Flow Matching Based Sequential Recommender Model

1 Introduction

Diffusion model (DM), owned to their great ability to generate high-quality image [Nichol and Dhariwal, 2021; Song et al., 2020], video [Ho et al., 2022; Harvey et al., 2022; Yu et al., 2024] and text [Gong et al., 2022; Wu et al., 2023], has inspired the development of innovative adaptations in sequential recommendation systems (e.g., DiffuRec [Li et al., 2023] and DiffRec [Wang et al., 2023]). Usually, the diffusion-based model consists of two main phases: the forward procedure and the reverse procedure. During the forward procedure of diffusion model, i.e., the training prob. Flow Matching based Sequential Recommender Model cedure, the model progressively adds noise to the real data based on a predefined noise schedule, eventually transforming it into random noise resembling that drawn from a normal distribution. In contrast, the reverse procedure, or the inference stage, iteratively removes the noise from the sampled noise using a reverse samplers, i.e., the SDE-based stochastic reverse sampler [Nakkiran et al., 2024]. This process is typically conditioned on both the random noise and additional inputs, allowing for the generation of realistic samples.

∗Corresponding Author.

Figure 1: An illustration that highlights the differences between Diffusion-based (a) and FlowMatching based (b) sequential recommender models in both the forward and reverse processes.

Following this paradigm, methods such as DiffuRec [Li et al., 2023], DreamRec [Yang et al., 2024], and DimeRec [Li et al., 2024b] have extended the diffusion model to the sequential recommendation. Specifically, these approaches generate next-item predictions by leveraging both random noise and user-item interactions. In the forward procedure, illustrated in Figure 1(a), these models progressively add noise to the target recommendation, transforming the actual next item into random noise. After integrating the random noise and historical interactions using a deep learning model, the reverse process utilizes a stochastic reverse sampler to progressively denoise the next item from the noise-perturbed user preferences.

Although effective, existing methods exhibit several limitations: (1) Inaccurate user preference modeling: The integration of random noise during the forward process, along with user preferences, can indeed compromise the accuracy of user preference modeling. Additionally, these methods [Li et al., 2023; Wang et al., 2023] typically utilize a curved noise schedule path during the noise addition process, which can result in error accumulation due to their long-curved trajectory, as illustrated by the curve in Figure 1(a). Consequently, during the reverse inference process, existing methods need to fit these curved paths, necessitating a greater number of diffusion steps to counteract the impact of these errors effectively. While existing methods might be aware of this issue, they often address it by selecting hyperparameters that introduce minimal random noise into user preference modeling. However, this merely treats the symptoms rather than resolving the underlying problem. (2) Randomness on recommendation generation: The stochastic reverse sampler used in reverse procedures introduces randomness into the recommendation generation process, potentially resulting in irrelevant suggestions. These stochastic samplers typically introduce extra noise-perturbations during the sampling phase, yielding diverse and varied samples that are beneficial in tasks like image generation (e.g., creating different cat breeds such as Ragdolls, Persians, and Folded-ear cats). However, in sequential recommendation systems, the primary goal is to accurately predict the next likely item while exploring diverse yet relevant topics. Unfortunately, these unintended perturbations often lead to irrelevant recommendations, which can ultimately degrade the user experience. In the example shown in Figure 1(a), such perturbations can shift recommendations from "watch" to "milk" and negatively impact users' preferences on the actual platform.

Towards this end, this work firstly adapts Flow Fatching, i.e., a simplified diffusion model, for the sequential recommendation and proposes FMREC. Specifically, in the forward process, we utilize a straight flow trajectory and derive a noise-free equivalent learning target for the sequential recommendation (the theoretical analysis of straight trajectory's advantage is provided in Appendix C). Thereby, FMREC minimizes error accumulation and achieves more precise recommendations. Additionally, we introduce a decoder architecture to reconstruct users' historical preferences, along with a corresponding interaction information reconstruction loss during the training process to enhance the model's robustness against noise perturbations. To control randomness in item generation, we implement a deterministic reverse sampler using an ODE-based deterministic sampling method, effectively eliminating random perturbations during inference. Figure 1(b) illustrates our proposed method. Finally, we conduct extensive experiments on four benchmark datasets and compare FMREC with state-of-the-art (SOTA) approaches. An average improvement of 6.53% over SOTA verifies the effectiveness of our proposed methods.

2 RelatedWork

2.1 Sequential Recommendation Systems

The rapid advancement of deep learning has significantly enhanced sequential recommendation systems [Zou et al.,

2019; Tang et al., 2024] through various architectures. Early pioneering works employ Recurrent Neural Network (RNN) [Donkers et al., 2017], such as GRU4Rec [Hidasi et al., 2015], RRN [Wu et al., 2017], and Convolutional Neural Networks (CNN) [de Souza Pereira Moreira et al., 2018], including RCNN [Xu et al., 2019] and Caser [Tang and Wang, 2018], which effectively capture user preferences and immediate interests. More recent approaches have enhanced sequence modeling using self-attention mechanisms [Zou et al., 2020; Tang et al., 2025; Yu et al., 2025], particularly through the transformer architecture [Vaswani, 2017]. Models like SASRec [Kang and McAuley, 2018], BERT4Rec [Sun et al., 2019], and STOSA [Fan et al., 2022] utilize self-attention to enhance performance on user interaction data, where SASRec focuses on sequential user behavior, BERT4Rec employs bidirectional self-attention with the Cloze objective for richer feature representation, and STOSA introduces uncertainty in capturing dynamic preferences using Wasserstein Self-Attention.

These methods discussed above form the backbone of sequential recommendation and are orthogonal to our proposed approach. Our work also uses a transformer-based architecture as foundation. Ideally, our proposed method is designed to be seamlessly integrated with all of these techniques.

2.2 Generative Recommender Systems

Generative recommender systems have gained significant attention due to their ability to model complex user-item interactions and generate diverse, innovative recommendations. Early works like AutoRec [Sedhain et al., 2015] apply autoencoder to collaborative filtering, while models like AutoSeqRec [Liu et al., 2023] and MAERec [Ye et al., 2023] enhance them with incremental learning and graph representations, boosting robustness to noisy, sparse data. Variational Autoencoder (VAE) introduces probabilistic latent space for modeling sparse user-item interactions, with innovations like dual disentanglement modules [Guo et al., 2024] and hierarchical priors [Li et al., 2024a] improving interpretability and addressing sparsity. Another popular direction is the use of Generative Adversarial Network (GAN), which enhances generative recommendation through adversarial training between generators and discriminators. Combined with traditional collaborative filtering [Dervishaj and Cremonesi, 2022], GAN captures complex user preferences and integrates them with techniques like Determinantal Point Processes (DPP)[Wu et al., 2019] to improve recommendation diversity. To address challenges like mode collapse, a newer GAN model [Jiangzhou et al., 2024] incorporates diffusion model for stable training and reliable recommendations.

While these methods advance generative recommendation by a large margin, our research focuses on tailoring more advanced diffusion model to generate even more diverse and innovative recommendations.

Diffusion-BasedRecommendation The usage of diffusion model in sequential recommendation is still in its early stages but rapidly gaining attention due to their success in various generative tasks [Ho et al., 2022; Gong et al., 2022]. Pioneering efforts like [Li et al., 2023; Yang et al., 2024] add noise to the target item and leverage user interaction history implicitly in the reverse process. Based on this, Huang et al. [2024] integrates both historical interactions and target items during noise addition, using both explicit sequence embeddings and implicit attention mechanisms to boost preference representation. Meanwhile, Wang et al. [2024] harnesses a Transformer as a conditional denoising decoder, embedding historical interactions into the model via cross-attention, thereby effectively guiding the denoising and enabling the model to focus on pertinent historical interactions.

Although these approaches are effective, they overlook the biases introduced by noise in diffusion model for recommendation task. In contrast, our model focuses on mitigating the distortion of user preferences caused by perturbations in both the forward and reverse diffusion processes.

3 Preliminaries

This section provides a brief overview of the Flow Matching to establish the necessary background. Particularly, Flow Matching can be considered a simplied diffusion model, designed to construct probabilistic path between two distinct distributions, thereby enabling the transformation from simple distribution, e.g., a simple normal distribution pn = N(0, I), to the complex and unknown distribution, denoted as pc [Esser et al., 2024; Nakkiran et al., 2024]. From the perspective of diffusion model, Flow Matching can also divide into the forward and reverse procedure.

Forward Procedure In the forward procedure, the model is required to map a real data point xc ∼ pc to a noise data point xn ∼ pn. It is defined as a time-dependent flow ϕ as

where t is a random variable uniformly sampled from the interval [0, 1]. The time-dependent hyperparameters at and bt follow two constraints: if a0 = 1, then b0 = 0, ensuring ϕ0(xc|xn) = xc at t = 0, and if a1 = 0, then b1 = 1, ensuring ϕ1(xc|xn) = xn at t = 1. These ensure a transition from the target distribution xc to the normal distribution xn over time. ϕt(xc|xn) provides a concise representation of the Flow Trajectory, illustrating the manner in which states change during the Flow process. To simplify the notation, we denote the intermediate variable perturbed by noise as

To characterize the flow ϕt(·|xn), a vector field ut is employed to construct the time-dependent diffeomorphic map ϕt as follows:

where ϕ-1 t (zt|xn) represents the inverse function, which computes xc based on the perturbed noise. The notation ϕ′t denotes the differential of ϕt.

The objective of the training process is to learn and predict this vector field with a Θ parameterized vector field vΘ(zt, t) as

Reverse Procedure In the reverse process, i.e., the inference procedure, the model reconstructs xc by solving the following ordinary differential equation (ODE):

where zt is the linear combination of xc and xn. In this work, we employ a deterministic reverse sampler, i.e., the Euler method, to solve this ODE.

Remarkably, the diffusion-based recommendation models [Li et al., 2023] and [Wang et al., 2023], typically employ an SDE-based stochastic reverse sampler, which introduces the stochastic disturbances, i.e., the variance, to the reverse diffusion process. However, this stochasticity deviates from the objective of sequential recommendation, potentially resulting in irrelevant predictions when attempting to accurately identify a user's next interaction item, which ultimately undermines the user experience. Contrarily, the ODE-based deterministic reverse sampler does not introduce any random perturbations during the generation process, ensuring that the generation results meet the user's personalized preferences.

4 Methodology

4.1 Problem Statement

For the sequential recommendation task, we define a set of users U = {u1, u2, . . . , u|U|} and a set of items I = {i1, i2, . . . , i|I|}, where |U| and |I| represent the total number of users and items, respectively. Each user u ∈ U has an interaction history represented as a chronological sequence of items S = (i1, i2, . . . , im), where im corresponds to the m-th item interacted with by the user. Here, m is the length of the interaction sequence. Formally, we aim at generating the next recommendation im+1, maximizing a specific metric Θ as

4.2 FlowMatching based Sequential Recommender Model

This section offers a detailed explanation of both the forward and reverse processes in FMREC. In the forward process, our designs include the development of flow trajectories, modification of the learning target, design of the parameterized vector field, and the corresponding learning loss. In the reverse process, we discuss the specific selection and implementation of the reverse sampler.

Forward Process

In the context of sequence recommendation task, we adopt the target distribution pc in FlowMatching as follows:

where em+1 represents the embedding of the target item, generated using the equation:

Here, Embedding denotes an embedding module that transforms discrete next item ID im+1 into dense vector representation. Notably, the parameter em+1 is trainable, which is different from the traditional FlowMatching settings.

Straight Trajectory Flow and Modified Loss The timedependent hyper-parameters at and bt in the forward process define the trajectory of the generated flow, enabling the flexible selection of flow paths to control the flow process. The Straight Trajectory, characterized by at = (1 - t) and bt = t, is known for its simplicity and computational efficiency, making it widely employed in various studies [Lipman et al., 2022; Liu et al., 2022]. Following this setting, we define the time-dependent flow of the next item recommendation as follows:

where I denotes the identity matrix. Further discussions regarding its effectiveness are provided in Section 5.3. The variable t is sampled using the Mode Sampling with Heavy Tails [Esser et al., 2024] method as

where the parameter s represents a scaling factor that governs the density distribution of the time step sampling. k ∈ U(0, 1) is a random variable. Further discussions on the sampling methods and the impact of the parameter s on model performance are provided in Appendix B.

To characterize the flow ϕt(·|xn), a vector field can be defined in the following form:

where ϕ-1 t (zt|xn) represents the inverse of the flow function at time t, conditioned on xn. By substituting into the Flow Matching loss defined in Equation (4), we can reformulate the training loss as follows:

The first derivation relies on the fact xn = ϕ-1 t (zt|xn). The second equation replaces the learning target vΘ(zt, t) with (-fΘ(zt, t) + xn), leading to the modified target in Equation (13). The reason for replacing the learning target is that predicting the intricate combination of real data and noise can be difficult and detrimental to recommendation systems, leading to the recommendation of diverse but irrelevant items.

Parameterized Vector Field To parameterize fΘ(zt, t) for predicting real data, we first integrate the historical interaction sequence with the noised data and then model the noised historical interactions using a robust transformer decoder. Specifically, we combine the historical interaction sequence with the noised data as follows [Li et al., 2023]:

where ⊙ denotes element-wise multiplication, λi is sampled from a Gaussian distribution, and δ is a hyperparameter representing both the mean and variance of the distribution. The term λi is instrumental in balancing the fusion ratio between the historical interaction sequence and the noised data.

Afterwards, we employ a decoder, denoted as Decoder1, which consists of n1 layers of unidirectional self-attention based transformer, to obtain the hidden states as follows:

where h1 to hm are the intermediate outputs that will be utilized for reconstructing the interaction matrix and calculating the reconstruction loss. Next, we model fΘ(zt, t) by taking the LAST token output from an additional n2-layer decoder, denoted as Decoder2, as follows:

Regularized Losses The modified FlowMatching loss presented in Equation (13) is not entirely suitable for recommendation task because the vector xc is trainable rather than fixed. This might lead to the problem that different xc converge to a single embedding, which is obviously meaningless. To mitigate this issue, we introduce a cross-entropy loss LCE as a regularizer that differentiates between various item embeddings, thereby preventing the aforementioned problem as:

where ˆ ym+1 is the normalized score of recommending im+1.

Furthermore, to mitigate the detrimental effects of noise perturbation on model performance, we incorporate an additional reconstruction loss that interprets the user's history from the final token of the hidden representation hm as follows:

where ˆ d ∈ R|I| is the predicted user-item interaction information. ω represents the parameters of the MLP-based Decoderω. We optimize these parameters using a Mean Squared Error (MSE) loss, enabling the generation of more accurate and personalized recommendations as follows:

where r ∈ R|I| is a binary vector, with 1 indicating an interaction between the user and the item, and 0 indicating no interaction.

Finally, the loss function utilized during training is formulated by integrating three distinct components, represented as follows:

where α and β are hyperparameters that govern their relative importance. A more detailed training procedure of FMREC is presented in Appendix A.

Reverse Process

In the reverse process, we employ a deterministic reverse sampler to model the generative process, thereby mitigating the errors introduced by the stochastic reverse sampler in the diffusion-based recommendation system. Specifically, the deterministic reverse sampler is defined as the updating function for the following ordinary differential equation:

To solve this equation, we iteratively apply the Euler method to compute the point transformations guided by the vector field, as given by

where ∆t is determined by a custom number of Euler method steps, denoted as q, with ∆t = 1q . This process is iteratively calculated from t = 0 to t = 1, resulting in z1 as the fully denoised generation ˆxc, which represents the corresponding item embedding. The inference procedure of FMREC is presented in Appendix A.

5 Experiment

This section presents comprehensive experiments to demonstrate the effectiveness of FMREC.

5.1 Experimental Settings

Dataset We evaluate FMREC's effectiveness using four widely recognized publicly available datasets: (1) Amazon Beauty [Ni et al., 2019] contains global purchasing interactions and user reviews for beauty products on the Amazon platform, documenting the purchasing history of 22,361 users across 12,101 products. (2) Steam is a leading PC game distribution platform with 3,480,000 interaction records from 280,000 gamers; (3) Movielens-100k [Harper and Konstan, 2015] is a widely used benchmark dataset in sequential recommendation research, providing ratings from 943 users on 1,682 movies from the Movielens platform; and (4) Yelp is a popular review site featuring user reviews and 1,764,589 ratings for various businesses, including restaurants, entertainment venues, and hotels. The statistics of the dataset are provided in Table 1.

Baselines We compare FMREC against both widely adopted and recently introduced baseline models: (1) GRU4Rec utilizes GRU to capture in-session behavioral patterns and predict subsequent user item preferences [Hidasi et al., 2015]; (2) Caser leverages convolutional neural networks to map user action sequences into both temporal and latent spaces [Tang and Wang, 2018]; (3) SASRec introduces a decoder architecture for sequential recommender model that effectively captures long-term dependencies in user behavior [Kang and McAuley, 2018]; (4) BERT4Rec employs a bidirectional Transformer architecture, coupled with a Cloze task for training, to effectively learn users' dynamic preferences [Sun et al., 2019]; (5) STOSA utilizes Wasserstein attention to introduce a degree of uncertainty within its model, allowing for the accurate representation of evolving user preferences [Fan et al., 2022]; (6) AutoSeqRec leverages a multiautoencoder framework to fuse long-term user preferences and short-term interests for sequential recommendation [Liu et al., 2023]; (7) DreamRec achieves direct generation of personalized oracle item embeddings through guided diffusion model [Yang et al., 2024]; (8) DiffuRec models items as distributions using diffusion model, capturing multi-faceted content and user preferences [Li et al., 2023].

Evaluation Protocol Following the procedures in [Sun et al., 2019; Li et al., 2023], we split user interaction sequences into three parts: the first m-2 sequences formed the training set, while im-1 and im served as targets for the validation and test sets, respectively. We evaluated performance using the metrics HR@K (Hit Rate@K) and NDCG@K (Normalized Discounted Cumulative Gain@K). Each baseline model generates a ranked list of items predicted for the next interaction based on user history, considering all dataset items as candidates, with K values set at {5, 10, 20}.

ImplementationDetails The implementation details are as follows: both Decoder1 and Decoder2 consist of 2 layers, with a hidden dimension of 128 and 4 attention heads. The item embedding dimension is also set to 128. The Decoderw is a three-layer MLP with tanh activations, featuring layer sizes of {128, 512, 2048, |I|}, mapping 128dimensional decoder outputs to the number of items. Hyperparameters include a batch size of 512, a learning rate of 0.001, and a maximum user interaction sequence length of 50. The loss weighting parameters α and β are set to 0.2 and 0.4, respectively. The scaling parameter s in the timestep schedule is set to 1.0. Besides, we use 30 Euler integration steps for generation. All experiments are conducted on a server with two Intel XEON 6271C processors, 256 GB of memory, and four NVIDIA RTX 3090 Ti GPUs.

5.2 Overall Performance Comparison

The experimental results and performance comparisons with baseline models are presented in detail in Table 2. From the table, we have the following observations: (1) The FMREC model shows a notable advantage over existing SOTA methods across four benchmark datasets. Significant improvements are consistently observed in the Beauty, Yelp, Movielens-100k, and Steam datasets with HR increasing by 5.5 - 8.9% and NDCG rising by 3.3 - 9.9%; (2) FMREC can effectively eliminate the negative influence of noise perturbations associated with diffusion-based recommendation methods. Specifically, we observe that FMREC outperforms both DreamRec and DiffuRec, achieving superior results with an average improvement of 12.44% on HR@5, which confirms that the deterministic reverse sampler and regularized loss are beneficial for generating more accurate recommendation. (3) Generative models demonstrate greater effectiveness in capturing user preferences and providing enhanced recommendations. Particularly, DreamRec, DiffuRec, and FMREC show marked improvements in performance over SOTA traditional sequential recommender models by large margins of 13.7%, 27.93%, and 34.98% on HR@5.

5.3 Analytical Experiment

This section evaluates the effectiveness of each design option in FMREC and analyzes the impact of hyperparameter configurations on model performance. We present results of Beauty and Movielens-100k as examples. Additional results are available in Appendix B.

Influence of Flow Matching Loss In this work, we have modified the Flow Matching loss from directing predicting the overall vector field, i.e., Equation (11), to the modified vector field, i.e., Equation (13). To demonstrate its effectiveness, Table 3 presents the comparison between the model using the modified loss and the naive loss (v-prediction in Table 3) from the Flow Matching loss. From the table, we observe a notable performance drop when using the naive Flow Matching loss. This suggests that directly predicting the vector field results in inaccurate modeling of user preferences, which significantly undermines the model's performance.

Effectiveness of Straight Trajectories To demonstrate the effectiveness of straight trajectories, we use the Cosine trajectory as a baseline for comparison. This trajectory has been employed in IDDPM [Nichol and Dhariwal, 2021] to achieve superior performance compared to DDPM [Ho et al., 2020]. In Figure 3, we present the results for both the Cosine trajectory (denoted as "Cosine") and straight trajectories (FMREC) on the Beauty and Movielens-100k datasets. The figure reveals a noticeable performance drop when using the Cosine trajectory, highlighting that straight trajectories improve robustness against error propagation. This robustness facilitates faster convergence to optimal results with fewer iterations.

Effectiveness of Regularized Lossesd FMREC incorporates two regularized loss functions to enhance the model's ability for recommendation. This section analyzes its effectiveness by examining the influence of the parameters α and β, as illustrated in Figure 4. Specifically, the figure displays the model's performance when trained with various combinations of α and β on the Beauty dataset. From Figure 4, we observe that an optimal setting of α = 0.2 and β = 0.4 yields the best performance. Low values of α can significantly impact model performance, while excessively high values of α can also adversely affect it to some extent. In contrast, choosing an appropriate β can enhance the overall performance of the model. These results indicate that both regularized loss functions are crucial for maintaining the effectiveness of the proposed FMREC model.

Influence of Noise Perturbation Before feeding the user's sequential interactions into the model, FMREC incorporates noise to meet the requirements of Flow Matching, where δ controls the fusion ratio's influence. To examine the impact of noise perturbation, we conduct experiments with δ ∈ {0.1, 0.01, 0.001}. Figure 5 presents the performance comparison across different δ values. From the figure, we observe that the best performance is achieved when δ was set to 0.001, demonstrating that an appropriate level of noise perturbation enhances performance. Conversely, too large noise perturbation might lead to deviations in the calculation of the vector field during the reverse process and negatively impact the generation performance.

6 Conclusion

In conclusion, this work introduces a sequential recommendation model using Flow Matching, a simplified diffusion model that effectively mitigates noise perturbations of the diffusion-based model. By employing a straight flow trajectory and deriving a noise-free training target, our method reduces error accumulation in modeling user preferences. Additionally, the integration of a decoder architecture with an interaction reconstruction loss increases robustness against noise, ensuring precise user preference modeling. Furthermore, our deterministic reverse sampler, utilizing the Euler method, removes random perturbations during recommendation generation. Extensive experiments on four benchmark datasets demonstrate that our method, FMREC, achieves an average improvement of 6.53% over SOTA techniques.

Acknowledgments

We express our sincere gratitude for the financial support provided by the National Natural Science Foundation of China (No. U23A20305, NO. 62302345), the CCFALIMAMA TECH Kangaroo Fund (NO. CCF-ALIMAMA OF 2024009), and the Natural Science Foundation of Wuhan (NO. 2024050702030136), Innovation Scientists and Technicians Troop Construction Projects of Henan Province, China (No. 254000510007), National Key Research and Development Program of China (No. 2022YFB3102900), the Xiaomi Young Scholar Program.