternatively, the user preferences can represent the non-binary trust between users, which can be provided explicitly by the user (Massa2007, ) or learned from user behavior (Chaney:2015, ); we use the latter in our analysis (sec:simulations).
Item attributes are then a representation of previous interactions, broken down by user, or an matrix where for each item and user ,
where is the set of users which have interacted with item as of time . The item representation can alternatively be a non-binary matrix, where is the number of interactions a user has with an item , or user ’s rating of item .
a.4. Collaborative Filtering
Collaborative filtering learns the representation of both users and items based on past user behavior, and is divided into roughly two areas: neighborhood methods and latent factor models.
The simplicity of neighborhood methods (RSH, , ch. 4) is appealing for both implementation and interpretation. These approaches find similar users, or neighbors, in preference space; alternatively, they can find neighborhoods based on item similarity. In either case, these methods construct similarity measures between users or items and recommend content based on these measures. We outline the user-based neighborhood paradigm, but the item-based approach has a parallel construction.
Users are represented in terms of their similarity to others, or
where the weight captures the similarity between users and . The similarity between users is typically computed using their ratings or interactions with items, and there are many options for similarity measures, including Pearson’s correlation, cosine similarity, and Spearman’s rank correlation (ahn2008new, ). These weights can be normalized or limited to the closest nearest neighbors.
Items are represented with their previous interactions or ratings, just as done for social filtering in eq:social_item. We can see that these neighborhood methods are very similar to social filtering methods—the biggest distinction is that in social filtering, the users themselves determine the pool of users that contribute to the recommendation system, whereas the collaborative filtering approach determines this collection of users based on similarity of behavior.
While neighborhood-based methods have some advantages, we focus our analysis of collaborative filtering approaches on latent factor methods for two main reasons: first, in the simulated setting, there is little distinction between social filtering and neighborhood-based collaborative filtering. Second, latent factor methods are more frequently used than neighborhood methods.
Latent Factor Methods.
Of the latent factor methods, matrix factorization is a successful and popular approach (Koren09, ). The core idea behind matrix factorization for recommendation is that user-item interactions form a matrix (as of time ) , which can be factorized into two low-rank matrices: a representation of user preferences and an representation of item attributes . The number of latent features is usually chosen by the analyst or determined with a nonparametric model. The multiplication of these two low-rank matrices approximates the observed interaction matrix, parallel to eq:rec_score, or
There are many instantiations of the user preferences and item attributes. Non-negative matrix factorization (Lee00, ) requires that these representations be non-negative. Probabilistic matrix factorization (PMF, )
assumes that each cell in the user preference and item attribute matrices are normally distributed, whereas a probabilistic construction of non-negative matrix factorization(CannyGaP, )
assumes that they are gamma-distributed. Under all of these constructions, these latent representations are learned from the data by following a sequence of updates to infer the parameters that best match the observed data.
Other latent factor methods, such as principal component analysis (PCA)(jolliffe2002principal, ) and latent Dirichlet allocation (LDA) (Blei:2003, ), similarly frame user and item representations in terms of a low dimension of hidden factors. These factors are then learned algorithmically from the observed interactions. We focus on matrix factorization in our simulations (sec:simulations), for simplicity.888Specifically, we use Gaussian probabilistic matrix factorization with confidence weighting, as described by Wang and Blei (CTR, ), with and .
To update a latent factor recommendation system with recently collected data, one has several options. Since these methods are typically presented without regard to time , one option is to refit the model entirely from scratch, concatenating old data with the new; in these cases, consistent interpretation of latent factors may be challenging. A second option is to hold fixed some latent factors (e.g., all item attributes ) and update the remaining factors according to the update rules used to originally learn all the latent factors. This approach maintains the ordering of the latent factors, but may break convergence guarantees, even if it works well in practice. This approach does not explicitly address new items or users, often called the “cold-start problem,” but can be adapted to account for them.
a.5. Modifications and Hybrid Approaches
The concepts of these systems can be modified and combined to create innumerable permutations. In considering collaborative filtering alone, neighborhood-based methods can be merged with latent factor approaches (Koren:2008, ). Full collaborative filtering system can be supplemented with content information (CTR, ) or augmented with social filtering (Chaney:2015, ). Any of these methods can be supplemented with a popularity bias. Under any of these modified or hybrid systems, the changes propagate to the representations of user preferences and item attributes , and the general framework for recommendation remains the same.
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