Many studies have focused on the complementary item recommendations. Embedding-based methods, such as Barkan and Koenigstein (2016) and Wan et al. (2018), collaboratively learn the complementary item relationship from the co-purchase data. Another way of using the co-purchase data is to construct the co-purchase graph and apply graph neural networks on it (McAuley et al., 2015; Wang et al., 2018; Liu et al., 2020). They use the co-purchase records as labels for link predictions based on the distance between item embeddings. In addition to vector item embeddings, Gaussian embedding is also explored in Ma et al. (2021) to address the noise in the co-purchase data for better complementarity learning. In addition to the co-purchase data, many types of auxiliary data are incorporated into the modeling, such as the multimodal data of items (Zhang et al., 2018) and the shopping context (Xu et al., 2019). Diversified complementary recommendation is studied in Hao et al. (2020) by leveraging the product-type information to improve the diversity. However, it focuses on the diversified recall process rather than the ranking process as our article targets.

In our study, we leverage the triple2vec in Wan et al. (2018) to learn the complementary item embedding due to its effectiveness in learning the item vector embeddings from the co-purchase data.

For a long time, not much importance was given to diversity in the recommendations, as it is challenging to achieve both high accuracy and diversity at the same time. This is called accuracy diversity dilemma (Liu et al., 2012). Novelty and diversity of items have been improved by penalizing accuracy (Díez et al., 2019). Diversity has also been captured in an entropy regularizer (Qin and Zhu, 2013). Post-processing methods for diversity have been proposed to improve the personalized recommendations generated by collaborative filtering (Adomavicius and Kwon, 2012). Determinantal point process (DPP) has been used for making personalized diversified recommendations and DPP models are probabilistic models with a lot of applications (Kulesza and Taskar, 2012). They have been incorporated with a tunable parameter allowing the users to smoothly control the level of diversity in recommendations and also, applied to large-scale scenarios with faster inference (Wilhelm et al., 2018). Deep reinforcement learning has utilized DPP to promote diversity to generate diverse, while relevant item recommendations. DPP kernel matrix is maintained for each user, which is constructed from two parts: a fixed similarity matrix capturing item-item similarity and the relevance of items dynamically learnt through an actor-critic reinforcement learning framework (Liu et al., 2019). However, they fail to give much attention to maintaining the delicate balance between the requirement of distinct diversity strategies for the exploratory and conventional shopping intents. Our proposed method focuses on the combined re-ranking strategy for exploratory and conventional user shopping intent on complementary recommendations.

Skip-gram-based methods for item embedding leverage the item co-occurrence signal (e.g., co-purchase of items). Models for complementary item recommendations such as McAuley et al. (2015) and Barkan and Koenigstein (2016) exactly use the item co-occurrence signal to model item complementarity. triple2vec in Wan et al. (2018) introduced the cohesion of (item, item, and user) triplets that reflect the co-purchase of two items by the same user in the same basket. This technique improves the performance of complementary item recommendations and triple2vec achieves the state-of-the-art performance. As we focus on the post-processing of the recommendations, we decide to leverage the item representations learned by triple2vec to generate item pools for downstream applications.

In triple2vec, a triplet of (q, r, u), q ∈ V, r ∈ V, u ∈ U, represents the user-item and the item-item relationship, where V is the set of items and U is the set of users. Here, q and r are two items purchased by the user u in the same basket. Particularly, we refer q to the query item and r to the recommended item. The relationship between q and r can be viewed in the way that r is the recommended complementary item for the query item q. The cohesion of (q, r, u) in triple2vec is computed by Equation (1), where f_q, g_r are two sets of representations for items (q, r) and h_u is the user embedding.

x and y in Equation (1) indicate the item-to-item complementarity and user-to-item compatibility, respectively. The loss function L in Equation (2) computes the likelihood of all possible triplets T and is optimized to learn representations of items and users.

Here, p(r|q,u)=exp(q,r,u)∑r′exp(q,r′,u), p(q|r,u)=exp(q,r,u)∑q′exp(q′,r,u), and p(u|q,r)=exp(q,r,u)∑u′exp(q,r,u′).

We leverage triple2vec to learn item representations and generate item pools of complementary recommendations for downstream processes. To recall the item pool of complementary recommendations, we consider the inner product score fqTgr for two items q, r. For each query item q, we select a pool of items R = {r₁, …, r_m} with the highest score of fqTgr.

Improving the diversity of recommendations benefits the recommender systems because it introduces novelty and better topic coverage (Ziegler et al., 2005). Many studies on diversification follow the setting of bi-criterion optimization problem, which balances the relevance (between the query and recalled elements) and diversity (Wu et al., 2019). Particularly, diversity can be further divided into two types, (1) individual diversity ¹ and (2) aggregate diversity ² (Wu et al., 2019). We focus on the individual diversity in this study to adjust the diversity of complementary recommendations, given a user's intent.

The determinantal point process (DPP) is a probabilistic model that is good at modeling repulsion. The recent study (Chen et al., 2018) applies DPP to diversification of item recommendations and develops the fast greedy MAP inference to generate diversified recommendations. Our study is based on DPP with the fast greedy MAP inference in Chen et al. (2018). We introduce details of DPP and the fast greedy MAP inference following the notation in Chen et al. (2018). For the rest of our article, we denote the fast greedy MAP inference as FG-MAP.

Formally, the DPP on a discrete set Z = {1, 2, …, M} is a probability measure P on 2^|Z| number of subsets of Z, where |Z| is the number of elements in Z. Because the empty set is also a subset of Z, when P does not give zero probability to the empty set, there exist a square, positive semidefinite (PSD) and real matrix L ∈ ℝ^M×M, which satisfies Equation (3) for each subset Y ⊆ Z.

L serves as a kernel matrix indexed by the elements in Z and det(L_Y) is the determinant of the sub-matrix extracted from L based on elements in Y. Equation (3) indicates that the probability of a subset Y is proportional to the determinant of the corresponding sub-matrix of the PSD kernel. The MAP inference of the aforementioned DPP P on Z is defined in Equation (4).

Unlike the other inference on DPP, the MAP inference of DPP is NP-hard. The algorithm FG-MAP approximates the MAP inference in a greedy approach. Equation (5) shows how to greedily select the next candidate item j that is added to the existing growing subset Y_g ⊆ Z built from the previous iterations. After the current iteration, Y_g grows and Y_g: = Y_g ⋃ {j} ³.

When Z becomes the item pools for complementary recommendations R = {r₁, …, r_m} recalled by the item representations (i.e., item embedding learned by triple2vec), the DPP on R maximizes the P(Y) and diversifies the recommendations by selecting r_i from R iteratively. Now, the kernel matrix L could be initialized by the item-to-item similarity matrix based on the item embedding. In our study, we adapt DPP and FG-MAP, with L defined in Equation (6).

H is a sub-matrix of the item embedding for the item pools R recalled by the triple2vec model. g_ri is normalized embedding of item r_i and the value of H^TH is shifted to ensure L is PSD. We only use one set of items embedding from the triple2vec model to compute item similarity, as the distance between f_q and g_r from two sets of embedding represents the complementarity of (q, r).

The heterogenization strategy for complementary item recommendation can be achieved by increasing the diversity in the complementary recommendations R recalled by a complementary item recommendation model, i.e., triple2vec. It could fulfill the users' intent on exploratory shopping by showing more diverse recommendations. We first generate R to ensure complementary recommendations and then re-rank items in R to surface more diverse but relevant items to the top. If we do not conduct diversification within the pool of pre-selected complementary items, the diversification logic could easily bias irrelevant items. We can re-rank the items in R by modifying FG-MAP into bi-criterion optimization. Specifically, we consider the score Sq,ri=1+fqTgri2 for complementarity of (q, r_i), where f_q and g_ri are normalized item embeddings. Equation (7) shows the modified objective function for diversification re-rank.

At tth iteration, R_t,d: = R_{t−1, d} + [r_j], where R_0,d = [] and R_{t − 1, d} + [r_j] means the newly selected recommendation r_j by the diversification re-rank is inserted at the end of the current item list R_{t−1, d}. The weight α controls the amount of diversity introduced to the re-ranked item list. Each selected item r_j can maximize the combined score of diversity and complementarity. The re-ranked item list R_d will surface more diversified recommendation to the top compared with the original item list R, in which items are simply sorted by the score S_q,ri in descending order.

The homogenization strategy is different from the heterogenization strategy. We need to surface more items that are related to the query items but under the same topic, instead of diverse results. For example, assume a query item milk has a list of recommendations R = {eggs, cheese, bread, margarine, banana, sausage, yogurt, cereal}. If we want to address the homogenization strategy, the re-ranked recommendations could be R_s = {eggs, cheese, margarine, yogurt, banana, bread, sausage, cereal} ⁴. We encourage more homogeneousness in this strategy while keeping the complementary relationship between recommendations and the query item. R_s surfaces more items under the Dairy & Eggs domain such as cheese and yogurt. The homogenization strategy can be promoted by the similarity of items among the recall set of complementary recommendations. Unlike diversification for the heterogenization strategy which is diverging the item relationship, boosting the homogeneity in the recommendations is more stable. We can mine candidate items in a bigger recall set. Formally, we recall extra complementary items R_x = {r_m+1, …, r_n} and insert them at the end of R. The new item list becomes R + R_x. To force the similarity between recommendations, we modify the kernel L in DPP by Equation (8) and apply DPP to the new dissimilarity matrix L′,

Where diag(L) is a diagonal matrix with all entries in the main diagonal equal to the diagonal of L and 1 is a square matrix with all entries equal to 1. Plug L′ into Equation (7), and we can have a new re-ranking logic on the extended item pool R + R_x, shown in Equation (9).

Here, the parameter β is used to control the degree of similarity between recommendations. At tth iteration of Equation (9), R_t,s: = R_{t − 1, s} + [r_h], where R_{0, s} = []. Both Equations (7), (9) can be optimized by the FG-MAP algorithm mentioned in Chen et al. (2018)⁵.

Only having two re-ranking strategies is not enough because we need to figure out the selection of two diversified re-ranking strategies for a certain user. We leverage the heuristic that users who usually add more diverse items during the next-k purchases would prefer more exploratory complementary shopping with the heterogenization strategy, while users who commonly add less diverse items during the next-k purchases would prefer more conventional complementary shopping with the homogenization strategy.

Formally, given a query item q at time t and a list of next-k items B_q = {b_t+1, …, b_t+k} purchased by a user u, we leverage the taxonomy information tax(·) ⁶ to estimate how much diversity the user u prefers. Let B_T,q = [tax(b_t+1), …, tax(b_t+k)] be the list of departments of the next-k items purchased by the user and |B_T,q| be the number of unique elements in B_T,q. We can estimate the degree of diversity for the query item q and the user u in Equation (10).

However, the score z_u,q is at user-item level and not stable due to the sparsity issue. We then extend it to a score at user-department level to reduce the sparsity, as shown in Equation (11), where dept_i is the department i and the score z_{u, depti} is an average of score z_u,q for any query items satisfying tax(q) = dept_i.

We treat the score z_{u, depti} as the user intent score of exploratory complementary shopping and apply a threshold T ∈ [0, 1] to binarize z_{u, depti} learnt from the training data. If z_{u, depti} < T, the user u prefers the more conventional complementary shopping under the department dept_i, otherwise, the user u might prefer more exploratory complementary items because u tends to add items from different departments during the next-k purchases. We can combine the score z_{u, depti} with the heterogenization and homogenization strategies to develop a dynamic re-ranking algorithm for complementary item recommendations (summarized in Algorithm 1), It provides either diversified re-ranking strategy based on the user intent on a certain department dept_i of the query item q.

We add z₀ as a default value for cold departments of query items which are not seen in the history. z₀ could be initialized by the average of all z_{u, depti}.

The Instacart dataset (Instacart, 2017) has 49,677 distinct items, 134 distinct aisles, 21 distinct departments, and 206,209 distinct users. We train the triple2vec model on the Instacart training dataset, with an embedding dimension of 100, a batch size of 128, an initial learning rate of 0.05, and a stochastic gradient descent optimizer. We also compute z_{u, depti} for each pair of (u, dept_i) for the next-5 purchase (k = 5 in Equation 10). When evaluating the re-ranking strategies, we compare the results before and after the re-rank. Given a query item q, a user u, and recommendations R, R_x generated by the triple2vec model, we compute the Hit-Rate@5 and Normalized Discounted Cumulative Gain (NDCG@5) for raw complementary recommendation R and the re-ranked complementary recommendations by (1) heterogenization only, (2) homogenization only, and (3) combining heterogenization and homogenization strategies with user intent scores dynamically on the task of next-item prediction. The reason why we focus on the next-5 purchase is because the user intent might last for a short period and we want to study the impact of two different complementary recommendations on the top recommendations. If we consider bigger k, it is likely to introduce diversity in recommendations. Here, we define R = {r₁, r₂, r₃, r₄, r₅} and R_x = {r₆, r₇, r₈, r₉, t₁₀} to cooperate the metrics of Hit-Rate@5 and NDCG@5.

Since it is a novel study on exploratory vs. non-exploratory user behaviors for complementary item recommendations, it is hard to find proper baselines. We choose three baseline models for comparison. (1) As aforementioned, we only consider the transaction data due to its high accessibility, we consider the raw recommendations from triple2vec for pure complementary item recommendations. (2) The second baseline model is the diversified recommendation by DPP for the pure heterogenization strategy. (3) Similarly, we use T = 1 to force homogenization and generate our third baseline model for comparisons.

To further understand the trade-off between heterogenization and homogenization strategies, we evaluate the combined strategy with T ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}. We use α = β = 0.01 for evaluations.

We evaluate our re-ranking strategies on the Instacart evaluation dataset and the detailed results are shown in Table 1. The heterogenization strategy improves the Hit-Rate@5 and NDCG@5 compared with the raw recommendations, while only using homogenization strategy reduces the performance. Combining both re-ranking strategies together with a proper T improves the overall performance. Particularly, T = 0.2 achieves the best Hit-Rate@5 with and T = 0.1 achieves the best NDCG@5. This result is reasonable because T = 0.2 means, on average, users purchase the next-5 items under the same department. The evaluation results show a better performance for covering users who prefer conventional complementary shopping with the homogenization strategy.

Note that, only applying the homogenization strategy reduces both Hit-Rate@5 and NDCG@5. It might be because only showing complementary recommendations in a narrow scope is likely to miss users' interests (see Section 5.3 for more details). If a user is not interested in the first recommended item, this user will likely not be interested in the following recommendations because they are similar. The heterogenization strategy improves this feature by surfacing different complementary items to the top. Now, the re-ranked recommendations are more likely to hit this user's interests. Combining these two strategies together actually covers the requirements of exploratory and conventional complementary shopping intents from users.

In summary, our results show that combining two strategies dynamically improves the overall performance, compared with only using a single diversification strategy.

To further indicate the necessity of personalized diversification strategies for complementary item recommendations, we visualize the distribution of user intent score z_{u, depti} by departments. Figure 3 summarizes the distribution. We can see that for a given department, the user intent scores distribute differently. For example, the majority of the user intent scores in Deli and Produce departments follow in the range [0.3, 0.6], which indicates that users tend to shop more diverse items when the query items are from these departments. Dairy and Beverage have similar results such as Deli and Produce. However, for departments such as Household and Pantry, the majority of the user intent scores are in the range [0.1, 0.4]. Compared with other departments, the users tend to purchase more homogeneous items when the query items are from Household and Pantry, which is reasonable because these departments usually cover most of the department-related shopping demands and correlate less with other departments. The aforementioned observation addresses the requirement of tuning the diversification of complementary item recommendations with personalization.