Title: Evidential Stochastic Differential Equations for Time-Aware Sequential Recommendation
Venue: NeurIPS 2024
Accepted: yes
Keywords: Sequential recommendation

time-evolving behavior
Abstract: Sequential recommender systems are designed to capture users' evolving interests over time. Existing methods typically assume a uniform time interval among consecutive user interactions and may not capture users' continuously evolving behavior in the short and long term. In reality, the actual time intervals of user interactions vary dramatically. Consequently, as the time interval between interactions increases, so does the uncertainty in user behavior. Intuitively, it is beneficial to establish a correlation between the interaction time interval and the model uncertainty to provide effective recommendations. To this end, we formulate a novel Evidential Neural Stochastic Differential Equation (*E-NSDE*) to seamlessly integrate NSDE and evidential learning for effective time-aware sequential recommendations. The NSDE enables the model to learn users' fine-grained time-evolving behavior by capturing continuous user representation while evidential learning quantifies both aleatoric and epistemic uncertainties considering interaction time interval to provide model confidence during prediction. Furthermore, we derive a mathematical relationship between the interaction time interval and model uncertainty to guide the learning process. Experiments on real-world data demonstrate the effectiveness of the proposed method compared to the SOTA methods.

Evidential Stochastic Differential Equations for Time-Aware Sequential Recommendation

Abstract

Sequential recommender systems are designed to capture users' evolving interests over time. Existing methods typically assume a uniform time interval among consecutive user interactions and may not capture users' continuously evolving behavior in the short and long term. In reality, the actual time intervals of user interactions vary dramatically. Consequently, as the time interval between interactions increases, so does the uncertainty in user behavior. Intuitively, it is beneficial to establish a correlation between the interaction time interval and the model uncertainty to provide effective recommendations. To this end, we formulate a novel Evidential Neural Stochastic Differential Equation ( E-NSDE ) to seamlessly integrate NSDE and evidential learning for effective time-aware sequential recommendations. The NSDE enables the model to learn users' fine-grained time-evolving behavior by capturing continuous user representation while evidential learning quantifies both aleatoric and epistemic uncertainties considering interaction time interval to provide model confidence during prediction. Furthermore, we derive a mathematical relationship between the interaction time interval and model uncertainty to guide the learning process. Experiments on real-world data demonstrate the effectiveness of the proposed method compared to the SOTA methods.

1 Introduction

Recommender systems have been used in various domains, such as e-commerce, entertainment, education, health care, social media, and many more [32, 11, 2, 44]. In these domains, users' interests and behaviors dynamically evolve over time. Therefore, capturing users' evolving behavior plays an essential role in effective recommendations. Various sequential recommendation models [35, 38, 40, 43] have been proposed accordingly. They mainly leverage the user's historical sequential interactions and aim to predict the next item that a user likes to interact with. These methods usually assume a uniform time interval of user interactions. However, in reality, the actual time interval between two consecutive interactions may vary dramatically, and a large time interval may be accompanied by a change in users' preferences.

The unrealistic assumption of a uniform interaction interval could significantly impact the model's capability to capture users' continuously evolving behavior and subsequently hurt the recommendation performance. Table 1 provides an illustrative example of the issues as outlined above. In this example, we follow the standard sequential recommendation similar to [19, 35] by predicting the next item considering

other 100 negative items from the item pool. We select a random user from the Movielens-100K dataset and provide recommended ranking results together with the interaction time intervals in the sequence. The recommendation performance for a sequential model BERT4Rec [35] deteriorates significantly when the gap between the consecutive interactions becomes large. For example, when the next interaction occurs soon after the first one, BERT4Rec usually ranks the ground-truth item in a top-10 list. However, for a much larger interval, the ground-truth item drops out of the top-20 or even the top-50 recommendation list.

Neural Ordinary Differential Equations (NODE) have been recently introduced that map the existing discrete neural networks to a continuous model [9, 36, 16], which can naturally capture users' continuously evolving preferences. However, one key limitation of these NODE models is the lack of uncertainty quantification capability, which is essential to understand user behavior when recommending the next item, especially when learning from extremely sparse user interactions given a large item space. Intuitively, the uncertainty of user behavior should increase along with the length of the interval since the last active interaction. For example, if the user has no activity for a long time, the user's prior interest tends to decrease. As a result, we become more uncertain about the user's preference. At the same time, a high uncertainty also presents a unique opportunity to effectively explore the item space by providing a diverse recommendation list to the users so that they may be attracted to a new category of items to keep them in the system.

Uncertainty can serve as a useful guidance to the recommender system to adapt to users' changing preferences and explore new items effectively. Table 2 further highlights the limitation of an existing ODE-based sequential recommendation method GRU-ODE [16], where recommended genres come from frequently watched 'Drama' and 'Romance' movies in the past. This issue becomes even worse when the interaction gap becomes large ( i.e., the fourth row of the GRU-ODE model), and the recommended genres significantly deviate from the ground truth. The recommendation list again concentrates on the same 'Drama' and 'Romance' genres, which are misaligned with the user's long-term interests. It is evident that conducting a more effective exploration is crucial to uncover the long-term and varied interests of users, aiming to optimize potential future benefits.

To address this critical gap as outlined above, we propose uncertainty augmented Neural Stochastic Differential Equations (NSDE) and integrate it into a novel sequential recommendation model. NSDE is a variant of NODE that adds Brownian

motion terms to incorporate stochasticity via a diffusion function. NSDE has been successfully applied in other fields, such as computer vision [20, 41]. It provides an effective means to capture users' continuously changing behavior, and also model the noise via stochasticity in user and item representations. However, recommender systems involve uncertainty from multiple sources: uncertainty caused by evolving user preferences and uncertainty arising from user-item interactions. While the standard NSDE can naturally capture the former, it is not designed to cover the latter. To this end, we further incorporate evidential deep learning (EDL) [33, 1] to gather evidence from user-item interactions and systematically capture uncertainty from multiple sources. Table 3 summarizes the key differences between the proposed model and existing relevant models.

The proposed E-NSDE seamlessly integrates an NSDE module and an EDL module, where the former is responsible for learning user and item representations over time and the latter utilizes these rich representations to identify important and diverse items that the model needs to learn to capture users' actual behavior with the help of uncertainty-aware exploration. Table 2 shows that E-NSDE places a pronounced emphasis on a wider array of genres, and a substantial portion of these manage

to capture sustained interest over the long term. In each sequence, E-NSDE attempts to provide diverse genres as shown in the first row ('Thriller') and second row ('Sci_Fi') of the E-NSDE result. When there is a long time interval, the model can leverage this opportunity guided by its predicted epistemic uncertainty and recommend a novel genre ('Crime') to explore the diverse items that can help to capture the user's long-term interest. The ability to quantify time-aware uncertainty also allows E-NSDE to outperform SOTA sequential recommendation models as shown in Table 1.

As a key innovation, we connect NSDE-based user-item representation learning with EDL using a monotonic network to provide an evidence-guided recommendation to capture time-sensitive rating prediction augmented with uncertainty estimation. Intuitively, the model's uncertainty of a user's interest should increase with respect to the time interval since the last observed user-item interaction. To model time-sensitive uncertainty, we integrate EDL using an uncertainty-aware regression model to infer evidential distributions that allow us to quantify both aleatoric and epistemic uncertainties. Similarly, the monotonic network captures the underlying constraint of monotonicity between the time interval and the predicted uncertainty: a longer time interval leads to a higher uncertainty. With accurately estimated uncertainty, the proposed framework effectively explores user preference from a large item space. The main contribution of this paper is fourfold:

a novel recommendation model that integrates neural stochastic differential equations with evidential learning for time-aware uncertainty quantification for effective sequential recommendations,

leveraging interaction and time-guided evidential uncertainty to maximize information gain through exploration of a large item pool,

a monotonic network to ensure a positive correlation between interaction gap and uncertainty,

an end-to-end integrated training process with seamless integration of NSDE and EDL modules.

To assess the feasibility of our method in comparison with existing state-of-the-art methods, we perform a wide range of experiments on publicly available real-world datasets. We also conduct ablation and case studies to analyze the effectiveness and interpretability of our method.

2 Related Work

In this section, we provide existing works most relevant to our proposed approach. Some additional related works are discussed in the Appendix.

Sequential recommendation models. Sequential models utilize users' historical interactions to capture users' preferences over time. Tang et al. utilized a CNN architecture to capture union level and point level contributions [37]. Kang et al. leveraged transformer-based user representation to better capture their interest [19]. Sun et al. utilized a bidirectional encoder for sequential recommendation [35]. Similarly, Zhou et al. [45] have leveraged auxiliary self-supervised objectives to learn correlation among attributes, items, sub-sequences, and sequences by utilizing mutual information maximization. Recently, contrastive learning has been used in sequential recommendation [40, 43] to learn highquality user representation leveraging different forms of data augmentation strategies. Recently, SAR [3] has leveraged an actor-critic network, where the action is generated as adaptive sequence length to better represent the user's sequential pattern. Similarly, ResAct [42] utilizes residual actornetwork to reconstruct policy that is close but better than online policy more efficiently in sequential recommendation. However, most existing sequential models are inadequate to capture long-term users' preferences, which is a critical gap that the proposed work aims to address.

NODE based recommender systems. Neural ODE solvers have recently been introduced into recommender systems [9, 36]. For example, the learnable time ODE-based collaborative filtering [9] redesigns linear graph convolution networks on top of the NODE that learns the optimal architecture and smooth ODE solutions for effective collaborative filtering. Similarly, [36] utilizes meta-learning enhanced neural ODE for citywide next POI Recommendation. It models city-invariant and city-specified information separately to achieve accurate citywide next POI recommendation. As discussed earlier, standard NODE models do not explicitly capture uncertainty, which is critical for a recommendation model to explore a large item space to capture user's long-term preference.

3 Preliminaries

Problem formulation. The input to a recommendation model include a user set ( U ) and item set ( I ), respectively. We represent a user as u t ∈ U and item as i t ∈ I at time t . In a sequential recommendation setting, users' interactions are organized in the chronological order and we use

( i 0 , i 1 , ..., i t -1 , i t ) ⊤ to represent interaction sequences for user u at time t . We perform recommendation and uncertainty quantification for each user using a function:

where γ ( u t ,i t ) is the recommendation score for item i assigned by user u , ν ( u t ,i t ) and α ( u t ,i t ) are the model evidence, and β ( u t ,i t ) is a total uncertainty arising from the user-item interaction at time t .

Evidential learning. Evidential learning is an evidence acquisition process where every training sample adds support to learn a higher order evidential distribution [33, 1]. Given that the target y n is drawn i.i.d. from a Gaussian distribution with unknown mean and variance ( µ, σ 2 ) the model evidence can be introduced by further placing a prior distribution on ( µ, σ 2 ) . Placing a Gaussian prior on the unknown mean and the Inverse-Gamma prior on the unknown variance, the posterior of ( µ, σ 2 ) is the Normal-Inverse-Gamma ( NIG ) distribution. The Gaussian and the Inverse-Gamma priors are chosen to ensure conjugacy:

where Inv-Gamma ( z | α, β ) = β α Γ( α ) ( 1 z ) α +1 exp( -β z ) with Γ( · ) being a gamma function; m = ( γ, ν, α, β ) are parameters of the corresponding prior distributions. The posterior of ( µ, σ 2 ) follows a Normal Inverse-Gamma (NIG):

Given a NIG posterior, one can derive the mean ( E [ µ ] ), aleatoric ( E [ σ 2 ] ) and epistemic (Var [ µ ] ) uncertainty as:

Neural ordinary differential equations (NODE). Ordinary Differential Equations (ODEs) are used to model continuous-time hidden dynamics in neural networks [7] that can be defined as:

where f ( · ) is a neural network with parameter ψ and h 0 is an initial value. Leveraging Eq (2) and integrating these dynamics forward, one can compute h ( t i +1 ) from h ( t i ) by solving the following Riemann integral problem:

Neural stochastic differential equations (NSDE). We could view Stochastic Differential Equations (SDE) as an ODE with infinitesimal noise added throughout time:

where f ( . ) and g ( . ) are drift and diffusion functions respectively, B t is a Brownian motion. Similar to NODE, we are also able to compute the forward dynamics of NSDE i.e. h ( T ) from initial value h ( t 0 ) integrating Eq (4) as:

4 Time-Aware Sequential Recommendations

Overview. We propose a novel time-aware sequential recommendation model as shown in Figure 1. The model leverages 1) NSDE to capture continuous time-evolving user dynamics and 2) an evidential module to capture uncertainty in user-item interactions and also to provide uncertainty-aware exploration that takes into consideration the interaction interval. The NSDE module takes the initial representations of users and items as inputs, and uses the interaction time gap to generate refined user and item representations. Subsequently, these improved user and item representations are fed into the EDL module. The rating network then generates a rating score, while the monotonic network produces evidential parameters that incorporate the interaction time gap, establishing a direct link to the model's predicted uncertainty. Our approach adheres to the conventional sequential training strategy and incorporates supervised signals derived from evidential learning. We delve into detailed discussions in the subsequent sections.

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4.1 NSDE Based User and Item Representations

The NSDE includes two key components: drift and diffusion functions. The drift component captures the system's evolving nature, and the diffusion component captures its stochasticity. The proposed NSDE for recommender system advances contemporary sequential models from the following aspects: 1) existing methods require partitioning the time into uniform intervals to support model training and inference, while NSDE removes this requirement, providing additional flexibility; 2) the existing methods largely negate stochasticity in the system, but the NSDE incorporates it into the form of inherent noise. The above issues suggest a better fit of the NSDE into sequential recommendation and essentially support generating richer user and item representations. This is because SDE can capture users' continuously evolving preferences over time, while previous discrete sequential and deterministic ODE methods cannot. The fine-grained user representation (based on NSDE formulation) can be written following (5):

where u ( t 0 ) is the user's initial representation which aggregates a set of initially interacted items: u ( t 0 ) = agg ( i 1 ..i k ) and T is the final time.

In the setting of recommender systems, the drift component naturally encodes the evolution of the user's preference, while the diffusion component captures inherent noise that occurs when the user interacts with the environment. In particular, we leverage the diffusion function with Brownian motion to capture the user's inherent noise over each interaction. This term is crucial in SDE to capture stochasticity in the system. By capturing stochasticity, we incorporate the impact of noisy user-item interactions. As it includes the Brownian motion for stochasticity, we relate this in recommender systems to incorporate noise considering the time interval of interaction. We first provide the standard definition and properties of the Brownian motion and then present its applicability in the recommendation setting.

Definition 1 (Standard Brownian Motion) . A standard Brownian motion B t is a stochastic process that satisfies the following properties: a) B t -B s is normally distributed with zero mean and ( t -s ) variance: N (0 , t -s ) for all t ≥ s ≥ 0 ; b) For every pair of disjoint time intervals [ t 1 , t 2 ] and [ t 3 , t 4 ] , with t 1 < t 2 ≤ t 3 ≤ t 4 , the increments B t 2 -B t 1 and B t 4 -B t 3 are independent random variables.

Theorem 1. For user representation defined by the Brownian motion in Eq. (6) , with two adjacent timestamps of interaction denoted as t 2 and t 1 , a larger interaction time interval ( t 2 -t 1 ) guarantees a higher uncertainty of user representation.

Given Theorem 1, the final term in Eq. (6) captures the user's time deviation and its impact in increasing large variance or noise in the system. Further, the first term captures the user's initial representation with some interactions, and the second drift component provides the user's evolving

interest. Considering all of these three components, the SDE solver captures the user's richer representation over time. Similarly, NSDE-based item representation processes can be formulated as:

where, i ( t 0 ) represents item's initial representation

4.2 Evidential Module

We leverage an evidential learning technique to provide an uncertainty-aware model prediction for effective recommendations. The evidential module consists of two key networks: Rating Network and Monotonic Network.

Rating Network. The rating network utilizes the fine-grained user u t and item i t representations from the NSDE and predicts the score γ ( u t ,i t ) of the corresponding user item interactions. We adopt an evidential loss as the marginal likelihood while computing the predicted loss. This includes the negative log-likelihood ( L NLL [ f Θ ] ) to maximize the marginal likelihood and an evidential regularizer ( L R [ f Θ ] ) to impose a high penalty on the predicted error with low uncertainty (i.e., high confidence). We first formulate the negative log-likelihood, given by

where, m ( u t , i t ) = ( γ ( u t ,i t ) , ν ( u t ,i t ) , α ( u t ,i t ) , β ( u t ,i t ) ) are model parameters at time t, and p ( r ( u t ,i t ) | m ( u t , i t )) = St ( r ( u t ,i t ) ; γ ( u t ,i t ) , β ( ut ,i t ) (1+ ν ( ut ,i t ) ) ν ( ut ,i t ) α ( ut ,i t ) , 2 α ( u t ,i t ) ) is a student t-distribution acquired after placing a NIG evidential prior on Gaussian likelihood function. We formalize our own evidence regularizer, which considers epistemic uncertainty to penalize confidently predicted errors. We multiply the predicted error with the inverse epistemic uncertainty that scales up the error, which encourages high inverse epistemic uncertainty when the predicted evidence is high (and vice-versa). Conversely, it will be less penalized if the prediction is close to the target score:

The regularized EDL loss for each sequential update is:

where λ is a regularization coefficient.

Monotonic Network. We adopt the concept of a monotonic network [34] in the context of building the relationship between the interaction time gap and model uncertainty. Intuitively, the monotonic network is designed in such a way that the increase in input, i.e., time interval ( ∆ t ), increases the output, i.e., the variance of the predicted rating. The variance or epistemic uncertainty is computed as given by Eq (1). For this, the nominator term, i.e., total uncertainty β ( u t ,i t ) should need to be increased with the increase in ∆ t , and the denominator terms, i.e., pseudo-observations ν and α should need to be decreased with the increase in ∆ t . We theoretically show this intuition in the following theoretical section and show a mathematical relation. We maintain this by performing exponential transformation of the network weights as: ϕ = exp( ϕ init ) , where ϕ init is the network's initial weight. To ensure β to monotonically increase, we assign all positive weights to the network layers. Similarly, to ensure ν and α to monotonically decrease, we assign negative weights to the last layer and positive weights to other layers of the network. We update the network utilizing the total loss similar to the rating network.

Lemma 1 (Monotonic increase of total uncertainty β ) . Let the total uncertainty of user-item interaction β ( u t ,i t ) be the output of the evidential monotonic network with weights ϕ = exp( ϕ init ) . Given a time interval ∆ t , the output of the network is guaranteed to monotonically increase.

Lemma 2 (Monotonic decrease of pseudo-observations α and ν ) . Let ∆ t be the increased time interval ∆ t , weights W L of the last layer be negative, and weights W 0 ,...,L -1 for other layers be positive, the output i.e., pseudo-observations α ( u t ,i t ) , and ν ( u t ,i t ) of the evidential monotonic network decreases monotonically.

Theorem 2. (Increased time interval ∆ t results in increased in epistemic uncertainty V ar [ µ ] ). Given the monotonic network formulated by Lemmas 1 and 2, an increase of the input time interval ( ∆ t ) of the evidential monotonic network guarantees an increase of the output epistemic uncertainty V ar [ µ ] .

Proof. The epistemic uncertainty equation from Eq (1):

Given the increase β ( u t ,i t ) from Lemma 1 and decrease in α ( u t ,i t ) and ν ( u t ,i t ) from Lemma 2 the nominator of the epistemic uncertainty increases, and the denominator decreases. This proves that the increase in the time interval ( ∆ t ) increases the epistemic uncertainty var [ µ ] of the evidential monotonic network. We enforce the constraints on ( β ( u t ,i t ) , α ( u t ,i t ) , ν ( u t ,i t ) ) with a soflplus activation and adding 1 to α ( u t ,i t ) since α ( u t ,i t ) > 1 ).

Interpreting hyper-parameters. Besides serving as the parameters of the evidential prior distributions, the hyper-parameters ( ν ( u t ,i t ) , α ( u t ,i t ) , β ( u t ,i t ) ) offer very intuitive meanings. First, both ν ( u t ,i t ) and α ( u t ,i t ) are essentially the 'pseudo' prior observations, and their posterior can be treated as the evidence to support a prediction. In the context of the recommendation, their relation with time interval is inverse, because the large time gap causes a decrease in the number of pseudo items, as mentioned in Lemma 2. Second, the β ( u t ,i t ) hyperparameter combines total uncertainty from pseudo samples and observed data. Lemma 1 shows that an increase in time interval will result in an increase in the uncertainty (due to a smaller number of pseudo and interacted items to the user), and therefore, the model will be less confident in providing an accurate prediction.

Weighted Bayesian personalized ranking (WBPR) loss. To leverage the effective exploration for the long-term, we formulate weighted BPR loss which is computed from non-interacted ( i.e., negative) items, j t ∈ N t that are similar to the user's future interacted items. We first select similar negative items from the user non-interacted item pool and then leverage cosine similarity with future positive item embeddings. Further, the model provides uncertainty-aware predicted rating score for those negative items leveraging both rating and monotonic network output as:

where j t represents non-interacted items at time t and η is scalar to control the influence of epistemic uncertainty. We then compute weight coefficients based on uncertainty-aware predicted scores with cosine similarity Sim ( · ) as:

where f _ emb,j _ emb are future and negative item embedding, respectively. We then formulate weighted BPR loss utilizing a negative log-likelihood function as:

where ˆ r ( i t ,j t ) = ˆ r ( u t ,i t ) -ˆ r ( u t ,j t ) , σ ( · ) is the sigmoid.

Remark. The intuition behind this weighted BPR formulation is to learn effective exploration by providing higher weight to the non-interacted items which are quite similar to user future positive items so that model can learn quickly a diverse range of evolving behavior to benefit the future.

The overall loss of the end-to-end model training is obtained by combining the EDL and WBPR loss:

where ζ represents the balancing factor between EDL and WBPR loss. Training and inference details are provided in Appendix D.

5 Experiments

Experimental setup. Our experiment setting of sequential recommendation is based on next-item recommendation tasks, which was used in [35]. We first split users by 70% into train and 30% in

test. For each user, we leverage the fixed sequence length and hold out the next item of the behavior sequence as the target item. We follow the standard strategy in [35] for easy and fair evaluation. We leverage the actual time of interactions (in UNIX timestamp) to provide user preference evolution.

Datasets and baselines. We conduct extensive experiments on four real-world datasets that contain explicit ratings: Movielens-100K , Movielens-1M , Netflix , and Amazon Book .

Movielens-100K 2 : This dataset contains 100,000 explicit ratings on a scale of (1-5) from 943 users on 1,682 movies. Each user at least rated 20 movies from September 19, 1997 through April 22, 1998.

Movielens-1M 3 : This dataset includes 1M explicit feedback (i.e. ratings) made by 6,040 anonymous users on 3,900 distinct movies from 04/2000 to 02/2003.

Netflix [4]: This dataset has around 100 million interactions, 480,000 users, and nearly 18,000 movies rated between 1998 to 2005. We pre-processed the dataset and selected 6,042 users with user-item interactions from 01/2002 to 12/2005.

Amazon Book [39]: This data set contains 2,984,108 ratings applied to 91,599 books by 52,643 users with at least ten interactions in each user sequence

For comparisons, we include a comprehensive list of SOTA baselines from diverse groups, including Dynamic models: timeSVD++ [22] and CKF [14]; Sequential models: CASER [37], SASRec [19], BERT4Rec [35], S 3 -Rec[45], CL4SRec [40], SAR [3], and ResAct [42]; Graph-based models: NGCF[39] and LightGCN[18]; ODE-based models: LT-OCF [9] and GRU-ODE [16].

Evaluation metrics. To evaluate the proposed and baseline recommendation models, we follow the sequential recommendation setup similar to [35]. We consider one ground truth item in each sequential recommendation. We use two standard metrics to measure the recommendation performance.

Precision@N (P@N) : It is the fraction of the topN items recommended in each sequence to the user . We reported the average overall sequence precision value as the final precision. Further, due to only one ground truth in the target, the P@N is equivalent to Recall@N.

nDCG@N : Normalized Discounted Cumulative Gain (nDCG) measures ranking quality, considering the relevant items within the topN of the ranking list in each recommendation.

For more details about datasets, and implementation please refer to the Appendix.

Recommendation performance comparison. Table 4 summarizes the recommendation performance from all models for four real-world datasets. The proposed model benefits from both the SDE module, which continuously captures user evolving preferences, and the evidential module, which estimates prediction confidence, and thus achieves better results in all four datasets. The dynamic models achieve less ideal performance due to their focus on discrete-term user interest and inability to provide continuous user preference. Graph-based methods take advantage of recently interacted items and have shown better performance than traditional dynamic methods. However, they may not be good enough to capture user sequential interest. Further, sequential methods benefit from sequential learning and have promising results. However, they do not consider the time component in the recommendation and are less effective than the proposed method. ODE-based methods have

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shown a clear advantage due to their focus on capturing users' continuous behavior over time, but they cannot estimate model confidence on predictions and hence have lower performance value than the proposed method. We provided a detailed plot of precision and nDCG @5 in Figure 2 considering test users in each training epoch. Further, we report the precision and NDCG @10 and @20 as the increased topN metric ( i.e., N = 10 , 20 ) for several baselines in the appendix. The results show the proposed model is consistently outperforming other baselines.

Uncertainty vs. Interaction gap We further investigate the impact of the user-item interaction gap and the corresponding uncertainty in providing important and diverse items. Table 5 (a) shows the proposed E-NSDE model provides diverse ( i.e., 'Thriller' and 'Sci_Fi') movies for the larger interaction time gap ∆ t = 896 , 291 seconds that help to capture the future interest of the user. On the other hand, the existing GRU-ODE model recommends only popular genres like 'Drama' and fails to explore others adequately. We further provided the simulation results of 5 users considering increasing time intervals:

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{ 10 2 , 10 3 , 10 4 , 10 5 , 10 6 , 10 7 } in seconds with random item interactions on those timestamps and the corresponding predicted uncertainty in Figure 3. The total uncertainty β monotonically increases with the time intervals which shows that E-NSDE effectively captures user's uncertainty coming from longer interaction gaps.

Effectiveness of exploration. To demonstrate the effectiveness of the uncertainty-based exploration, we measure the diversity of the recommended items. We consider three commonly used diversity measures: Gini index ( G ), Coverage ( C ), and Novelty ( N ):

Gini index can be calculated by leveraging the different genres as categories for each recommended top K items in a sequence: G = 1 -∑ C c =1 P ( c ) 2 , where C is the number of categories and P ( c ) , c ∈ [1 , C ] is the probability for each category.

Coverage represents the number of categories that are included in the recommendation list. For the recommended top K items, we compute item coverage as: C = # categories / total categories.

Novelty is defined by the ratio of recommended items from new categories not interacted by the user. It is computed as : N = # new categories /K .

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0.4026

0.2402

0.2236

16

32

64

128

256

0.25

0.30

0.35

0.40

0.45

0.50

nDCG@5

ML-100K

ML-1M

Netflix

(a) Embedding size

1

0.1

0.01

0.001

0.0001

η

0.25

0.30

0.35

0.40

0.45

0.50

nDCG@5

ML-100K

ML-1M

Netflix

(b) Uncertainty-aware rating

1

0.1

0.01

0.001

0.0001

ζ

0.25

0.30

0.35

0.40

0.45

0.50

nDCG@5

ML-100K

ML-1M

Netflix

(c) WBPR loss

Table 6 shows that E-NSDE recommends more diverse items based on all three evaluation metrics as compared with the two competitive baselines thanks to its uncertainty-guided exploration strategy.

Ablation study. We conduct an ablation study and details are summarized below:

Impact of key components. Weevaluate the impact of each key component in the proposed E-NSDE. Table 5 (b) shows each component contributes to improved recommendation performance.

Evidential regularization parameter. One of the key hyperparameters of the E-NSDE model is the regularizer constant ( λ ) for the evidential learning. We cross-validated this parameter with empirical results of the model for the different λ values in two datasets as shown in Table 7. From the table, our model achieves the best performance in both datasets with λ = 0 . 001 .

Embedding dimension. We generate user and item embeddings using the embedding network. We perform a grid search for the embedding dimension ( d ) of the user and item representation in E-NSDE model as shown in Figure 4a. From the plot, it shows that E-NSDE has the best performance with d = 64 .

Balancing factors. We leverage grid search on uncertainty-aware ranking factor η , and WBPR loss balancing factor ζ on three datasets as shown in Figure 4b and Figure 4c, respectively. The figure shows a clear advantage with η = 0 . 01 , which indicates that the uncertainty-aware exploration component takes an effective role in providing the best performance for our proposed E-NSED model. Similarly, for ζ balancing factor integrated overall loss has the best performance when it is equal to 0 . 001 .

6 Conclusion

We propose a novel evidential stochastic differential equations (E-NSDE) model for the time-aware sequential recommendations. E-NSDE seamlessly integrates an NSDE module and an EDL module to capture users' continuously evolving behavior and model predictive uncertainty at the same time. Our proposed model effectively leverages the interaction time gap and provides uncertainty-aware recommendations with diverse items to the user. Further, we theoretically derive mathematical relationships between the interaction time gap and model uncertainty to enhance the learning process, as demonstrated in our extensive experiments on multiple real-world datasets.

Acknowledgments

This research was supported in part by an NSF IIS award IIS-1814450. The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing any funding agency. We would like to thank the anonymous reviewers for their constructive comments.

References

Alexander Amini, Wilko Schwarting, Ava Soleimany, and Daniela Rus. Deep evidential regression. Advances in Neural Information Processing Systems , 33:14927-14937, 2020.

Anitha Anandhan, Liyana Shuib, Maizatul Akmar Ismail, and Ghulam Mujtaba. Social media recommender systems: review and open research issues. IEEE Access , 6:15608-15628, 2018.

Stefanos Antaris and Dimitrios Rafailidis. Sequence adaptation via reinforcement learning in recommender systems. In Proceedings of the 15th ACM Conference on Recommender Systems , pages 714-718, 2021.

James Bennett, Stan Lanning, et al. The netflix prize. In Proceedings of KDD cup and workshop , volume 2007, page 35. New York, NY, USA., 2007.

Rianne van den Berg, Thomas N Kipf, and Max Welling. Graph convolutional matrix completion. arXiv preprint arXiv:1706.02263 , 2017.

Laurent Charlin, Rajesh Ranganath, James McInerney, and David M Blei. Dynamic poisson factorization. In Proceedings of the 9th ACM Conference on Recommender Systems , pages 155-162, 2015.

Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems , 31, 2018.

Heng-Tze Cheng, Levent Koc, Jeremiah Harmsen, Tal Shaked, Tushar Chandra, Hrishi Aradhye, Glen Anderson, Greg Corrado, Wei Chai, Mustafa Ispir, et al. Wide & deep learning for recommender systems. In Proceedings of the 1st workshop on deep learning for recommender systems , pages 7-10, 2016.

Jeongwhan Choi, Jinsung Jeon, and Noseong Park. Lt-ocf: learnable-time ode-based collaborative filtering. In Proceedings of the 30th ACM International Conference on Information & Knowledge Management , pages 251-260, 2021.

Gideon Dror, Noam Koenigstein, Yehuda Koren, and Markus Weimer. The yahoo! music dataset and kdd-cup'11. In Proceedings of KDD Cup 2011 , pages 3-18. PMLR, 2012.

Lian Duan, W Nick Street, and E Xu. Healthcare information systems: data mining methods in the creation of a clinical recommender system. Enterprise Information Systems , 5(2):169-181, 2011.

Ziwei Fan, Zhiwei Liu, Yu Wang, Alice Wang, Zahra Nazari, Lei Zheng, Hao Peng, and Philip S Yu. Sequential recommendation via stochastic self-attention. In Proceedings of the ACM web conference 2022 , pages 2036-2047, 2022.

Simon Funk. Incremental svd method, 2006.

San Gultekin and John Paisley. A collaborative kalman filter for time-evolving dyadic processes. In 2014 IEEE International Conference on Data Mining , pages 140-149. IEEE, 2014.

Huifeng Guo, Ruiming Tang, Yunming Ye, Zhenguo Li, and Xiuqiang He. Deepfm: a factorization-machine based neural network for ctr prediction. arXiv preprint arXiv:1703.04247 , 2017.

Jiayan Guo, Peiyan Zhang, Chaozhuo Li, Xing Xie, Yan Zhang, and Sunghun Kim. Evolutionary preference learning via graph nested gru ode for session-based recommendation. In Proceedings of the 31st ACM International Conference on Information & Knowledge Management , pages 624-634, 2022.

Qingyu Guo, Fuzhen Zhuang, Chuan Qin, Hengshu Zhu, Xing Xie, Hui Xiong, and Qing He. A survey on knowledge graph-based recommender systems. IEEE Transactions on Knowledge and Data Engineering , 2020.

Xiangnan He, Kuan Deng, Xiang Wang, Yan Li, Yongdong Zhang, and Meng Wang. Lightgcn: Simplifying and powering graph convolution network for recommendation. In Proceedings of the 43rd International ACM SIGIR conference on research and development in Information Retrieval , pages 639-648, 2020.

Wang-Cheng Kang and Julian McAuley. Self-attentive sequential recommendation. In 2018 IEEE International Conference on Data Mining (ICDM) , pages 197-206. IEEE, 2018.

Lingkai Kong, Jimeng Sun, and Chao Zhang. Sde-net: Equipping deep neural networks with uncertainty estimates. arXiv preprint arXiv:2008.10546 , 2020.

Yehuda Koren. Factorization meets the neighborhood: a multifaceted collaborative filtering model. In Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining , pages 426-434, 2008.

Yehuda Koren. Collaborative filtering with temporal dynamics. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining , pages 447-456, 2009.

Yehuda Koren, Robert Bell, and Chris Volinsky. Matrix factorization techniques for recommender systems. Computer , 42(8):30-37, 2009.

Jiacheng Li, Yujie Wang, and Julian McAuley. Time interval aware self-attention for sequential recommendation. In Proceedings of the 13th international conference on web search and data mining , pages 322-330, 2020.

Xuechen Li, Ting-Kam Leonard Wong, Ricky TQ Chen, and David Duvenaud. Scalable gradients for stochastic differential equations. In International Conference on Artificial Intelligence and Statistics , pages 3870-3882. PMLR, 2020.

Zihao Li, Aixin Sun, and Chenliang Li. Diffurec: A diffusion model for sequential recommendation. ACM Transactions on Information Systems , 42(3):1-28, 2023.

Yifang Qin, Wei Ju, Hongjun Wu, Xiao Luo, and Ming Zhang. Learning graph ode for continuous-time sequential recommendation. IEEE Transactions on Knowledge and Data Engineering , 2024.

Ruihong Qiu, Zi Huang, Hongzhi Yin, and Zijian Wang. Contrastive learning for representation degeneration problem in sequential recommendation. In Proceedings of the fifteenth ACM international conference on web search and data mining , pages 813-823, 2022.

Steffen Rendle. Factorization machines. In 2010 IEEE International Conference on Data Mining , pages 995-1000. IEEE, 2010.

Steffen Rendle, Christoph Freudenthaler, Zeno Gantner, and Lars Schmidt-Thieme. Bpr: Bayesian personalized ranking from implicit feedback. arXiv preprint arXiv:1205.2618 , 2012.

Nachiketa Sahoo, Param Vir Singh, and Tridas Mukhopadhyay. A hidden markov model for collaborative filtering. Mis Quarterly , pages 1329-1356, 2012.

J Ben Schafer, Joseph Konstan, and John Riedl. Recommender systems in e-commerce. In Proceedings of the 1st ACM conference on Electronic commerce , pages 158-166, 1999.

Murat Sensoy, Lance Kaplan, and Melih Kandemir. Evidential deep learning to quantify classification uncertainty. Advances in neural information processing systems , 31, 2018.

Joseph Sill. Monotonic networks. Advances in neural information processing systems , 10, 1997.

Fei Sun, Jun Liu, Jian Wu, Changhua Pei, Xiao Lin, Wenwu Ou, and Peng Jiang. Bert4rec: Sequential recommendation with bidirectional encoder representations from transformer. In Proceedings of the 28th ACM international conference on information and knowledge management , pages 1441-1450, 2019.

Haining Tan, Di Yao, Tao Huang, Baoli Wang, Quanliang Jing, and Jingping Bi. Meta-learning enhanced neural ode for citywide next poi recommendation. In 2021 22nd IEEE International Conference on Mobile Data Management (MDM) , pages 89-98. IEEE, 2021.

Jiaxi Tang and Ke Wang. Personalized top-n sequential recommendation via convolutional sequence embedding. In Proceedings of the Eleventh ACM International Conference on Web Search and Data Mining , pages 565-573, 2018.

Shoujin Wang, Liang Hu, Yan Wang, Longbing Cao, Quan Z Sheng, and Mehmet Orgun. Sequential recommender systems: challenges, progress and prospects. arXiv preprint arXiv:2001.04830 , 2019.

Xiang Wang, Xiangnan He, Meng Wang, Fuli Feng, and Tat-Seng Chua. Neural graph collaborative filtering. In Proceedings of the 42nd international ACM SIGIR conference on Research and development in Information Retrieval , pages 165-174, 2019.

Xu Xie, Fei Sun, Zhaoyang Liu, Shiwen Wu, Jinyang Gao, Jiandong Zhang, Bolin Ding, and Bin Cui. Contrastive learning for sequential recommendation. In 2022 IEEE 38th international conference on data engineering (ICDE) , pages 1259-1273. IEEE, 2022.

Winnie Xu, Ricky TQ Chen, Xuechen Li, and David Duvenaud. Infinitely deep bayesian neural networks with stochastic differential equations. In International Conference on Artificial Intelligence and Statistics , pages 721-738. PMLR, 2022.

Wanqi Xue, Qingpeng Cai, Ruohan Zhan, Dong Zheng, Peng Jiang, Kun Gai, and Bo An. Resact: Reinforcing long-term engagement in sequential recommendation with residual actor. arXiv preprint arXiv:2206.02620 , 2022.

Yuhao Yang, Chao Huang, Lianghao Xia, Chunzhen Huang, Da Luo, and Kangyi Lin. Debiased contrastive learning for sequential recommendation. In Proceedings of the ACM Web Conference 2023 , pages 1063-1073, 2023.

Shuai Zhang, Lina Yao, Aixin Sun, and Yi Tay. Deep learning based recommender system: A survey and new perspectives. ACM computing surveys (CSUR) , 52(1):1-38, 2019.

Kun Zhou, Hui Wang, Wayne Xin Zhao, Yutao Zhu, Sirui Wang, Fuzheng Zhang, Zhongyuan Wang, and Ji-Rong Wen. S3-rec: Self-supervised learning for sequential recommendation with mutual information maximization. In Proceedings of the 29th ACM international conference on information & knowledge management , pages 1893-1902, 2020.

Cai-Nicolas Ziegler, Sean M McNee, Joseph A Konstan, and Georg Lausen. Improving recommendation lists through topic diversification. In Proceedings of the 14th international conference on World Wide Web , pages 22-32, 2005.

Appendix

Organization of Appendix In this Appendix, we first summarize all notations used in our paper. Next we provide additional related work. Further, we provide the theoretical proof for Theorem 1, Lemma 1, and Lemma 2. Followed by that, we provide algorithms for the training and inference process. Further, we provide additional results for the extra baselines in terms of precision, NDCG@10 and NDCG@20. Finally, we provide a broader impact, limitation, future work, link to the source code, and checklist.

A Summary of Notations

We summarize the major notations used throughout the paper in Table 8.

B Additional Related Work

In this section, we discuss some commonly used reocmmendation models to complement the related works covered in the main paper.

Static and dynamic recommendation models. Matrix Factorization (MF) leverages user and item latent factors to infer user preferences [23, 13, 21]. MF is further extended with Bayesian Personalized Ranking (BPR) [30] and Factorization Machine (FM) [29]. Recently, deep learningbased recommender systems [8, 15] have achieved impressive performance. DeepFM [15] integrates traditional FM and deep learning to learn low- and high-order feature interactions. Both wide and deep networks are jointly trained in [8] for better memorization and generalization. The dynamic model shifts latent user preference over time to incorporate temporal information. TimeSVD++ [22] considers time-specific factors, which uses additive bias to model user and item-related temporal changes. Gaussian state-space models introduce time-evolving factors with a one-way Kalman filter [14]. To process implicit data, Sahoo et al. extended the hidden Markov model [31], and Charlin et al. [6] further augmented it with the Poisson emission. Nevertheless, these models grasp the evolving preferences of users, yet they exhibit limited foresight into future interactions, leading to poor recommendations performance.

Graph-based recommendation models. Another popular line of recommendation systems is graph-based models. A graph captures high-order user-item interactions through an iterative process to provide effective recommendations [17]. Users and items are represented as a bipartite graph in [5] and links are predicted to provide recommendations. Similarly, a graph-based framework called Neural Graph Collaborative Filtering (NGCF) [39] explicitly encodes the collaborative signal in the form of high-order connectivities in a user-item bipartite graph via embedding propagation. However, these methods are unable to capture long-term user preferences or deal with cold-start problems.

C Proof of Theoretical Results

In this section, we provide proofs of the theoretical results in the main paper.

C.1 Proof of Theorem 1

Proof. Given Definition 1, we can show that the increase in interaction time gap ( t 2 -t 1 ) increases the chance that the user may deviate from the current interest. It turns out that the last term of Eq (6) has a larger variance, i.e., there is a higher deviation in the user interest, if there is a longer time gap in the interaction. Suppose t 3 > t 2 , then we have

where we applied the Ito Isometry in the second step. This result indicates that if there is a longer interaction gap t 3 -t 1 , then the state is more uncertain.

C.2 Proof of Lemma 1

Given the user final representation u t , item final representation i t , and interaction time gap ∆ t . The monotonic network produces β ( u t ,i t ) as:

Hence ∂β ∂ (∆ t ) = W ∆ t ≥ 0 .

C.3 Proof of Lemma 2

Given the same setup as Lemma 2, the monotonic network outputs α ( u t ,i t ) , and ν ( u t ,i t ) . We first consider α ( u t ,i t ) :

where h i ( x ) is the transformation induced by the i th affine layer, h i ( x ) = W i x + b i , ◦ is the composition of functions, and g is an increasing activation function.

According to the backpropagation, we have



W



W

W

u

T

T

i

T

∆

t

where we rewrite W T 1 the same blocks as that in Lemma 2. Therefore,

Since the entries in W 1 , . . . , W L -1 are all negative, and g ′ ( h 1 ) , . . . , g ′ ( h L -1 ) are all positive. Therefore, if entries in W L are all negative, we have ∂α ( ut ,i t ) ∂ (∆ t ) ≤ 0 .

Similarly, we can show that ∂ν ∂ (∆ t ) ≤ 0 . We leverage the monotonic non-linear activation function i.e. ELU ( . ) to satisfy this condition.





where ⊙ is element-wise product of two matrices with the same size. The matrix is just

W

T

1

,

D Training and Inference Processes

The training procedure involves the end-to-end parameter updates associated with the NSDE module and evidential module. Both modules utilize the overall loss mentioned in Eq (13), which includes a supervised signal from evidential loss and ranking loss from WBPR loss. The rating network ( θ ) and monotonic network ( ϕ ) are updated with the Adam optimizer, and the SDE module with model parameters ( ψ, ω ) is updated with SDE adjoint method [25]. Algorithm 1 shows the training process that learns the model parameters. During inference, we consider test users ( i.e., distinct users from the training set) and perform standard sequential recommendations respecting the time interval of interactions.

E Additional Experimental Results

Higher order topN results. We report the results using topN metric with a larger N value (i.e., @10 and @20) in Table 9. We reported those results on two sparse datasets: Movielens-1M with 95.75% sparsity and Amazon Book with 99.98% sparsity. In both datasets, our proposal models outperform other baselines in those metrics and show consistent performance over metrics.

Additional datasets. We conduct experiments on two additional datasets from the book and music domains to cover a broader range. The Book-Crossing dataset is a collection of book ratings, including both explicit ratings (1-10 stars) and implicit ratings (interaction) [46]. The Yahoo! Music dataset is a collection of user ratings for songs, albums, artists, and genres, with fine-resolution timestamps for ratings [10]. The proposed E-NSDE consistently outperforms the competitive baselines.

F Broader Societal Impact, Limitation, and Future Work

This paper presents a novel stochastic neural ODE and evidential learning-based recommendation model to capture users' continuously evolving behavior. This work aims to understand the user's dynamic interest considering an actual time interval of user interaction with the system. This is very useful in many time-series user interaction fields like e-commerce, social media, health care, and gaming sectors.

As a potential limitation of this work, if user-item interactions have uniform time intervals like in click-through, then the impact of time-aware interval is less effective in capturing uncertainty.

Future work could be applying the E-NSDE method to other areas of recommender systems, such as healthcare, where patient behaviors evolve over time, and capturing their real-time behavior is crucial to handling their problem. Similarly, understanding user behavior in the gaming sector is also a potential area to explore in the future.

G Source Code

For the source code, please click this link: https://github.com/ritmininglab/ENSDE

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: Introduction section includes contribution and scope of the work.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: Please refer Appendix F for limitation of our work.

3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: We provide assumptions and a complete proof of lemmas and theorems in Appendix.

4. Experimental Result Reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [Yes]

Justification: We fully disclose the dataset, and experiment settings in Appendix.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [Yes]

Justification: We provide data open access with in the anonymous source code link in Appendix.

6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: We provide all details of experiment settings in Appendix.

7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: We provide statistical testing results in main table.

8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We provide the computing details in the Appendix.

9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: We confirm that we follow the NeurIPS Code of Ethics.

10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We mention broader societal impacts in Appendix.

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: We exclusively utilize public data and release our models to promote research in recommender systems.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: We acknowledge the contributions of open-source baselines and public datasets by citing them as references.

13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [Yes]

Justification: We provide a ccode asset via an anonymized URL in Appendix G.

14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: We haven't used human-subject based crowd sourcing dataset to our experi- ment.

15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: We haven't done any human subjects experiment in our work.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors note that existing methods assume a uniform time interval among user behaviors. This paper posits that the time intervals in sequential recommendation increase the uncertainty of users' behavior. Therefore, it proposes NSDE to learn users’ fine-grained time-evolving behavior, while evidential learning quantifies both aleatoric and epistemic uncertainties. The authors believe that this is something other sequential learning methods have not considered.\n\nSTRENGTHS:\n1. The paper starts from the uncertainty of time intervals to capture the evolution and uncertainty of user behavior, which is a valuable insight. However, this perspective lacks innovation, as \"Sequential recommendation via stochastic self-attention\" (2022) also approaches the problem from the same insight.\n2. The straightforward derivation of the relationship between increasing time intervals and uncertainty to guide learning demonstrates a degree of innovation and robustness in modeling.\n\nWEAKNESSES:\n1.\tThere are formatting issues in lines 99-105. The abbreviation for Normalized Discounted Cumulative Gain is generally NDCG.The combination of metrics chosen in the paper is uncommon and lacks persuasiveness. According to line 294, the metrics are derived from Bert4Rec. However, Bert4Rec's metrics are Hit Ratio (HR), Normalized Discounted Cumulative Gain (NDCG), and Mean Reciprocal Rank (MRR). \n\n2.The selected baselines could include more recent state-of-the-art models, such as STOSA, DuoRec, CoSeRec, GDERec,TiCoSeRec and GCG-ODE. The baselines particularly lack ODE-related sequential recommendation models. Among the two ODE-related models, LT-OCF is not specifically designed to address sequential recommendation.\n\n3.The increase in time intervals may lead to changes in user interests. Addressing this issue within NODE-based recommender systems, the paper proposes corresponding solutions and offers a relatively novel perspective.\n\nQUESTIONS:\n1.Why do the authors believe that sequential recommendation methods typically use uniform time intervals and overlook changes in user interests? Many earlier works have addressed this issue, such as \"Uniform Sequence Better: Time Interval Aware Data Augmentation for Sequential Recommendation\".\n2.See weeknesses\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the modeling of interaction time intervals in sequential recommendation. Considering both time interval and model uncertainty, this paper formulates E-NSDE to integrate NSDE and evidential learning to model effective time-aware sequential recommendation.  Experimental results on four real-world datasets prove the effectiveness of proposed method.\n\nSTRENGTHS:\n1. The problem is interesting. The time interval between interactions is really an important factor in the sequential recommendation. Meanwhile, time-aware uncertainty guides the recommendation to adapt to users' changing preferences.\n\n2. Combining two existing techniques is well-motivated. Overall, both user NSDE and item NSDE are introduced here to encode the evaluation of the preference and inherent noise. The evidential module then provides uncertainty-aware rating prediction. All the stack techniques are combined reasonably.\n\n3. Experiments are conducted to verify the effectiveness. And the results are statically reliable.\n\nWEAKNESSES:\n1. The motivation in the introduction is unclear. As the authors argue, GRU-ODE recommends genres that come from past behaviors, while NSDE recommends genres which have potential future benefits in Table 2. It is not sure if some genres recommended by NSDE are in accordance with long-term interests. For example, Sci_Fi, Mystery, and Crime in Table 2. My concern is whether motivation is not the real need of the user.\n\n2.  As to method, in Figure 1, users also have evolved into representation. It is clear that the user interfaces with items in an evolving manner. But not so clear that the users also evolved into different user.\n\n3. The experiments in Table 4 report the P@5, nDCG@5. However, in the movielens datasets, 1m and 100k are used, so more top-n results should be given to verify the effectiveness. In line 316, \"Uncertainty vs. Interaction gap vs.\" may be a typo.\n\nQUESTIONS:\n1. Please refer to weakness 1, and give more clarification on Table 2.\n\n2. Please explain the relationship between this work and [1],[2],[3]. The baselines lack the time-interval awarded by the sequential model and diversity model.\n\n3. Pleas refer to weakness3, more n in top-n results should be given.\n\n\n\n[1]Time interval aware self-attention for sequential recommendation,WSDM20\n\n[2]Learning Graph ODE for Continuous-Time Sequential Recommendation,TKDE\n\n[3]Temporal Conformity-aware Hawkes Graph Network for Recommendations,Webconf 24\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper revolves around enhancing sequential recommendation systems by incorporating time-awareness through the utilization of Evidential Neural Stochastic Differential Equations (E-NSDE). Traditional recommendation systems often overlook the temporal dynamics of user interactions, leading to suboptimal recommendations. The authors aim to bridge this gap by developing a novel framework that captures the evolving behavior of users over time intervals, thereby improving the accuracy and reliability of recommendations.\nIt acknowledges the limitations of existing methods in effectively modeling temporal dynamics and uncertainty in user behavior. By integrating NSDE and evidential learning, the authors build upon the foundations laid by previous works on stochastic differential equations and recommendation systems. This integration allows for a more robust modeling of user preferences and interactions over time, setting the stage for more accurate predictions.\n\nSTRENGTHS:\n1.\tThe E-NSDE framework dynamically models user behaviors over time, effectively addressing a crucial gap in traditional recommendation systems that often overlook temporal dynamics.\n2.\tBy incorporating NSDEs to capture the continuous-time dynamics of user interactions, along with evidential learning to assess recommendation uncertainty, this dual approach ensures that the recommendations are not only time-sensitive but also carry a higher degree of confidence.\n3.\tThe integration of NSDE and evidential learning into a unified framework facilitates comprehensive modeling of both the evolution of user preferences and the inherent uncertainty in predicting user behavior.\n\nWEAKNESSES:\n1.\tWhen user-item interactions have uniform time intervals, such as in click-through scenarios, the model's time-aware approach may be less effective in capturing uncertainty.\n2.\tThe paper acknowledges the interpretability of the E-NSDE framework but falls short in providing detailed explanations of the decision-making process and incorporating feature importance analysis, which would significantly enhance transparency and build user trust.\n3.\tThe discussion regarding the generalizability of the E-NSDE framework across various domains and datasets is inadequate. Providing more detailed insights into the model's performance in diverse settings would clarify its robustness and broader applicability.\n\nQUESTIONS:\n1.\tCould the authors provide a detailed explanation of how NSDE and evidential learning are integrated within the E-NSDE framework?\n\n2.\tCan the authors elaborate on the characteristics of the real-world datasets used in their experiments? How diverse are these datasets, and do they encompass a broad spectrum of recommendation scenarios to validate the generalizability of the E-NSDE framework?\n\n3.\tAre there specific techniques or methodologies incorporated to enhance the transparency and explainability of the recommendation process?\n\n4.\tIn the context of large-scale recommendation systems, how does the E-NSDE framework address challenges related to scalability and computational efficiency? Have any experiments been conducted to evaluate the framework's scalability as dataset sizes or user interactions increase?\n\nLIMITATIONS:\nThe authors have adequately addressed the limitations and identified no potential negative societal impacts of their work.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}

comments: SUMMARY:
This work investigates sequential recommendation, and proposes a new method that utilizes stochastic differential equations (SDEs) to model dynamic time intervals and estimate uncertainty.

Overall, this study addresses an engaging problem and provides a novel and reasonable solution. Extensive experiments have been conducted. Consequently, I lean towards accept.

STRENGTHS:
1.	This work studies on an interesting and important problem --- how to capture dynamic time interval as well as estimate model uncertainty. 

2.	The paper is well-written, with clear motivations.

3.	The application of stochastic differential equations to model sequential recommendation is both novel and reasonable.

4.	Extensive experiments are conducted to validate the effectiveness of the proposal.

WEAKNESSES:
1.	It would be advantageous to include diffusion model-based sequential recommendation baselines for comparison in the experiments, such as [a1][a2], especially since SDEs have been utilized in these methods as well.

2.	The work reports performance metrics only for P@5 and NDCG@5. It would be better to include the results with different @N, particularly @20, which is commonly adopted by recent work.

3.	A discussion on the limitations and future directions of the research in Section 6 would be beneficial.

4.	There are some typos. For example, in the eq.(13), ‘BPR’->’WBPR’; In the line 316, removing ‘vs’. 


5.	Some important relate work is omitted:
[a1] TOIS’23: Diffurec: A diffusion model for sequential recommendation
[a2] NIPS’23: Generate what you prefer: Reshaping sequential recommendation via guided diffusion

QUESTIONS:
Please refer to weaknesses.

LIMITATIONS:
No concern.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-13