Evidential Stochastic Differential Equations for Time-Aware Sequential Recommendation

Authors: krishna prasad neupane
ervine  zheng
qi  yu
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-09 03:25:14

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.

Main paper: 
 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. [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. 
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. 
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:f Θ (u t , i t ) = γ (ut,it) , ν (ut,it) , α (ut,it) , β (ut,it)where γ (ut,it) is the recommendation score for item i assigned by user u, ν (ut,it) and α (ut,it) are the model evidence, and β (ut,it) 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:p(y n |µ, σ 2 ) = N (µ, σ 2 ), p(µ|γ, σ 2 ν -1 ) = N (γ, σ 2 ν -1 ), p(σ 2 |α, β) = Inv-Gamma(α, β)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):p(µ, σ 2 |m) = β α √ ν Γ(α) √ 2πσ 2 1 σ 2 α+1 exp - 2β + ν(γ -ν) 2 2σ 2Given a NIG posterior, one can derive the mean (E[µ]), aleatoric (E[σ 2 ]) and epistemic (Var[µ]) uncertainty as:EQUATIONNeural 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:EQUATIONwhere 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:EQUATIONNeural stochastic differential equations (NSDE). We could view Stochastic Differential Equations (SDE) as an ODE with infinitesimal noise added throughout time:EQUATION) 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:h(T ) = h(t 0 ) + T t0 f (h(t), t; ψ)dt + T t0 g(h(t), t; ω)dB t (5)4 Time-Aware Sequential RecommendationsOverview. 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.  
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):EQUATIONwhere 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. 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:i(T ) = i(t 0 ) + T t0 f (i(t), t; ψ)dt + T t0g(i(t), t; ω)dB t (7) where, i(t 0 ) represents item's initial representationWe 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 (ut,it) 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 N LL [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 byEQUATIONwhere, m(u t , i t ) = (γ (ut,it) , ν (ut,it) , α (ut,it) , β (ut,it) ) are model parameters at time t, andp(r (ut,it) |m(u t , i t ))= St(r (ut,it) ; γ (ut,it) , β (u t ,i t ) (1+ν (u t ,i t ) ) ν (u t ,i t ) α (u t ,i t ) , 2α (ut,it)) 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:EQUATIONThe regularized EDL loss for each sequential update is:EQUATION)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 (ut,it) 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 (ut,it) 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 (ut,it) , and ν (ut,it) of the evidential monotonic network decreases monotonically. Proof. The epistemic uncertainty equation from Eq (1):U (ut,it) = Var[µ] = β (ut,it) ν (ut,it) (α (ut,it) -1)Given the increase β (ut,it) from Lemma 1 and decrease in α (ut,it) and ν (ut,it) 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 (ut,it) , α (ut,it) , ν (ut,it) ) with a soflplus activation and adding 1 to α (ut,it) since α (ut,it) > 1).Interpreting hyper-parameters. Besides serving as the parameters of the evidential prior distributions, the hyper-parameters (ν (ut,it) , α (ut,it) , β (ut,it) ) offer very intuitive meanings. First, both ν (ut,it) and α (ut,it) 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 (ut,it) 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: r(ut,jt) = γ (ut,jt) + ηU (ut,jt) (11) 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:w (it,jt) = max[Sim(f_emb,j_emb)], if r(ut,jt) > τ min[Sim(f_emb,j_emb)], otherwisewhere f _emb, j_emb are future and negative item embedding, respectively. We then formulate weighted BPR loss utilizing a negative log-likelihood function as:EQUATIONwhere r(it,jt) = r(ut,it) -r(ut,jt) , 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:EQUATIONwhere ζ represents the balancing factor between EDL and WBPR loss. Training and inference details are provided in Appendix D. 
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-100K2 : 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 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] , S3 -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 top-N 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 top-N 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 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 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: {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: 2 , where C is the number of categories and P (c), c ∈ [1, C] is the probability for each category.G = 1 - C c=1 P (c)• 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. 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. We evaluate 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. 
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. 
 
 Appendix: In this section, we provide proofs of the theoretical results in the main paper.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 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.Given the user final representation u t , item final representation i t , and interaction time gap ∆t. The monotonic network produces β (ut,it) as:β (ut,it) = (W ) concat(u t , i t , ∆t) = W T u W T i W ∆t u t i t ∆t . Hence ∂β ∂(∆t) = W ∆t ≥ 0.Given the same setup as Lemma 2, the monotonic network outputs α (ut,it) , and ν (ut,it) . We first consider α (ut,it) :α (ut,it) = [h L • g • h L-1 • g • • • • • g • h 2 • g • h 1 ](concat(u t , i t , ∆t)), where h i (x) is the transformation induced by the ith 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∇ concat(ut,it,∆t) α (ut,it) =   W T u W T i W T ∆t   W T 2 W T 3 • • • W T L-1 W T L ⊙ g ′ (h L-1 ) ⊙ g ′ (h L-2 ) • • • ⊙ g ′ (h 2 ) ⊙ g ′ (h 1 ) ,where ⊙ is element-wise product of two matrices with the same size. The matrix  W T u W T i W T ∆t   is just W T 1 ,where we rewrite W T 1 the same blocks as that in Lemma 2. Therefore,∂α (ut,it) ∂(∆t) = W T ∆t W T 2 W T 3 • • • W T L-1 W T L ⊙ g ′ (h L-1 ) ⊙ g ′ (h L-2 ) • • • ⊙ g ′ (h 2 ) ⊙ g ′ (h 1 ) .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 (u t ,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.MovieLens-1M Amazon Book P@10 P@20 N@10 N@20 P@10 P@20 N@10 N@20 BERT4Rec [35] 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.Require: Hyperparameters: λ, η, ζ, α, β, γ, ν Initialize both NSDE and EDL modules:Θ = (ψ, ω, θ, ϕ) while not converge do Sample train user T u from user pool U for all u ∈ T u do Compute user and item final representations using Eq (6) and Eq (7) respectively from SDE module.Compute interaction time gap ∆t with target item Compute EDL loss for each sequence using Eq (10) . Compute weighted BPR loss for each sequence using Eq (12) . Perform end-to-end update using overall loss Eq (13) end for end whileHigher order top-N results. We report the results using top-N 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. 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.For the source code, please click this link: https://github.com/ritmininglab/ENSDEhttps://grouplens.org/datasets/movielens/100k/https://grouplens.org/datasets/movielens/1M/ 


Decision: Very slightly on the positive side according to review scores, though arguably mostly saved by one reviewer. Reviewers find the problem interesting, straightforward, and reasonably convincing in terms of experiments. Reviewers complain about interpretability issues, lack of metrics (and other experimental details), and some problems with the motivation.

All of the reviews (including the most positive one) aren't particularly detailed, so it's hard to be confident of the recommendation. Not that that's the authors' fault.

Reviews: review: paper_summary: paper_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.
strength_weakness: strength_weakness: 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: questions: Please refer to weaknesses.
limitations: limitations: No concern.
paper_summary: paper_summary: The 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.
It 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.
strength_weakness: strength_weakness: 1.	When 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.
2.	The 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.
3.	The 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.
questions: questions: 1.	Could the authors provide a detailed explanation of how NSDE and evidential learning are integrated within the E-NSDE framework?

2.	Can 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?

3.	Are there specific techniques or methodologies incorporated to enhance the transparency and explainability of the recommendation process?

4.	In 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?
limitations: limitations: The authors have adequately addressed the limitations and identified no potential negative societal impacts of their work.
paper_summary: paper_summary: This 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.
strength_weakness: strength_weakness: 1. 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.

2.  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.

3. 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.
questions: questions: 1. Please refer to weakness 1, and give more clarification on Table 2.

2. 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.

3. Pleas refer to weakness3, more n in top-n results should be given.



[1]Time interval aware self-attention for sequential recommendation,WSDM20

[2]Learning Graph ODE for Continuous-Time Sequential Recommendation,TKDE

[3]Temporal Conformity-aware Hawkes Graph Network for Recommendations,Webconf 24
limitations: limitations: Yes
paper_summary: paper_summary: The 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.
strength_weakness: strength_weakness: 1.	There 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). 

2.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.

3.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.
questions: questions: 1.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".
2.See weeknesses
limitations: limitations: N/A

Comments: comment: Dear Reviewer V8wq,

Thank you again for reading our responses and providing quick feedback. We would like to clarify that we have clearly described the fundamental differences between the three mentioned related works and ours. We have also experimentally compared the most relevant method i.e. TiSASRec, with our proposed ENSDE model and the result is shown in the table of the general response. We will appreciate if you could be more explicit about your remaining concerns about the related work.
Thank you for your response; I will keep my score due to concerns about related work.
Dear Reviewer zE4M,

Thank you once again for your insightful comments and questions. In our rebuttal, we have:

- provided a comparison with the suggested baselines (STOSA, DuoRec, and GDERec).

- compared the paper "Uniform Sequence Better" with our work, focusing on the varying uniform time intervals of user interactions.

- offered a detailed explanation of the metrics used.

We believe that addressing your comments will improve the paper's readability, and we appreciate your support. We hope you find our responses satisfactory and consider updating the score accordingly. We are more than happy to address any further questions you may have.
Dear Reviewer V8wq,

Thank you again for your insightful comments and questions. In our rebuttal, we have:

- clarified the motivation behind the paper and emphasized the importance of exploring new tastes for users who have not been active for a long time.

- explained the evolving nature of user behavior and how the drift and diffusion components of the NSDE module help capture these changes.

- incorporated more top-n results on both datasets with competitive baselines.

- explained relationships between suggested papers and further included one paper i.e. TiSASRec as a baseline comparison.

We believe that addressing your comments will enhance the paper's readability, and we appreciate your support. We hope you find our responses satisfactory and consider updating the score accordingly. We are more than happy to address any further questions you may have.
Dear Reviewer LeEA,

Thank you once again for your thoughtful comments and questions. In our rebuttal, we have:

- explained the impact of capturing uncertainty when user interactions occur at uniform time intervals,

- provided a detailed explanation of the interpretability, generalizability, and transparency aspects of our work, specifically focusing on the two key components of the ENSDE model.

- offered a thorough discussion on handling sparse datasets and the diverse settings of our experiments.

We believe that addressing your suggestions has significantly strengthened our paper, and we appreciate your support. We hope you find our responses satisfactory and consider updating the score accordingly. We are more than happy to answer any additional questions you may have.
Dear Reviewer p7WH,

Thank you once again for your insightful comments and questions. In our rebuttal, we've clarified the key differences of suggested baselines (DiffuRec and DreamRec) from our work and further included DiffuRec in baseline comparison with higher ranking metrics. We believe that addressing your feedback has significantly improved and strengthened our paper. We hope you find our responses satisfactory and are more than happy to address any further questions you may have.