We propose a model which extends variational autoencoders by exploiting the rich information present in the past preference history. We introduce a recurrent version of the VAE, where instead of passing a subset of the whole history regardless of temporal dependencies, we rather pass the consumption sequence subset through a recurrent neural network. At each time-step of the RNN, the sequence is fed through a series of fully-connected layers, the output of which models the probability distribution of the most likely future preferences. We show that handling temporal information is crucial for improving the accuracy of recommendation.
Learning of accurate models able to suggest interesting items to the user is an important problem for many scienti c and industrial applications. To address this problem,
collaborative ltering approaches to recommendation were extensively
investigated by the current literature. Among these, latent variable models [
The aforementioned approaches rely on the \bag-of-word" assumption: when
considering a user and her preferences, the order of such preferences can be
neglected and all preferences are exchangeable. This assumption works with general
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user trends which re ect a long-term behavior. However, it fails to capture the
short-term preferences that are speci c of several application scenarios,
especially in the context of the Web. Sequential data can express causalities and
dependencies that require ad-hoc modeling and algorithms. And in fact, e orts
to capture this notion of causality have been made, both in the context of latent
variable modeling [
In this paper, we consider the task of sequence recommendation from the
perspective of combining deep learning and latent variable modeling. Inspired by
the approach in [
Our contribution can be summarized as follows: (i ) we extend the framework to the case of sequential recommendation, where user's preferences exhibit temporal dependencies; (ii ) we evaluate the proposed framework on standard benchmark datasets, by showing that (a) approaches not considering temporal dynamics are not totally adequate to model user preferences, and (b) the combination of latent variable and temporal dependency modeling produces a substantial gain, even with regard to other approaches that only focus on temporal dependencies through recurrent relationships.
The paper is organized as follows. Sections 2 proposes the modeling of user preferences in a variational setting, by describing how the framework can be adapted to the case of temporally ordered dependencies. The e ectiveness of the proposed modeling is illustrated in section 3, and pointers to future developments are discussed in section 4. 2
The reader is referred to [
We also consider a precedence and temporal relationships within X . First of all, the preference matrix induces a natural ordering relationship between items: i u j has the meaning that xu;i > xu;j in the rating matrix. Also, we assume the existence of timing information T 2 IR+M N [f;g, where the term tu;i represents the time when i was chosen by u (with tu;i = ; if xu;i = 0). Then, i <u j denotes that tu;i < tu;j , With an abuse of notation, we also introduce a temporal mark in the elements of x u: the term x u(t) (with 1 t Nu) represents the t-th item in Iu in the sorting induced by <u, whereas x u(1:t) represents the sequence x u(1); : : : ; x u(t).
The reference model is the Multinomial variational autoencoder (MVAE)
proposed in [
N (0 ; I K ) ; (z u)
exp ff (z u)g ; x u
Multi (Nu; (z u)) : (2.1) The underlying \decoder" is modeled by log P (x ujz u) = P xu;i log i(z u) ; i thus enabling a complete speci cation of the overall variational framework. Prediction for new items is accomplished by resorting to the learned functions f and Q : given a user history x , we compute z = (x ) and then devise the probabilities for the whole item set through (z ). Unseen items can then be ranked according to their associated probabilities.
Ideally, latent variable modeling should be able to express temporal dynamics and hence causalities and dependencies among preferences in a user's history. In the following we elaborate on this intuition. Within a probabilistic framework, we can model temporal dependencies by conditioning each event to the previous events: given a sequence x (1:T ), we have
T 1 Y t=0 P x (1:T ) =
P x (t+1)jx (1:t) : This speci cation suggests two key aspects: { There is a recurrent relationship between x (t+1) and x (1:t) devised by P x (t+1)jx (1:t) , upon which the modeling can take advantage, and { each time-step can be handled separately, and in particular it can be modeled through a conditional VAE.
Let us consider the following (simple) generative model z u(t)
N (0 ; I K ) ; z u(t) exp f z u(t)
; x u(t)
z u(t) ; which results in the joint likelihood Here, we can approximate the posterior P (z u(1:T )jx u(1:T )) with the factorized proposal distribution
Q (z u(1:T )jx (1:T )) =
Y q (z u(t)jx (1:t 1)) ; t where q (z u(t)jx (1:t 1)) is a gaussian distribution whose parameters (t) and (t) depend upon the current history x u(1:t 1), by means of a recurrent layer h t: (t); (t) = ' (h t) h t = RNN h t 1; x u(t 1) : (2.2) (2.3) The resulting loss function can be formalized as follow:
Nu ( L( ; ; X ) = X X
The proposal distribution introduces a dependency of the latent variable from a recurrent layer, which allows to recover the information from the previous history. We call this model SVAE.
Notably, the prediction step can be easily accomplished in a similar way as for MVAE: given a user history x u(1:t 1), we can resort to eq. 2.3 and set z = (t), upon which we can devise the probability for the x u(t) by means of (z ).
The generative model of eq. 2.2 only focuses on the next item in the sequence. The base model however is exible enough to extend its focus on the next k items, regardless of the time: x u(t:t+k 1) Multi k; z u(t) . Again, the resulting joint likelihood can be modeled in di erent ways. The simplest way consists in considering x as a time-ordered multi-set, Alternatively, we can consider the probability of an item as a mixture relative to all the time-steps where it is considered: where P (x u(t)jz u(j)) is the probability of observing x u(t) according to (z u(j)). In both cases, the variational approximation is modeled exactly as shown above, and the only di erence lies in the second component of the loss function, which has to be adapted according to the above equations.
There is an interesting analogy between eq. 2.5 and the attention
mechanism [
We evaluate SVAE on some benchmark datasets, by comparing with various baselines and the current state-of-the-art competitors, in order to assess its capabilities in modeling preference data. Additionally, we provide a sensitivity analysis relative to the con gurations/contour conditions upon which SVAE is better suited.
We evaluate our model along with the competitors on two popular publicly available datasets, namely Movielens-1M and Net ix. Movielens-1M is a time series dataset containing user-item ratings pairs along with the corresponding timestamp. Since we work on implicit feedback, we binarize the data, by considering only the user-item pairs where the rating provided by the user was strictly greater than 3 on a range 1:5. Net ix has the same data format as Movielens-1M and the same technique is used to binarize the ratings. We use a subset of the full dataset that matches the user-distribution with the full dataset. The subset is built by stratifying users according to their history length, and then sampling a subset of size inversely proportional to the size of the strata.
Since we are considering implicit preferences, the evaluation is done on top-n recommendation, and it relies on the following metrics.
Normalized Discounted Cumulative Gain. Also abbreviated as NDCG@n, the metric gives more weight to the relevance of items on top of the recommender list and is de ned as Precision. By de ning Hits = Pi ri as the number of items occurring in the recommendation list that were actually preferred by the user, we have
Precision@n = n Recall. de ned as the percentage of items actually preferred by the user that were present in the recommendation list: jRj In our experiments, we use the above metrics with two values of n, respectively 10 and 100. The evaluation protocol works as follows. We partition users into training, validation and test set. The model is trained using the full histories of the users in the training set. During evaluation, for each user in the validation/test set we split the time-sorted user history into two parts, fold-in and fold-out splits. The fold-in split is used as a basis to learn the necessary representations and provide a recommendation list which is then evaluated with the fold-out split of the user history using the metrics de ned above. We believe that this strategy is more robust when compared to other methodologies wherein the same user can be in both the training as well as testing sets.
It is worth noticing that Liang et. al [
We compare our model with current state-of-the-art models including
recently published neural architectures and we now present a brief summary about
our competitors to provide a better understanding of these models.
{ POP is a simple baseline where the most popular items are recommended.
{ BPR is a state of the art model based on Matrix Factorization, which ranks
items di erently for each user [
The evaluation takes place on their corresponding fold-out sets.
{ FPMC [
The experiments consider the basic SVAE model and the extensions to the next-k prediction, as discussed in the previous section. The model is trained end-to-end on the full histories of the training users. Model hyperparameters are set using the evaluation metrics obtained on validation users. The SVAE architecture includes an embedding layer of size 256, a recurrent layer realized as a GRU with 200 cells, and two encoding layers (of size 150 and 64) and nally two decoding layers (again, of size 64 and 150). We set the number K of latent factors for the variational autoencoder to be 64. Adam was used to optimize the loss function coupled with a weight decay of 0:01. As for RVAE, the architecture includes user/item embedding layers (of size 128), two encoding layers (size 100 and 64), and a nal layer that produces the score f (z u;i). Both SVAE and RVAE were implemented in PyTorch and trained on a single GTX 1080Ti GPU. The source code is available on GitHub.
In a rst set of experiments, we compare SVAE with all competitors described above. Table 1 shows the results of the comparison. SVAE outperforms the competitors on both datasets with a signi cant gain on all the metrics. It is important here to highlight how the temporal fold-in/fold-out split is crucial for a fair evaluation of the predictive capabilities: MVAE was evaluated both on temporal and random split, exhibiting totally di erent performances. Our interpretation is that, with random splits, the prediction for an item is easier if the encoding phase is aware of forthcoming items in the same user history. This severely a ects the performance and overrates the predictive capabilities https://github.com/noveens/svae_cf. of a model: the accuracy of MVAE drops substantially when a temporal split is considered.
By contrast, SVAE is trained to capture the actual temporal dependencies and ultimately results in better predictive accuracy. This is also shown in g. 1, where we show that SVAE outperforms the competitors irrespective of the size of fold-in. The only exception is with very short sequences (less than 10 items), where MVAE gets better results with respect to the sequential models. It is also worth noticing how the performance of both sequential models tend to degrade with increasing sequences, but SVAE maintains its advantage over CASER.
We discussed in the previous section how the basic SVAE framework can be extended to focus on predicting the next k items, rather then just the next item. We analyse this capability in g. 2, where the accuracy for di erent values of k is considered according to the modeling in 2.4. On Movielens, the best value is achieved for k = 4, and acceptable values range within the interval 1 10.
Model
Combining the representation power of latent spaces, provided by variational
autoencoders, with the sequence modeling capabilities of recurrent neural
networks is an e ective strategy to sequence recommendation. To prove this, we
[
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devised SVAE, a simple yet robust mathematical framework capable of modeling temporal dynamics upon di erent perspectives, within the fundamentals of variational autoencoders. The experimental evaluation highlights the capability of SVAE to consistently outperform state-of-the-art models.
The framework proposed here is worth further extensions that we plan to accomplish in a future work. In particular, from an architectural point of view, the attention mechanism, outlined in section 2.5, requires a better understanding and a more detailed analysis of its possible impact in view of the recent developments in the literature.
However, di erent architectures (e.g. based on convolution or translation invariance are worth being investigated within a probabilistic variational setting.