Data sparsity is a well-known historical limitation of recommender systems that still impacts the performance of state-of-theart approaches. One practical technique to mitigate this issue involves transferring information from other domains or tasks to compensate for scarcity in the target domain, where the recommendations must be performed. Following this idea, we propose a novel approach based on Neuro-Symbolic computing designed for the knowledge transfer task in recommender systems. In particular, we use a Logic Tensor Network (LTN) to train a vanilla Matrix Factorization (MF) model for rating prediction. We show how the LTN can be used to regularize the MF model using axiomatic knowledge that permits injecting pre-trained information learned by Collaborative Filtering on a diferent task or domain. Extensive experiments comparing our model with a baseline MF on two versions of a novel real-world dataset prove our proposal's potential in the knowledge transfer task. In particular, our model consistently outperforms the MF, suggesting that the knowledge is efectively transferred to the target domain via logical reasoning. Moreover, an experiment that drastically decreases the density of user-item ratings shows that the benefits of the acquired knowledge increase with the sparsity of the dataset, showing the importance of exploiting knowledge from a denser source of information when training data is scarce in the target domain. knowledge transfer, matrix factorization, neuro-symbolic integration, logic tensor networks, rating prediction, explicit Recommender systems (RSs) have recently become essen- the context of recommender systems, knowledge transthe user's historical data, these tools mitigate informa- categories: feature-based models and fine-tuning ∗Corresponding author.
longs to the research field of transfer learning [ 14]. In
fer techniques [
ful recommendation approaches. Latent Factor models,
in particular Matrix Factorization (MF), have dominated
the CF scene [6, 7, 8] for years, and this has been further
emphasized with the deep learning rise [
performance, state-of-the-art models still sufer from a plicability in real-world scenarios. One way to address data sparsity consists of leveraging models pre-trained on other sources of information (i.e., source domains) to make the final model more accurate in the target do
Despite Neuro-Symbolic (NeSy) [19] approaches having been successfully applied in many AI fields [ 20, 21, 22], including RSs [23, 24, 25, 26], they have not yet been investigated in the task of knowledge transfer for recommender systems, where we believe their application is particularly suited. In particular, NeSy aims at integrating knowledge, usually expressed using logical axhistorical issue, i.e., data sparsity, that limits their ap- recommendation task, namely the target domain. The main, where the recommendations must be performed. transferred to the target domain via logical reasoning. to be particularly beneficial in tasks with poor training 31, 32, 33]. Among them, some directly use logical fordata [27], giving insights that this paradigm can help deal mulas to learn a recommendation model [23, 26, 31]. The with data sparsity in recommendation datasets. seminal approach that applied NeSy to RSs has been
Following this intuition, we propose using a Logic Ten- Neural Collaborative Reasoning (NCR) [26]. In NCR, the sor Network (LTN) [28] to encode axiomatic knowledge sequential recommendation task is formalized as a logical to enable efective knowledge transfer and injection for reasoning problem. In particular, the user’s ratings are a vanilla MF model1 trained on movie ratings. LTN is a represented using propositional variables, and logical opNeSy framework that efectively integrates logical rea- erators (e.g., ∧, ⟹ ) are used to construct propositional soning and neural networks. Our approach uses it as the formulas that express sequential patterns between them. interface between the model pre-trained on the source Then, NCR maps the variables to logical embeddings and domain and the final model trained on the target domain. the operators to neural networks (NNs) that act on those We called it an interface as it allows the explicit transfer embeddings. The NNs are forced to behave as classical of information between domains via logical reasoning. logic operators through logical regularization. By doing To perform our experiments, we use MindReader [30], so, the formulas can be organized as a neural network to a novel recommendation dataset containing explicit rat- conduct logical reasoning and prediction in a continuous ings from real users on movies and non-recommendable space. They compared NCR with many linear and deep entities, such as movie genres, actors, and producers. In baselines, showing it can reach state-of-the-art perforparticular, we use ratings on movie genres to learn our mance. However, this approach is not properly NeSy as pre-trained model via Collaborative Filtering and ratings neural networks implement the symbolic part (i.e., logion movies to train the final model to provide accurate cal connectives). LTN uses fuzzy logic semantics instead, recommendations in the target domain. The pipeline of making the framework theoretically sound. Moreover, the proposed approach consists of two steps. In the first NCR uses propositional logic, which makes it impossible step, we train a genre classifier using an MF model to to encode complex and expressive knowledge due to the learn which genres the users like and dislike in the source simplicity of the language syntax. domain. In the second step, we use LTN to transfer the In Graph Collaborative Reasoning (GCR) [31], NCR pre-trained knowledge to an MF model trained on the is extended to work with knowledge graphs. In partictarget domain for the movie rating prediction task. ular, they provide a simple approach for translating the
We compare our approach with a baseline MF model to graph structure into logical expressions to convert the understand if knowledge transfer is successfully2 reached link prediction task into a logical reasoning problem. As thanks to Neuro-Symbolic integration. The results show in NCR, they use logically constrained neural modules that our model consistently outperforms the MF, prov- to build the network architecture according to the logiing its ability to transfer knowledge across domains. In cal expression. They conducted experiments similar to addition, an experiment that drastically reduces the den- NCR, showing that GCR can improve NCR on knowledge sity of user-item ratings shows that the benefits of the graphs. knowledge increase with the sparsity of the dataset. This In Counterfactual Collaborative Reasoning (CCR) [32], gives insight that our model can successfully deal with NCR is used to perform data augmentation based on logdata sparsity thanks to Neuro-Symbolic reasoning. To ical reasoning for sequential recommendation. Specifithe best of our knowledge, this is the first time a NeSy cally, counterfactual logic reasoning is exploited to generapproach has been successfully applied to the knowledge ate counterfactual examples for data augmentation. The transfer task for recommender systems. examples are generated by discovering slight changes in users’ explicit feedback (i.e., the sequence of purchases) by solving a counterfactual optimization problem. They 2. Related works showed that these new examples, together with the original examples, can alleviate scarcity and enhance the In the last two years, the RS community has seen the performance of sequential recommendations. emergence of some Neuro-Symbolic approaches [25, 24, Another approach that successfully integrated logical reasoning and learning has been HYbrid Probabilistic 1Note we selected Matrix Factorization for our experiments because, Extensible Recommender (HyPER) [25], which is based tdhees-pairtet aitpspsriomapchliecsity[2,9it].isTshtiilsl oisnneootfttohebemionsttenpdoewderafsualsltiamteit-ooff- on Probabilistic Soft Logic [ 34]. In particular, HyPER our approach, as any other state-of-the-art model could be used in exploits the expressiveness of First-Order Logic (FOL) principle. to encode knowledge from a wide range of information 2Note the objective of the experiment is not to obtain state-of-the-art sources, such as multiple user and item similarity meaperformance. Instead, our goal is to show that a NeSy approach sures, content, and social information. Then, Hinge-Loss tchains benedu,stehde foonrlykndoifewrelendcgeebetrtawnesefnert hine rmeocodmelsminentdheercsoymstpeamriss.onTo Markov Random Fields are used to learn how to balance is the addition of knowledge via LTN. the diferent information types. HyPER is highly related considered more theoretically sound compared to previ- is defined as a set of − couples − = {(, ) () |(, , ) () ∈ to our work with LTN since the logical formulas we use resemble the ones used in HyPER.
formulas to enhance knowledge graph embeddings [35] and provide accurate knowledge-aware recommendations. Their approach consists of three steps: () the FOL formulas are automatically extracted from a recommendation knowledge graph, then () the knowledge graph embeddings are learned jointly with the extracted formulas using a NeSy approach. Finally, () the useritem embeddings are fed to a neural architecture to get predictions. Specifically, in the second step, they used KALE [36], a NeSy approach that allows learning knowledge graph embeddings jointly with FOL formulas. In particular, the learning can be unified as the graph triples can be interpreted as FOL atoms. As KALE uses fuzzy semantics to learn graph embeddings, this approach can be ous deep approaches, as the symbolic component remains symbolic during learning, as it happens with LTN and
related to ours. Specifically, they try to mitigate data sparsity by using LTN to inject content information into an MF model. In particular, they encode FOL formulas to use side information as a regularizer for the latent factors of the MF model. They show that the proposed NeSy approach can outperform the MF. However, the improvement is poor, and the model has some scalability issues due to the number of times the formulas have to be evaluated during training. Our approach difers from [23] on how axiomatic knowledge is used. In [23], the knowledge extends the MF model using content information, while our model uses it to enable the transfer of Bold notation diferentiates vectors, e.g., x = [3.2, 2.1], and scalars, e.g., = 5 . Matrices and tensors are denoted with upper case bold notation, e.g., X. X is used to denote the -th row of X, while X, to denote the item at row and column . We refer to the set of users of an RS with , where | | =
. Similarly, the set of items is denoted as ℐ, such that |ℐ | = . We use to denote a dataset. is defined as a set of triples = {(, , ) R ∈ ℕ× , such that R, = if (, , ) ∈ ∈
, ∈ ℐ , and ∈ {0, 1} is a binary explicit rating. can be reorganized in the so-called user-item matrix () }
=1 , where , 0 otherwise.
Then, since we work with binary feedback, we refer to + (resp. −) as the dataset of positive (resp. negative) user-item pairs. + is defined as a set of + couples + = {(, ) () |(, , ) () ∈ ,
() = 1}=1 . Similarly, − ,
() = 0}=1 . Finally, ? denotes the dataset of useritem pairs for which the rating is unknown, i.e., ? = {(, ) ∈ × ℐ |(, , ) ∉ } . Clearly, ? = ⋅ − .
Matrix Factorization (MF) is a Latent Factor Model that aims at factorizing the user-item matrix R into the product of two lower-dimensional rectangular matrices, U ∈ ℝ× and I ∈ ℝ× , containing the users’ and items’ latent factors, respectively. represents the number of latent factors. More formally, the objective of MF is to find U and I such that R ≈ U ⋅ I⊤. An efective way to learn the latent factors is by using gradient-descent optimization.
Given the dataset , an MF model seeks to minimize the Mean Squared Error (MSE) between predicted and 1 ∑(,,)∈
|| −̃ || 2 + || ||2. In U ⋅ I⊤ + u + i , where u and i pre-trained knowledge learned via Collaborative Filter- target ratings, defined as ing on another task (i.e., movie genre rating prediction). the formulation, ̃ = Moreover, our model provides a logical formalization for are bias terms for user and item , respectively, and the rating prediction task3, while [23] proposes a ranking- = {U, I, u, i}. is a hyper-parameter to set the strength based method. Finally, note all the presented models use some form of knowledge (e.g., knowledge graphs, logic) to improve the recommendation performance. However, none of them have designed experiments to understand if the advantages of a NeSy system can help mitigate one or some of the historical issues of recommender systems (e.g., data sparsity, cold-start, explainability). Hence, it is unclear if the properties (e.g., few-shot learning, interpretability) of a NeSy approach are totally exploited. of the 2
regularization.
In our setting, we use a diferent implementation of MF since we treat the recommendation problem as a binary classification task. Specifically, we need to recommend if a user likes (1) or dislikes (0) an item. Hence, the focal loss is used in place of MSE for the training, and the logistic function is applied to the prediction of MF to restrict the output between 0 and 1. Focal loss is defined as
This section provides useful notation and terminology used in the remainder of the paper. 3Note our approach can be easily extended to the top-n recommendation task by changing the formalization of the knowledge base. 1 (,,)∈ ∑
− (1 − ) log + || ||2 = { = { 1 − main of real numbers. We refer to the term grounding4, tor, generally defined using allows mapping every symbolic expression into the do- SatAgg ∶ [0, 1]∗ ↦ [0, 1] is a formula aggregating operaME . items of the dataset), with ∈ ℕ+, > 0. As a conse- ternal source of information (e.g., content information, quence, a term ()
or a formula P() , will be mapped to a sequence of values too. Afterward, connectives additional ratings), a pre-trained model learned to generate features5 for users and items in the (, | 1) is are grounded using fuzzy semantics (i.e., operators deal- source domain. In the notation, is a user index, is an for gradient-based optimization [37]. In the notation, source domain via Matrix Factorization; hence, the outclassify, and = ( logistic function.
U ⋅ I⊤ + u + i ), where is the
that is fully diferentiable and has concrete semantics that formally denoted by , as the function that defines this mapping. Real Logic allows LTN to ground logical formulas into computational graphs, enabling gradient-based optimization.
In particular, maps individuals (e.g., users) to tensors of real features (e.g., users’ demographic information), functions (e.g., Score( , ) ) as real functions (e.g., inner product), and predicates (e.g., Likes( , ) real functions with output in [0, 1]. Then, a variable is mapped to a sequence of individuals (e.g., some ) as
Given a Real Logic knowledge base = { 1, … , }, where 1, … , are closed formulas, LTN allows learning the grounding of constants, functions, and predicates appearing in them. In particular, if constants are grounded as embeddings and functions/predicates onto neural networks, their grounding depends on some learnable parameters . We denote a parametric grounding as (⋅| ). by finding parameters of , namely ∗ = argmax SatAgg∈ ∗ that maximize the satisfaction (| ), where
Our model is inspired by feature-based knowledge transfer models [13]. In these approaches, a pre-trained model is learned on some additional source of information, usually side information. Then, the acquired knowledge is transferred to the target domain during the training of the final model to help deal with the data sparsity of user-item interactions. More formally, given the exwhere is a hyper-parameter to give diferent weights to the two classes, is a hyper-parameter that represents the penalty assigned to the examples that are hard to individual of each variable. quantifying over specific tuples of individuals of variables 1, … , , such that the -th tuple contains the -th item index, and 1 are the parameters of the pre-trained model. Instead of generating features, our approach learns
to predict user-genre preferences in the put of this model is a score for the given user-genre pair in the source domain. Then, we assume a model that learns to predict user-movie preferences in the target domain. In the notation, is a user index, is an item index, and 2 are the parameters of the final (, | 2) is recommendation model. Specifically, score for the given user-movie pair in the target domain. outputs a In our approach, knowledge from is to maximize function Neuro-Symbolic computing. The objective of our model is learned while transferring6 via logical reasoning thanks to ℱ ( 1, 2) = ( (, | 1), (, | 2), (, )) where is a logic-based aggregation function, is a user Note
is a function that relates movies and genres
to obtain features for users and items from the content or side information. In our scenario, it learns to predict user-item ratings ing with fuzzy values), while quantifiers are grounded as special aggregation functions (e.g., generalized means).
, , 1, … , ∈ [0, 1] and ≥ 1 .
(∧) =
T
( , ) = ∗ ( ⟹ ) = I ( , ) = 1 − + ∗ (¬) =
N ( ) = 1 − (∀) =
ME ( 1, … , ) = 1 − ( (∃) =
M ( 1, … , ) = ( 1 =1 1
∑ =1 ∑(1 − ) )
1 ) 1 ) The intuition behind the choice of hyper-parameter is that the higher that is, the more weight that M
converging to the max (resp. min) operator. Real Logic also provides a special type of quantification, called
diagonal quantification , denoted as Diag( 1, … , ). It allows (resp. ME ) will give to true (resp. false) truth-values, index, is a movie genre index, and is a movie index. 4Notice that this is diferent from the common use of the term
grounding in logic, which indicates the process of replacing the variables of a term or formula with constants or terms containing no variables. via Collaborative Filtering. 6Note that parameters 1 (of parameters 2 (of ).
) are frozen during the training of using content information. It can be seen as a lookup 4.2. Grounding of symbols table denoting which movies belong to a specific genre in the dataset. In our approach, this function is used by The grounding defines how logical symbols are mapped iItnno crpeoagmrutbilcaiunrilazateri,ottnhhewetiartaghigntrhieneggapotriefodnicftuionncvstiioaonfl o gciacanl braeenadsseoenninags. okLnnTtoNowmtlheodedgreeel.balaIsfienelddtheaifinnsedwthhoeernkcc,oe m(hpouwtat∗tih)oen=aalx⟨gi or(am)p|sh(, io)nf (tt)hhee∈ the logical knowledge base that LTN seeks to maximally ∗⟩=∗1 and ( ∗) = ⟨ () |(, ) () ∈ ∗⟩=∗1 , namely satisfy during training. Specifically, is a composition of ∗ and ∗ are grounded as a sequence of the ∗ logical formulas that defines how the source and target user and movie indexes in ∗, with ∗ ∈ {+, −, ?}. Indomain interact during training. In other words, makes stead, ( ) = ⟨1, … , ⟩, namely is grounded it possible to transfer knowledge between domains via as a sequence of genre indexes, where is the logical reasoning and is carefully formalized in Section 4. number of movie genres in the dataset. Afterward, ( Likes |U, I, u, i) ∶ , ↦ ( U ⋅ I⊤ + u + i ), namely Likes is grounded onto a function that takes as input a 4. Method user index and a movie index and returns the prediction in [0, 1] of the MF7 model for the given useritem pair. U ∈ ℝ× , I ∈ ℝ× , u ∈ ℝ , and i ∈ ℝ are the matrices of the users’ and items’ latent factors, and the vectors of users’ and items’ biases, respectively. Notice Likes is the predicate that implements . Then, Our approach uses an LTN to enable domain adaptation for efective knowledge transfer. Specifically, the LTN is trained using a Real Logic knowledge base containing facts designed to intuitively transfer information about movie genre preferences (i.e., the source domain) to a Ma- ( LikesGenre) ∶ , ↦ G, , where G ∈ {0, 1}× , trix Factorization model trained on movie ratings (i.e., the namely LikesGenre is grounded onto a function that target domain). In the next subsections, we will present takes as input a user index and a genre index and our knowledge base, how is used to convert it into returns the prediction contained in matrix G for user a computational graph suitable for gradient-based opti- and genre . In particular, G can be seen as a lookup table mization, and how the learning of the LTN takes place. containing the binarized8 predictions of a pre-trained genre classifier. LTN has shown to work better with 4.1. Real Logic knowledge base binarized outputs as the classifier was returning preThe objective of our LTN model is the satisfaction of the dictions too near the decision boundary for LTN to unfollowing Real Logic knowledge base. derstand9 the diference between like and dislike. Note LikesGenre is the predicate that implements . Finally, ( ) ∶ , ↦ {0, 1} , namely HasGenre is ∀ Diag( +, +) Likes( +, +) (2) grounded onto a function that takes as input a movie index and a genre index and returns one if the movie ∀ Diag( −, −)¬ Likes( −, −) (3) belongs to genre , zero otherwise. Note HasGenre ∀ Diag( ?, ?)(∃ ¬ LikesGenre( ?, ) is the predicate that implements . Intuitively, ∧ HasGenre( ?, )) ⟹ ¬ Likes( ?, ?()4) t(heLsikoeusrGceendroem)caoinn.taIinnscothnetrkasnto, w( leLdigkeesp|rUe,-tIr,aui,nie)dreopnresents the MF model we need to train on the target
Specifically, + and + are variable symbols de- domain. noting positive user-item pairs, − and − are vari- Intuitively, Axiom (2) forces Likes to be true for each able symbols denoting negative user-item pairs, ? and positive user-item pair in +, while Axiom (3) forces ? are variable symbols denoting user-item pairs for Likes to be false for each negative user-item pair in −. which the rating is unknown, and is a variable sym- In other words, by maximizing the satisfaction of Axbol denoting the genres of the movies. Then, Likes(, ) iom (2) and Axiom (3), the model learns to factorize the is a predicate symbol denoting whether a user likes a user-item matrix using the ground truth. In contrast, movie , LikesGenre(, ) is a predicate symbol denoting Axiom (4) is designed to transfer knowledge from the whether a user likes a movie genre , and HasGenre(, ) is a predicate symbol denoting whether a movie belongs 7iNteomticpeatihr.atInLitkheiss cwaonrbke, waneyufsuenacntioMnFremtuordneiln.gTahsiscohraesfonrotatuosebreto genre . intended as a limit of our approach as any other state-of-the-art
Intuitively, Axiom (2), Axiom (3), and Axiom (4) are ap- model could be used in principle. plied to user-item pairs in +, −, and ?, respectively. 8A binarized prediction is obtained by using the decision boundary Diag is used to quantify over the desired user-item pairs 9Foofrthaebcilnaassriyfiercloansstihfieer, o0u.4t5puatnodf 0th.5e5maroedpelretodigcetitovnaslubeesloinngin{0g, 1to}. rather than quantifying over all possible combinations diferent classes. For a logical framework, those values represent of user and item indexes in the dataset. similar truth values. source domain to the target domain through logical rea- 5.1. Datasets soning. Specifically, it forces Likes to be false whenever a user does not10 like at least one genre of a movie . Note this axiom is applied only to unknown user-item pairs in ?. In fact, when no movie ratings are available on the target domain, knowing something about movie genre preferences is better than knowing nothing. In other words, we believe transferring knowledge from the source domain is crucial when data is missing in the target domain.
To perform our experiments, we selected MindReader13 (MR) [30], a novel dataset containing ratings from real users for movies and non-recommendable entities, such as movie genres, actors, and producers. We performed experiments on both MR-100k and MR-200k, the two available versions of the dataset. We used the ratings on movie genres as the source domain14 and the movie ratings as the target domain. To guarantee the users in the source and target domains totally overlapped, we removed the users that only rated movie genres or movies. After this 4.3. Learning of the LTN pre-processing, MR-100k (resp. MR-200k) comprised 962 The objective of our model is to learn ( Likes |U, I, u, i) (resp. 2,182) users, 3,034 (resp. 3,806) movies, and 140 by maximizing the satisfaction of the knowledge base. In (resp. 159) movie genres. The density of user-movie other words, LTN seeks to minimize the following loss ratings was 0.62% (resp. 0.58%), while for user-genre ratfunction: ings was 8.09% (resp. 6.37%). Selecting ratings on movie genres as the source domain allowed us to use particularly dense information15 for knowledge transfer. When L( ) = (1 − SatAgg∈ ( +, +)←ℬ+(| )) + || ||2 ( ) ≫ ( ) , knowledge transfer is ( −, −)←ℬ− more likely to be efective [ 39].
( ?, ?)←ℬ? (5) MR provides three types of ratings: likes (1), unknown where ℬ∗ denotes a batch of training examples randomly (0), and dislikes (-1). As in [30], we removed the unknown sdaemnoptleesdthfraot mva ria∗b.leTs he n∗oatnatdi o n ( ∗ a∗r,e groun∗)de←d wℬith∗ irnagtisngfrso.mAfte-1r tthoa0t,. wAes cthheandgaetadstehteplraobveildfeosrbnineagraytiveexprlaitc-it actual user-movie pairs coming from the corresponding feedback, we treated the recommendation problem as batch ℬ∗, where ∗ ∈ {+, −, ?}. Notice the loss does not a binary classification task 16, where one has to predict specify how the variable is grounded. At each whether a user likes or dislikes an item. This choice altraining step, we ground it with the sequence of all the lowed us to use the focal loss (Equation (1)) to train the movie genre indexes in the dataset. Note ℬ? is created MF models and the F-measure as an evaluation metric. by uniformly sampling user-item pairs from ? at each This helped in dealing with class imbalance. The class imtraining step. While all the user-item pairs in + and − balance ratio in MR-100k (resp. MR-200k) is 21%(-)/79%(+) are iterated at each epoch, going through all the possible (resp. 20%(-)/80%(+)) for movie genres, and 38%(-)/62%(+) unknown pairs is unnecessary and would be unfeasible. (resp. 36%(-)/64%(+)) for movies. In both cases, the negaIn this sense, ℬ? has not to be considered a mini-batch tive class is the minority one. Hence, we used it as the in the usual sense. positive one to compute evaluation metrics in Table 2.
As the splitting strategy for the target domain, we randomly sampled 20% of the movie ratings from each user 5. Experiments to construct the test set. Then, we randomly sampled 10% of the remaining movie ratings from each user to construct the validation set. Instead, for the source domain, we only created the validation set by randomly sampling 20% of the movie genre ratings from each user. The test
our method. They have been executed on an Apple MacBook Pro (2019) with a 2,6 GHz 6-Core Intel Core i7. The models have been implemented in Python using PyTorch.
In particular, we used the LTNtorch11 library [38]. Moreover, we used Weights and Biases (WandB) for hyperparameter optimization. Our source code is freely available12. 13https://mindreader.tech/dataset/ 14Notice that our approach is flexible on the type of knowledge that has to be transferred. In this work, we use movie genre ratings, but every type of rating (e.g., ratings on books, actors) or side information can be used in principle. One has just to change the 10Note the negated formula is used on purpose, as it is likely that knowledge base formalization accordingly. a user dislikes the majority of movies belonging to a genre she 15Notice ratings on movie genres are dense as they are easily obtaindislikes. For example, if does not like horror, likely, she will not able. It is more likely a user will provide a rating about some genre like all the horror movies. Moreover, it is more critical to avoid over hundreds rather than some movie (or actor) over thousands. recommending something users do not like than not recommend- 16MindReader provides binary explicit ratings rather than usual ing something they like. This will restrict the recommendation to 1-5 star ratings. For this reason, the recommendation task can a few positive movies. be interpreted as a binary classification problem rather than a 11https://github.com/logictensornetworks/LTNtorch regression one. Finally, we work in a rating prediction task rather 12https://github.com/tommasocarraro/NESYKnowledgeTransfer than ranking as the feedback is clearly explicit. set is not needed in the source domain, as we only need to computational time, the searches have been conducted a validation set to find the optimal hyper-parameters for only for the first seed of the experiment and just for the the pre-trained model.
complete dataset (i.e., 100% ratings). The best hyperparameters found for the models have been then used in the rest of the experiment. For all the models, we tried a number of latent factors ∈ {5, 10, 25, 50} , regularization coeficient ∈ {0.01, 0.001, 0.0001, 0.00005} , learnOur experiment compares the proposed Neuro-Symbolic approach, denoted as NESYMF, with a baseline MF model, ing rate ∈ {0.01, 0.001, 0.0001} , training batch size ∈ denoted as MF, to check if NESYMF can efectively trans{64, 128, 256}. For the MF models, we additionally tried fer knowledge from source to target domain and im- focal loss hyper-parameters ∈ {0.05, 0.1, 0.2, 0.3, 0.4, 0.5} prove the performance when training data becomes scarce. Specifically, the experiment consists of the following pipeline: (1) additional training sets are generated by randomly sampling the 50%, 20%, 10%, and 5% and ∈ {0, 1, 2, 3} . For the MF model trained on the source domain, we also tried diferent thresholds for the decision boundary ∈ {0.3, 0.4, 0.5, 0.6, 0.7} . Due to the huge class imbalance in the source domain, we preferred of the movie ratings17 from the entire training set, re- finding a threshold to maximize ferred to as 100%. Notice ratings are sampled indepen- recall19. Finally, for NESYMF, we additionally tried difprecision rather than dently from the user, diferently from the splitting strategy explained previously. Then, (2) for each training set ∈ {100%, 50%, 20%, 10%, 5%} and for each model ferent values for hyper-parameter ∈ {2, 4, 6, 8, 10} of ME and M . Table 1 presents the best hyper-parameters
found for the MF model trained on the source domain ∈ { MF, NESYMF}: (2) is trained on
using hyper- (pre-trained model) and the models trained on the target parameters found through a bayesian search. Finally, (2) is evaluated on the test set. Note that for NESYMF, step (2) consists of two steps: () a standard MF model is pretrained on the source domain to populate matrix G, then () NESYMF is trained on the target domain, namely .
the Adam optimizer. Early stopping has been used to stop the training if no improvements were found on the validation set for ten epochs. For all the models, the user and item latent factors, U and I, and the user and item biases, u and i, have been randomly initialized using the
using Equation (1), while NESYMF using Equation (5). For NESYMF, Axiom (4) has been added to the loss from epoch five 18, allowing LTN to learn something about the latent factors before starting reasoning on the acquired knowledge.
hyper-parameters for our models. We executed every hyper-parameter search for 150 runs and selected the configuration that led to the best validation score. Due domain (i.e., MF and NESYMF).
For the MF model trained on the source domain, we used F0.5-measure20 as the validation metric, while in all the other cases, we used F1-measure. Clearly, the performance of NESYMF depends on the quality of G (i.e., the pre-trained model). Using F0.5-measure allowed us to obtain more precise predictions for G. In particular, we reduced the number of false positives, namely cases in which G erroneously predicts that a user dislikes a genre.
rized in Table 2. The discussion is limited to MR-200k as for MR-100k we obtained similar results. Note the results for MR-100k are better since the dataset is slightly denser.
NESYMF outperforms MF on all five folds. In particular, the performance gap increases with the sparsity of the user-item ratings, starting from a 1.04% improvement on the full dataset (i.e., 100% fold) and ending with a 6.69% improvement on the most sparse dataset (i.e., 5% fold).
This shows the benefits of transferring knowledge from a denser domain when training data is poor. Moreover, it suggests our proposal can be efectively used in the task 17Notice the ratings on movie genres are kept untouched as they are of knowledge transfer for recommendation.
used for accurate pre-training. 18Notice this is an arbitrary choice. The idea of using knowledge transfer is to correct the misclassifications made by the MF model when training data are sparse. The more the data sparsity, the more likely the MF will erroneously classify user-item pairs. During the ifrst steps of learning, the MF has not learned enough information to predict accurately. For this reason, it is likely knowledge transfer is applied to random predictions, hence useless.
Interestingly, by looking at recall, it is possible to observe that the addition of knowledge helps NESYMF in 19By maximizing precision, we obtained a more accurate pre-trained model for predicting user-genre preferences. This helped in transferring knowledge more efectively. 20F0.5-measure gives more weight to precision than recall. It is used when avoiding false positives is particularly important. Best hyper-parameters (referred to as h-p) found through Bayesian optimization for the pre-trained model (left) and the models learned on the target domain (right). The hyper-parameters are subdivided by dataset. Comparison of MF and NESYMF on the selected datasets. The test metrics are averaged across 30 runs. confidence) that a user really dislikes an item, only know
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