Gradient-Based Diversity Optimization with Differentiable Top-$k$ Objective
Authors: Tianyi Zhou
Sebastian Dalleiger
Ece Calikus
Aristides Gionis
Venue: ICLR 2026 Poster (2026)
Keywords: Diversity Optimization
Gradient-based learning
Recommendation
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: We introduce a differentiable top-k diversity objective with direct and indirect optimization, showing fine-tuning quickly adds diversity at scale with negligible accuracy loss.

Abstract: Predicting relevance is a pervasive problem across digital platforms, covering social media, entertainment, and commerce. However, when optimized solely for relevance and engagement, many machine-learning models amplify data biases and produce homogeneous outputs, reinforcing filter bubbles and content uniformity. To address this issue, we introduce a pairwise top-k diversity objective with a differentiable smooth-ranking approximation, providing a model-agnostic way to incorporate diversity optimization directly into standard gradient-based learning. Building on this objective, we cast relevance and diversity as a joint optimization problem, we analyze the resulting gradient trade-offs, and propose two complementary strategies: direct optimization, which modifies the learning objective, and indirect optimization, which reweights training data. Both strategies can be applied either when training models from scratch or when fine-tuning existing relevance-optimized models. We use recommendation as a natural evaluation setting where scalability and diversity are critical, and show through extensive experiments that our methods consistently improve diversity with negligible accuracy loss. Notably, fine-tuning with our objective is especially efficient, requiring only a few gradient steps to encode diversity at scale.

Published as a conference paper at ICLR 2026
GRADIENT-BASED DIVERSITY OPTIMIZATION WITH DIFFERENTIABLE TOP-k OBJECTIVE
Tianyi Zhou' Sebastian Dalleiger' KTHH Royal Institute of Technology. Stockholm, Sweden Uppsala University, Uppsala, Sweden {t (tz,o, sdall, argionijokth..o
Ece Calikus²
Aristides Gionis'
ece. se
ABSTRACT
Predicting relevance is a pervasive problem across digital platforms, covering social media, entertainment, and commerce. However, when optimized solely for relevance and engagement, many machine-leaming models amplify data biases and produce homogeneous outputs, reinforeing filter bubbles and content uniformity. To address this issue, we introduce a pairwise top-k diversity objective with a differentiable smooth-ranking upproximation, providing a model-agnostic way incorporate diversity optimization directly into standard gradient- based learning. Building on this objective, we cast relevance and diversity as a joint optimization problem, we analyze the resulting gradient trade-offs, and propose two comple- mentary strategies: direct optimization, which modifies the learning objective, and indirect optimization, which reweights training data. Both strategies can be applied either when training models from scratch or when fine-tuning existing relevance- optimized models. We use recommendation as natural evaluation setting where scalability and diversity are critical, and show through extensive experiments that our methods consistently improve diversity with negligible accuracy loss. Notably, fine-tuning with our objective is especially efficient, requiring only a few gradient steps encode diversity scale.
INTRODUCTION
Modern prediction models are typically evaluated by their ability to maximize accuracy, that is, to rank or classify items in line with ground-truth labels. However, optimizing exclusively for accuracy often yields homogeneous outputs: models repeatedly select similar items, overlook long- tail instances, and fail to provide novel or diverse options. This challenge is particularly evident in recommender systems, which influence decisions in shopping, entertainment, and news (Ricci et al., 2010). However, it also arises broadly in ranking and selection tasks across machine learning (Wang et al., 2023). When diversity is neglected, models risk reinforcing bias, amplifying popularity effects, and reducing the utility of top-k prediction sets (Nguyen et al., 2014; Lambrecht & Tucker, 2019).
Diversity has thus emerged as an important complementary criterion. Recent user studies confirm that diverse outputs can improve satisfaction and engagement when achieved without major loss in rele- vance (Chen et al., 2018; Holtz et al., 2020; Anderson et al., 2020). The trade-off between relevance and diversity is mainly tackled by either post-hoc re-ranking methods or learning-based methods. Post-hoc re-ranking methods (e.g.. MMR, DPP) modify top-k sets to improve diversity (Carbonell & Goldstein, 1998: Chen et al., 2018), but typically suffer from degrading accuracy when diversity increases (Chen et al., 2017). Model-specific learning based methods integrate diversity into training objectives (Borodin et al., 2017: Hurley, 2013; Wang et al., 2023), achieving strong gains with modest accuracy loss. However, these methods rely on opaque models that obscure the source of diversity gains, exhibit slow convergence, and are sensitive to the choice of trade-off parameter balancing relevance and diversity. Additionally, data-centric approaches such as augmentation, reweighting, and debiasing (Wang et al., 2021; Lai et al., 2023; Ren et al., 2018; Rastegarpanah et al., 2019) address bias in data distributions, but do not explicitly target diversity. Despite these efforts, still lack a unified, differentiable, and model-agnostic framework for optimizing both relevance and diversity directly during training.

---

Published as a conference paper at ICLR 2026
To address these challenges, we propose unified framework that leverages differentiable ranking to optimize diversity in top-k prediction sets in a scalable and model-agnostic way. At the core, we use an effective diversity objective that can be integrated into the gradient-based training without requiring architecture changes or post-preessing. Building on this objective, we introduce two diversification methods. (i) direct diversät-guudder tuning (DDT), which augments the loss with a joint relevance- diversity term, and (ii) meta-diversity reweighting (MDR), which preserves relevance-only training while reweighting data points using the joint loss as a meta-objective. Our approach offers flexible alternative to post-hoc or model specificecciicc diversification without compromising efficiency.
Our contributions are threefold: (1) We propose unified differentiable framework for optimizing relevance and diversity in top-k prediction sets, applicable to both end-to-end training and fine- tuning: (2) we provide theoretical analysis of gradient conflicts, deriving feasible intervals for the trade-off parameter |3 and showing that an adaptive update coincides with the two-objective solution of multi- multi-gradient descent algorithm (MGDA),, guaranteeing convergence to Pareto-stationary points; (3) we empirically validate the framework five benchmark datasets and two model architectures, demonstrating that DDT and MDR achieve substantial diversity improvements with minimal relevance loss, outperforming strong baselines. Notably, the diversity gains extend beyond the explicitly optimized top-k range, reshaping subsequent predictions as well.
2 RELATED WORK
Our work is related to diverse recommender systems and multi-objective learning.
Diversity in recommender systems. Among the vast literature on recommender systems, the closest are post-hoe and learning-basse diversification methods (Zhao al., 2025); see the survey for a broader overview. Post-hoc methods re-rank the output of relevance-only model to balance relevance and diversity. Representative approaches include maximal marginal relevance (MMR) (Carbonell & Goldstein, 1998), diversity-weighted utility maximization (DUM) (Ashkan al., 2015), and determinantal point processes (DPP) (Chen et al., 2018). These methods are model-agnostic and easy to implement, but their performance is limited by the quality of the initial relevance ranking, and diversity gain usually comes at a cost of reduced accuracy (Chen et al., 2017).
Learning-based approaches incorporate diversity objectives directly into training, including penalties for similarity among recommended items (Hurley, 2013; Wasilewski & Hurley, 2016), formulations that optimize relevance-diversie that encourage coverage of item categories or long-tail exposure (Zheng et al., 2021; Yang et al., 2023). While they often outperform post-hoc re-ranking, they require architectural modifications or adversarial training, making them model-specific and computationally heavy. In contrast, our framework is differentiable and model-agnostic: it can be integrated into standard training pipelines without altering architectures or adding inference overhead.
trade-offs (Wang al., 2023) list-wise, and graph-based models
Multi-objective learning. Related the study of multi-objective optimization for balancing goals such as accuracy, fairness, and revenue (Zheng & Wang, 2022). Classical approaches include scalarization (Paul et al., 2022; Di Noia et al., 2017), which reduces multiple objectives to a single weighted loss, and population- based heuristics such as evolutionary algorithms (Cai et al., 2020), which approximate the Pareto front. While effective in some cases, these approaches either rely on carefully tuned weights or suffer from high computational cost.
More recently. gradient-based methods such as the multi-gradient descent algorithm (MGDA) (Désidéri, 2012) have been applied to recommendation. For instance, MGDA has been used to balance accuracy with fairness (Du et al., 2025; Wu et al., 2022) and with revenue (Milojkovic et al., 2019). These methods guarantee convergence Pareto-stationary solutions, but their application has so far been limited to objectives other than diversity.
3 PROBLEM FORMULATION
In this section, we introduce top-k diversity and our objectives, starting with notation. Given a set of candidate items I {i1. ig. ...,.... )m) and collection of users 2 = {ur, ., Un }. the goal is to identify those items in I that are most relevant for each u E U. We assume access to a partially-observed supervision matrix R E Raxm, where entries R.,; represent relevance scores
2

---

Published as a conference paper at ICLR 2026
(e.g., rating, label, or interaction). The set of all observed scores is Ω {(,,i, i, R,,,)),, where N < nm is the number of observations. We use a relevance prediction model Fe with parameters θ to estimate the remaining scores R = Fe(u, i). To learn such a model, we consider two widely-used approaches. First, we consider matrix factorization (MF) (Koren et al., 2009) which predicts ratings R,,; x ys for user and item i using the embeddings X Vr E RR Second, consider neural network models (He et al., 2017), which predict ratings R = MLP((x,) yxl) with a multi-laye perceptron (MLP), allowing non-linear interactions.
In both cases, the models are trained minimize the regularized mean squared ervor (MSE)
min
(R (R (u.))E
Σ
RR,,)² +|
between observed relevance scores and predictions. While this approach yields models that predict accurately, neither MF nor MLP is optimized for diversity. Top-k diversity seeks to predict relevance scores F., E T of user u, that the top-k scores correspond to a of diverse items (k) {z1...., 2 indicated by lu top; (T.). To measure the diversity of the top- (highest scoring) items, we take a distance- based approach (Hassin et al., 1997). In particular, for given pairwise item-item affinity matrix S E mm we define diversity as the average pairwise dissimilarity
2
(k))
(0L - S...)
(2)
k-K
j=
of items in Zk)).. Then, the average top-k diversity is simply
DDO = Σ Ds((.(()) a=1
(3)
While this serves as a natural diversity objective, it cannot be directly optimized with gradients as (-p involves non-differentiable operations. Our goal, however, is to optimize for relevance and top-k diversity simultaneously as part of gradient-based optimization. To achieve this, we relax the non-differentiable diversity reward objective (DRO) using differentiable surrogate (DDRO) next.
Differentiable diversity. To overcome the non-iifferentiabiiity challenge in t-px(-), we adopt differentiable ranking (Blondel al., 2020), which continuous relaxation of sorting. The key
idea is to replace the discrete permutation z, with a soft ranking vector 7(() E TR"" obtained by projecting the predicted scores F. onto the permutahedron the convex hull of all permutations of (1, 2, ,, m) embedded in an m-dimensional space. This projection is computed by solving the following entropy-regularrized optimization problem (Blondel al., 2020):
{ (t) H(r)
% softrank(F..) == arg min rEPm
where H (r) is the entropy regularizer and E > 0 controls the approximation smoothness. This definition enables the top-k soft indicator (t) a- (i)) using scaled sigmoid function for user-defined smoothhesss-harppess т, often set to our experi- ments. In turn, this makes it possible to train the prediction model end-to-end using diversity-aware gradient updates. Like the discrete counterpart, soft ranking operates with O(n log n) time and O(n) space complexity. The obvious question is, does soft ranking lead sufficiently accurate top-k recommendations? The better soft ranking approximates the hard ranking, the more reliable are our top-k recommendations, formally summarized in Lemma. 1.
(x) = [1 + exp(-x/7)
Lemma (Soft rank approximation (Blondel et al., 2020)). Given a rating vector i.. let 7((²) IR" be the soft rank vector obtained from optimizing (4). Then, - 0. the soft ranks converge to the true ranks of lime-so z!e)) rank(f.), where rank(f.)) E (1...., n." denotes the discrete ranks (breaking ties arbitrarily).
Replacing indicator in Equation (3) with a the soft-ranking-derivel indicator i, yields the differen- tiable diversity reward objective
m M
CDDRO
n ΣΣΣ LOi0))
(5)
3

---

Published as a conference paper at ICLR 2026
To achieve both a high relevance and high diversity in top-k outputs, we balance both the relevance objective Crel and diversity objective Laiv introducing our problem below.
Problem 2. For given model class Fe, an item affinity matrix S. user-defined relevance-diversty trade-off B E [0, 11; find parameters θ that minimize the joint loss
LIOINT (3, ) - 3 L + (1 B) (+).
In practice, we take relevance objective Lrol = LMSF and diversity objective Laiv = - While leading efficiently optimizable objective (6). joining them combines two diametrically opposed goals: diversity and relevance. Diverse outputs are not necessarily the most "relevant" ones, and vice versa. We discuss how to deal with this balance in the following section.
Our framework is agnostic to the specific choice of relevance objective. In the main experiments, we adopt a mean-squared error (MSE) loss for concreteness, but any suitable pointwise ranking loss (e.g.. Bayesian Personalized Ranking (BPR)) can be used instead. To demonstrate this flexibility, we report additional results with a BPR- based relevance objective in Appendix
JOINT GRADIENT-BASED TRAINING
In this section, discuss how this objective be incorporated into practical training. We propose two strategies: one direct, by optimizing the joint loss explicitly, and one indirect, by using the joint loss meta-objective to reweigh training samples, starting with the direct approach.
4.1 BALANCING RELEVANCE AND DIVERSITY
The first approach, called direct diversity tuning (DDT), optimize the joint loss in Eq. (6). By utilizing the differentiability of our joint loss, we take an efficient gradient- based optimization approach in which the model parameters are updated using the gradient VeLJOINT.
As diversity opposes relevancy, we need to ensure that the optimization converges solution that is both relevant and diverse. That is, if both relevance and diversity objectives have gradients that point in similar directions, we say that they are 'aligned'. the two gradient directions are aligned, any / decreases both terms. However, when the gradients Jrot = V and Odiv VLivv are misaligned, any linear combination will necessarily favor one objective at the expense the other. We want to ensure a descent that ensures a good diversity and accuracy balance for which we realign the gradients. For this, we adaptively compute the optimal balance parameter 3 that ensures simultancous descent during optimization. Formally, for gradient norms ||9reil| and ||gaiv ||. we denote the cosine similarity by p p= (grel, 9 The combined gradient is g = 3grel + (1 - 3) For a step along 93 to decrease both objectives simultaneously, the following conditions
(B) > 0
(grel: 93) > 0, (B)
=
(gaiv: 9a) > 0
(A-B)
must be satisfied (Désidéri, 2012). Equivalently, the projections of the combined gradient Is onto Grel and Jdiv should be positive. This guarantees that a step along -ga decreases both losses at once. We give the feasible region of that satisfies (A-B):
Lemma 3 (Common descent (Désidéri, 2012)). For any a, b > O and p E [-1, 1). the feasible region of 3 E [o, 1] satisfying (A-B)
1. If 0 (aligned), all 8 E [0. 1] are feasible.
2. If p (orthogonal), the feasible set is 3 E (0, 1).
3. If p (opposing), the feasible set is < b+a pl
h
Proof sketch. The result follows from expanding the directional derivatives (,0) = -Baa (1 B)bbp and ddiv (3) -(1 B36² - Babp, and solving the inequalities (A-B) in the three cases p> 0. O. and p < O. Full details are provided in Appendix A.
Even when 3 lies in the feasible interval, different values may still lead to unbalanced overall progress, with one loss improving much more than the other. To avoid this, we select the parameter
8 arg max min (3)), - -aiv (B)} $E
4

---

Published as a conference paper at ICLR 2026
that maximizes the minimum per-step decrease, yielding multiple gradient descent algorithm (MGDA) (Désidéri, 2012) specialized to two objectives. MGDA finds a convex combination of gradients that minimizes the maximum directional derivative across tasks using the closed-form
b(b - ap)
8"
a² + 62 2abp
When o S the ideal balance 3 lies in the feasible interval described in Lemma 3, ensuring valid common descent. Intuitively, 3 equalizes the first-order decreases of relevance and diversity, yielding a balanced update. Finally, when 3e is chosen adaptively by the MGDA rule, we obtain convergence to Pareto-stationn
solutions.
Corollary 4 -Pareto-stationaiity with adaptive 3t (Sener & Koltun, 2018)). Under diminishing step sizes, if each Be is chosen by the minimax rule (or projected variant), then every accumulation point of (0') is Pareto-stationary for (freh, Ldiv).
While the adaptive choice provides balanced progress at each step, a fixed 3 remains usefull, as it avoids per-iteration computation. To contextualize our results, we introduce an additional alternative approach which operates with fixed [3 next. We provide details, incl. pseudocode, in Appendix B.
4.2 OPTIMIZING DIVERSITY BY EXAMPLE REWEIGHTING.
A complementary line of work on machine learning faimess and robustness has shown that reweighting training examples can effectively mitigate inherent data bias. The central idea is to retain the standard prediction objective, but assign adaptive weights individual samples so that the resulting model better aligns with a criterion (Ren et al., 2018; Rastegarpanah et al., 2019). This approach is naturally formulated as meta-learning: the inner loop minimizes a weighted relevance loss, while the outer loop adjusts weights using a meta-objective that encodes the criterion.
Inspired by this paradigm, we aim improve diversity by reducing bias the data level through per-sample weights. Intuitively, increasing the importance of 'minor' items increases the chance of their exposure. brief, the idea is to learn the model parameters by optimizing a weighted relevance loss, while simultaneously learn the weights that balance relevance and diversity. More concretely, our mela-diversit reweighting (MDR) introduces a weight Wo,i E [0, 1] for each user-item pair (u, i) in mini-batch B and optimizes the reweighted relevance loss
MSE
Σ (B
Wazi (Ru,i RR,,)²
(8)
The algorithm starts with obtaining temporary model O' (initializing Wa,i 0) using one-step inner update. We then re-evaluate the predictions with the updated parameters and compute the joint meta-loss LIOINT. Next, we compute the gradient of LJOINT with respect w to obtain a utility score for each sample, and finally normalize the weights of all data points in B so that they sum to 1. The joint relevance-diversity objective is used only as a meta-loss to update the weights w, and is never applied directly to model parameters. as detailed in Apx. B.
In contrast to direct diversity-guidde tuning, which modifies the training objective itself, MDR pre- serves the standard relevance-oriented loop while implicitly reshaping the effective data distribution. The meta-objectivv encourages weights that downplay biased interactions and upweight samples that contribute to both accuracy and diversity. This reweighting perspective allows to test whether diversity gains can obtained not only by altering the optimization objective, but also by correcting data imbalance through implicit meta-optimization.
To illustrate the mechanism of MDR, we construct a controlled case study where all items have identical ratings but belong to distinct diversity clusters. After one-step meta update, the gradient of the joint objective with respect to sample weights assigns larger normalized weights to items that increase inter-cluster coverage, while suppressing redundant items within the same cluster. This example demonstrates that MDR automatically emphasizes samples that contribute most to diversity without explicit cluster supervision (see Appendix C for details).
We evaluate both DDT and MDR in two settings: from-scratch training, where relevance and diversity optimized jointly, and fine-tuning, where deleaance-traieed model is adapted for diversity.
5

---

Published as a conference paper at ICLR 2026
Table Statistics of datasets and diversity metrics.
Dataset
[
[2]]
DRO(I)
|C|
Coat
290
300
6960
0.73
33
KuaiRee
1411
3
676
0.91
31
Netflix
4 4999
112
557 176
0.83
27
Yahoo-R2
4.0
5000
684782
0.26
58
MovieLens
6040
706
000 208
0.83
18
EXPERIMENTAL EVALUATION
We describe the experimental setup evaluate the effectiveness of our proposed solutions, introducing the datasets and evaluation criteria in this section, and provide further details in Appendix D and E.
Datasets. We evaluate our methods across three domains:
entertainment, product, and social recommendations. For entertainment, we use MovieLens, Netflix, and Yahoo-R2, which contain user ratings on movies or music with genre/category annotations. For product recommendation we use the Coat dataset, which includes user ratings and product attributes. For social recommendation we use KuaiRec, large-scale mobile video dataset with watch-time based ratings. The item affinity matrices S are pre-computed by Jaccard similarity scores based on genre/category information of items. Basic statistics and detailed preprocessing are given in the Appendix D.
Baselines. We compare against relevance- and diversity aware approaches: (i) Non-negative Matrix Factorization (NMF) (Lee & Seung. 1999) as relevance-only baseline; (ii) greedy post-processing methods such as Maximal Marginal Relevance (MMR) (Carbonell & Goldstein, 1998) and Diversity- weighted Utility Maximization (DUM) (Ashkan et al., 2015); (iii) probabilistic diversification via Determinantal Point Processes (DPP) (Chen et al., 2018); 8); and (iv) the graph- based Diversified GNN Recommender (DGRec) (Yang et al., 2023). We use the authors' publicly-available implementations.
Evaluation criteria. We evaluate accuracy using hit rate, precision, and recall. For each user u, let R. denote the top-k recommended items and Tu the set of ground-truth relevant items (i.e., rated above threshold), the Hit measures whether least is the fraction of items in R, that are in Tw: and recall is the fraction of relevant items in Tu that retrieved in Ry When ground-truth unknown, report the estimated satisfaction using preference likelihoods p (R.... .) E [0, 1] and threshold 0.8 as the relevance score
relevant item appears in Rus precision
Relevance(u) 1 Σ ll.(R..)) 7].
h iER.
5.1 EXPERIMENTAL ANALYSIS
Having introduced our setup, we now introduce our research questions and experimental analysis.
Q1 Does adaptive B behave as predicted by our theory?
Q2 How does the choice of 3 affect the accuracy-diveet diversity trade-off?
Q3 How do our approaches compare with established diversification methods?
Q4 How do diversity gains evolve with varying A:?
In the following, experimentally answer all these questions in detail.
Evaluating adaptive optimization. We first study the direct optimization approach using both from-scratch training and fine-tuning of gelevance-pretrannee model. At each step, adopt the adaptive 3 rule, computing both the optimal coefficient 3* and the feasible interval from Lemma Figure la shows from-scrate training and fine-tuning for 100 epochs MovieLens. We employ the neural collaborative filtering (NCF) model with an MLP architecture, and consider top-10 diversity.
In the top row of Figure 1a, observe that the adaptive coefficient 3 remains consistently within the feasible interval defined by Lemma 3. We see how 3* gradually shifts from favoring relevance toward favoring diversity, confirming that the adaptive rule preserves common descent direction.

---

Published as a conference paper at ICLR 2026
D
Lower bound
Upper bound
MF DDT
NCF+
Lacsc
DDRO
DRO
+
NCF
NCF Train
NCF + Line-tune
Netflix
MovieLLens
0.8
0.5
0.5
0.0
0.0
30
45
75 90
0
15
30
45
60
75
90
0.0
0.5
1.0
0.0
0.5
1.0
Losc Diverityy
1.0
1.0
1.0
Diverstyy
0.9
0.9
0.8
0.8
0.8
0.6
Epoch
Epuch
(a) Optimization with adaptive / (MovieLens).
(b) Optimization with varying 'static' Bs.
Figure 1: Optimization of relevance-diversity joint objective. Figure 1a shows results of training and fine-tuning of NCF models with adaptive 33 on the Movielenns dataset. Figure 1b shows the performance comparison of DDT and MDR applied to two models (NMF and NCF) across two datasets with varying fixed 3 E [0, 1].
Table 2: Performance of DDT, MDR, and five competitors in terms of diversity and relevance across Coat, Yahoo-R2. Neuflxx MovieLens, and KuaiRec. We highlight the the best results with bold, and underline the second best, reporting mean and standard deviation over 10 trials.
Algorithm
Coal
Yahoo-R2
Notflis
MovicLons
KuaiRee
Diversity
Relevance
Diversity
Relevance
Diversity
Relevance
Diversity
Relevance
Diversity
Relevance
(0.04)
0.09
(0.02)
0.84 (0.01)
(0.08)
(0.01)
(0.01)
MMR
0.80 (0.01)
0.40 (0.04)
0.80 (0.06)
0.68 (0.03)
0.93 (0.01)
0.85 (0:04)
0.94 (0.03)
0.95 (0.01)
0.99 (0.00)
0.76 (0.01)
DUM
0.81 (0.01)
0.31 (0.04)
0.98 (0.02)
0.60 (0.02)
0.93 (0.00)
0.71 (0.03)
0.93 (0.01)
0.91 (0.01)
0.95 (0.00)
0.42 (0.01)
DPP
(0.01)
0.39 (0.04)
1.00
(0.01)
(0.00)
(0.04)
(0.01)
(0.00)
0.75 (0.01)
DGRec
0.71 (0.01)
0.69 (0.02)
0.33 (0.01)
0.83 (0.02)
0.76 (0.00)
0.83 (0.01)
0.75 (0.01)
0.47 (0.02)
0.91 (0.02)
0.18 (0.04)
DDT
0.83 (0.01)
0.50 (0:05)
0,98 (0.01)
0.85 (0.01)
0.98 (0.00)
0.97 (0.01)
1.00 (0.00)
1.00
0.98 (0.02)
0.95 (0.00)
(0.01)
0.47 (0.03)
0.56 (0.05)
(0.02)
0.98 (0.01)
0.93 (0.02)
(0.02)
0.99 (0.00)
(0.01)
0.85 (0.01)
The bottom row reports the optimization trajectories. The relevance loss (LMSE) decreases steadily while diversity (DRO and DDRO) increases, showing that adaptive 3 achieves a balanced trade-off in both training objectives. We see that fine-tuning preserves the relevance of the pretrained model, whereas from-scratch converges to a worse stationary point.
We further observe that fine-tuning preserves the relevance performance of the pretrained model, whereas training from scratch reduces LMSE but converges to a worse stationary point. In contrast, both approaches reach a similar level of diversity, suggesting that regularization may hinder the convergence towards more 'relevant' models during from-scratch optimization. Fine-tuning, on the other hand, starts with relevance-optimized model, which we effectively tune for diversity. Finally, because DRO and DDRO are close, we empirically validated the accuracy of our relaxation. We discuss the impact of the parameter e on approximation accuracy later. Overall, we saw that adaptive 3 effectively delivers simultaneous improvements in relevance and diversity.
Relevance-diversity trade-off. We next examine how the choice of static 3 affects relevance and diversity. Figure lb reports recall and diversity for DDT and MDR with MF and NCF on Netflix and MovieLens. Starting from a relevance-trainen model, we fine-tune with the joint objective (6) while varying 3 [0, 1]. ranging from B = 1 (pure relevance) to 3 0 (pure diversity). We run 10 random trials for each experiment and report the mean and variance. We observe a stable low-variance recall across wide range of 3, showing that introducing diversity does not substantially compromise accuracy with fixed 3. A noticeable drop occurs only when 3 approaches zero, where the objective focuses almost exclusively Moreover, DDT consistently outperforms MDR in terms of diversity with a neural network model, while the two strategies behave more similarly with matrix factorization model, suggesting that
diversity. In contrast, diversity improves as B decreases.
7

---

Published as a conference paper at ICLR 2026
reweighting is particularly beneficial for simpler low-rank models. These findings demonstrate that fixed-B optimization offers a practical means to explore the -ivyesstdiivesitt trade-off, enabling substantial diversity gains with minimal loss of accuracy. Similar patterns observed for MSE loss, hit rate, and precision, with detailed results on additional datasets provided in Appendix E.
Table 3: Diversity gain achieved by Direct Diversity Tuning (DDT) top-1 ~ k and k + 1 2k: recommendations across five datasets Coat, Yahoo-R2, Netflix, Moviel and KuaiRec. We vary the diversity reward parameter & E (5, 10, 20, 30, 40) and apply DDT to pre-trained NMF and NCF, reporting diversity gain. We report the mean and standard deviation across 10 runs.
Diversity gain (%)
Dataset
1 k
I++ 2h
Am5
K-10 10
k-20
Ie-30
1--40
km-
10
k-20
k-30
Coal
(1.3
(1.8)
(1.7)
(1.1
(.6(
-0.5 (1.7)
(4.4)
(1.3
(1.5
(1.9)
NMP
Yahoo-R2
(1.0
78.5 (8.7)
78.4
54.9 (1.)
(9.9)
47.1 (2.7)
(2.2
41.9(1.3)
Netflix
(2.3)
(1.0)
11.3:007)
(0.6
9.6(...)
(2.6
(1.)77)
3.1 (2.1)
2.22
2.2(1.8)
MovicLons
(13.(13.1)
(8.)
28.4
21.6 (3.2)
25.6 (10.)
10.1 (6.1
8.3
6.8(1.))
KuuiRee
9.2(1.7)
(.5)
9.4.0.8)
10.6 (0.7)
(0.9)
(1.(
(2.3
6.0.
3.3
1.4(1.)
Coat
9,7(2.4)
12(18))
6.9(1.))
(0.8)
(.5.)
(..6
(..4
-1.1 (0.7)
(.7
(1.0
Yahoa-R2
(9.5(12.8)
9.4(17755
(7.3
(6.0)
(8.7
(13.7)
(4.6)
(260))
(6.0
(.3)
Netflix
13.7
(129)
(1.44))
(19)1.))
(.5)
MovieLens
(13.0)
19.6 (6.6)
(7.8)
26.9 (3.0)
(1.9)
23.0
9.6(1.5)
20.9 (2.5)
(.0
KumiRee
8.8(1.3)
(1.8)
6.3.0..3
10.7 (2.6)
(..))
8.0(1.2)
(0.7)
(.290
(0.5)
(.39
Diversity-relevane performance of all approaches. Next, we compare the proposed methods against alternative diversification approaches across all datasets. Fixing A = 10 and 3 = 0.2, we fine-tune a pre-trained relevance-optimizee matrix factorization model for 100 epochs and select the most diverse checkpoint. In Table 2 we report the mean and standard deviation of diversity and relevance scores for all methods from 10 trials.
We see that our direct diversity tuning approach (DDT) demonstrates strong performance in both relevance and diversity, achieving either the best second-best results across nearly all metrics and datasets, showing the effectiveness of jointly optimizing. Similarly, we see that the implicit data reweight approach (MDR) shows highly competitive performance. Its performance particularly notable on Netflix, MovieLens and KuaiRec, where it approaches or matches the performance of DDT. This supports our earlier observation that implicit, data-driven reweighting can offer strong benefits. On the other hand, while post-hoc diversification of NMF using MMR, DUM, and DPP, considerably improve top-k diversity, they often notably reduce relevance, aligning with the results reported in previous studies (Chen et al., 2017). For example, DPP occasionally achieves the highest on Yahoo-R2 and KuaiRec-but at a cost of low relevance. Greedy methods like MMR and DUM yield moderate diversity improvements but underperform in relevance. DGRec excels in diversity on certain datasets but suffers from severe relevance degradation, especially on KuaiRec, with diversity of 0.91 and relevance of 0.18. In contrast, DDT and MDR maintain significantly better trade-off. In summary, fine-tuning with direct and implicit methods both achieve superior relevance-diverte trade-offs across the board, outperforming the competitors.
diversity-
Diversity gain with varying k. We examine how diversity changes across the growing number of recommended items. Starting with relevance-optimized matrix factorization and neural network models, we fine-tune with the joint objective LJOINT with DDT for 10 epochs at fixed 3 = 0.5, and vary l E (5, 10, 20, 30, 40). selecting the best results. Since larger top-k sets approach coverage of the entire item space, it becomes impossible to improve diversity beyond the dataset's 's average. To account for this, we report relative diversity gains over the relevance-only baseline, normalized by the maximum achievable score. Specifically, we measure DRO gains as the difference between fine-tuned and pre-trained models, normalized lie in (0, 1] and reported as percentages. We distinguish between in-objective gains (top-k) and out-of-obeective gains (the subsequent k+1 ~ 24 items).
In Table 3, we observe positive in-objective diversity gains across datasets. In Coat, Netflix, and MovieLens, we see decreasing gain as & increases. The improvement is especially pronounced on Yahoo-R2 -2, with diversity gains exceeding 78.5 = 3.8% in NMF and 79.3 = 7.6 in NCF at = 20. due to its low initial diversity (e.g., 0.09 in the base model). Substantial gains are also observed on MovieLens (e.g., 41.4+ 13.1% and 46.8 133% at l = 5) and Netflix, demonstrating the effectiveness of our method across both sparse and dense recommendation scenarios.
8

---

Published as a conference paper at ICLR 2026
10
1.0
Approx inarion erear [|z- --1
107
0.8
101-
0.6
107-
Eirox
102
0.4
1
10'
0.2
1- 100
500
10
100
10
10 10
10
(a)
(b)
Ablation study on approximation errors for ranking and diversity. (a) Rank approximation with varying parameter and item size. The y-axis reports the approximation error between the exact rank and the softrank approximation, IZ 2 ////,, and the z-axis reports the parameter Setting & = 1/n yields zero approximation error, and there is a tunable range of e that achieves small error while allowing larger values. (b) Diversity approximation error of DDRO with different and on MovieLens. The y-axis reports the approximation error [DRO LDDRO CDRO-
Figure 2:
Generality beyond the optimized range. We next ask whether the benefits of diversity optimization extend beyond the explicitly encoded objective. To this end, evaluate diversity only the target top-k set but also in the subsequent k++1 ~ 2k: items in Table We observe broadly positive out-of-objective gains, which again diminish & increases. This effect is especially strong on Yahoo-R2 and MovieLens, showing that optimization reshapes the ranking itself: improvements are limited the items directly optimized, but generalize deeper parts the recommendation list. In contrast, results on Coat fluctuate around zero, likely due to its small scale and limited item pool. These findings highlight the generality our diversity-gune guidded optimization: improves diversity not only within the objective's 's explicit target (top-k) but also beyond, demonstrating that the learned updates capture a broader notion of diversity than what is directly optimized.
Approximation error with varying In Figure 2a, first examine the rank approximation error induced by the softrank operator. We report the discrepancy between the exact rank and its differentiable approximation, measured as I / Vni, as a function of e and the number of items. Consistent with the theoretical construction, setting approximation error. Moreover, there exists a range of larger e values that still achieve small error, indicating that can trade off modest loss in approximation accuracy for improved numerical stability and smoother gradients. In Figure 2b, we report the approximation error between the exact diversity DRO and its differentiable surrogate DDRO the top-k size varies. We define the error as [DRO - LDDRO|/ CDRO. For small k (c.g., k < 100), setting < 10 already yields a close approximation. As k increases, however, smaller required to maintain the same accuracy. This highlights that the choice of is critical: an improper value can lead to arbitrarily large errors, rendering the diversity measure unreliable for optimization.
1/n yields essentially zero
Case study. To demonstrate the utility of our approaches, we examine the recommendations generated from an NMF model and its diversity-optimizze variant. In Table 4, we observe that NMF produces homogeneous items where six out of ten items are labeled as drama, with remaining entries only marginally extending into romance, documentary, comedy or War. In contrast, the DDT- generated list spans much broader range of genres, including horror, thriller, animation, crime, and adventure. Despite diversity gains, DDT retains three of the top items in relevance-optimizee recommendation (Mamma Roma, Smashing Time, and Gate of Heavenly Peace), which collectively represent the relevant genre themes (drama, comedy, and documentary, respectively). suggesting that DDT successfully preserves relevant while enhancing diversity.

---

Published as a conference paper at ICLR 2026
Table Comparison of recommendation lists for user from DDT and NMF. Movies appearing in both lists are bold.
DDT
NMF
Rank
Genres
Movie Title
Rank
Genres
Movic Title
Horror
Vampyros Lesbos (Las Vampiras)
Drama
Mamma Roma
Thriller
The Spiral Stairease
2
Drama
Foreign Student
War
Prisoner Mountains
3
Drama
The Apple
Animation. Musical
Melody Time
Drama, Romance
Leather Jacket Love Story
Documentary
The Gate of Heavenly Peace
5
Comedy
Smashing Time
Crime
Lured
Documentary
The Gate Heavenly Peace
7
Drama
Mamma Roma
7
Documentary
Modulations
Comedy
Smashing Time
Comedy, Remance,
Forrest Gump
Adveniure
Ulysss (Ulisse)
9
Drama
Schlafes Bruder (Brother of Sleep)
Romance
Persuasion
Schindler's
6 CONCLUSION
We addressed the issue of limited top-k diversity of relevance prediction models, which contributes echo chambers, reduced novelty, and social polarization. By integrating diversity into gradient- based optimization, we presented unified framework for diversity- recommendation by introducing differentiable diversity objective that enables end-to-end optimization of both relevance and diversity. We proposed two complementary, model- algorithms to support explicit and implicit integration of diversity into standard recommender systems. With extensive experiments on real-world datasets we demonstrated that our methods consistently improve diversity, converge efficiently, and introduce minimal computational overhead.
Limitations and future work. While our framework effectively promotes both relevance and diversity in top-k recommendations, several challenges remain. First, the adaptive 3 is applicable only for direct joint objective optimization, while it is optimal in each step, but the optimization trajectory may converge to a compromised stationary point, which not globally optimal. The mechanism that leads to global optimal solution remains unexplored. Future work includes developing new ways to balance our competing goals via multi-objective formulations. Second, we primarily relies on categorical similarity to quantify diversity. While this provides interpretability. does not capture more nuanced relationships or learned semantic embeddings. Future work extends the research into more expressive and context-sensitive diversity metrics. Finally, as diversity bias stems from the training data iitta-implicit and explicit approaches often achieve similar performance levels we see significant potential in the data-driven diversification, such as counterfactual data augmentation or diversity-awe sampling.
Despite these research opportunities, we observe that our methods significantly increase diversity with only an imperceptible decrease in relevance.
10

---

Published as a conference paper at ICLR 2026
ETHICS STATEMENT
This work is primarily theoretical and focuses on incorporating diversity into machine-learning models for ranking. All datasets used our experiments are publicly available, and no personally identifiable sensitive information was collected or processed. No user studies or interventions involving human participants were conducted.
While our approach is motivated by the goal of improving diversity in ranking outcomes, acknowl- edge that any deployment of such methods may have broader societal implications. Potential concerns include including fairesss considerations, potential biases in the underlying data, unintended ef- fects depending on the application context. We leave a deeper exploration of these implications to future research.
REPRODUCIBILITY STATEMENT
Our code is available in online repository!.
AI USAGE DISCLOSURE
I Large language models (LLMs) were employed during the research phase to assist with surveying related literature, including identifying and summarizing relevant papers and methods, as well as sketching and testing proposed approaches described in prior work. Generative models were also used to draft scripts for data processing (c.g.. formatting and visualizing data, preliminary experimental validation). These scripts pipeline or released codebase. In addition, ChatGPT and Grammarly were used to assist with grammar and phrasing in the manuscript. All outputs from these tools were reviewed, edited, and verified by the authors, who take full responsibility for the final content.
only used for exploration and not part of the final experimental
ACKNOWLEDGEMENTS
We thank all anonymous reviewers and the area chair for their valuable feedback and constructive suggestions. This research supported by the ERC Advanced Grant REBOUND [834862], the Swedish Research Council (VR) [2024-05603], the European Commission MSCA DN [101 168951], and the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
REFERENCES
Ashton Anderson, Lucas Maystre, Ian Anderson, Rishabh Mehrotra, and Mounia Lalmas. Algorithmic effects on the diversity of consumption on spotify, In Proceedings of The Web Conference, pp. 2155-2165, 2020.
Azin Ashkan, Branislav Kveton, Shlomo Berkovsky, and Zheng Wen. Optimal greedy diversity for recommendation. IJCAI, volume 15, pp. 1742-1748, 2015.
James Bennett and Stan Lanning. The Netflix prize, 2007.
Mathieu Blondel, Olivier Teboul, Quentin Berthet, and Josip Djolonga. Fast differentiable sorting and ranking. In Proceedings of the 37th International Conference Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, pp. 950-9599 9099 2020.
Allan Borodin, Aadhar Jain, Hyun Chul Lee, and Yuli Ye. Max-sum diversification, monotone submodular functions, and dynamic updates. ACM Transactions Algorithms (TALG), pp. 1-25, 2017.

---

Published as a conference paper at ICLR 2026
Xingjuan Cai, Zhaoming Hu, Peng Zhao, Wensheng Zhang, and Jinjun Chen. A hybrid recommenda- tion system with oajy-objective evolutionary algorithm. Expert Systems with Applications, 159: 113648, 2020.
Jaime Carbonell and Jade Goldstein. The use of mmr, enversit-bbaser reranking for reordering documents and producing summaries. In Proceedings of the 21 annual international ACM SIGIR conference on Research and development in information retrieval, pp. 335-336 1998.
Laming Chen, Guoxin Zhang, and Hanning Zhou. Improving the diversity of top-n recommendation via determinantal point process. In Large Scale Recommendation Systems Workshop, pp. 3, 2017.
Laming Chen, Guoxin Zhang, and Eric Zhou. Fast greedy map inference for determinantal point process to improve recommendation diversity. Advances in Neural Information Processing Systems, 2018.
Jean-Antoine Désidéri. Multiple-gradient descent algorithm (mgda) for multiobjective optimization. Comptes Rendus Mathematique, 350(5-6)313-3188, 2012.
Tommaso Di Noia, Jessica Rosati, Paolo Tomeo, and Eugenio Di Sciascio. Adaptive multi multi-attribute diversity for recommender systems. Information Sciences, 382:234 082:234-253,, 2017.
Gideon Dror, Noam Koenigstein, Yehuda Koren, and Markus Weimer. The yahoo! music dataset and kdd-cup' 11. In Proceedings of KDD Cup 20/ I, pp. 3-18. PMLR, 2012.
Yongkang Du, Jieyu Zhao, Yijum Yang, and Tianyi Zhou. Controllable pareto trade-off between fairness and accuracy. arXiv preprint MMXiv:2509.136// 2025.
Chongming Gao, Shijun Li, Wenqiang Lei, Jiawei Chen, Biao Li, Peng Jiang, Xiangnan He, Ji- axin Mao, and Tat-Seng Chua. Kuairec: A fully-observed dataset and insights for evaluating recommender systems. In CIKM, pp. 540-550, 2022.
Maxwell Harper and Joseph A Konstan. The moviclens datasets: History and context. ACM transactions on interactive intelligent systems (TIIS), 2015.
Refael Hassin, Shlomi Rubinstein, and Arie Tamir. Approximation algorithms for maximum disper- sion. Operations Research Letters, pp. 133-137, 1997.
Xiangnan He, Lizi Liao, Hanwang Zhang, Liqiang Nie, Xia Hu, and Tat-Seng Chua. Neural collaborative filtering. In Proceedings of the 26th International Conference on World Wide Web, pp. 182-182,2011 2017.
David Holtz, Ben Carterette, Praveen Chandar, Zahra Nazari, Henriette Cramer, and Sinan Aral. The engagement-diversity connection: Evidence from field experiment spotify. In Proceedings of the 2 ACM Conference on Economics and Computation, pp. 75-76, 2020.
Neil Hurley. Personalised ranking with diversity. In Proceedings of the 7th ACM Conference on Recommender Systems, pp. 379-382, 2013.
Yehuda Koren, Robert Bell, and Chris Volinsky. Matrix factorization techniques for recommender systems. Computer, pp. 30-37, 2009.
Riwei Lai, Chen, Yuhan Zhao, Rui Chen, and Qilong Han. Disentangled negative sampling for collaborative filtering. In Proceedings of the Sixteenth ACM International Conference Web Search and Data Mining, pp. 96-104,7 2023.
Anja Lambrecht and Catherine Tucker. Algorithmic bias? an empirical study of apparent gender- based discrimination in the display of stem career ads. Management science, pp. 2019.
Daniel D Lee and H Sebastian Scung. Learning the parts of objects by non non-negative matrix factoriza- tion. nature, 1999.
Nikola Milojkovic, Diego Antognini, Giancarlo Bergamin, Boi Faltings, and Claudiu Musat. Multi- gradient descent for multi-objective recommender systems. arXiv preprint arXiv:2001.008. 2019.
12

---

Published as a conference paper at ICLR 2026
Tien T Nguyen, Pik-Mai Hui, F Maxwell Harper, Loren Terveen, and Joseph A Konstan. Exploring the filter bubble: the effect of using recommender systems on content diversity. In Proceedings of the 23rd international conference on World wide web, pp. 677-686, 2014.
Agyemang Paul, Zhefu Wu, Kai Liu, and Shufeng Gong. Robust multi-objective visual bayesian personalized ranking for multimedia recommendation. Applied Intelligence, 52(4):3499-351 2022.
Bashir Rastegarpanah, Krishna P. Gummadi, and Mark Crovella. Fighting fire with fire: Using antidote data to improve polarization and fairness of recommender systems. In Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining, pp. 23 231-239, 2019.
Mengye Ren, Wenyuan Zeng, Bin Yang, and Raquel Urtasun. Learning to Reweight Examples for Robust Deep Learning. In Proceedings of the 35th International Conference Machine Learning, pp. 4334-4343, PMLR, 2018.
Francesco Ricci, Lior Rokach, and Bracha Shapira. Introduction to recommender systems handbook. In Recommender systems handbook, pp. 1-35. Springer, 2010.
Tobias Schnabel, Adith Swaminathan, Ashudeep Singh, Navin Chandak, and Thorsten Joachims. Recommendations treatments: Debiasing learning and evaluation. In international conference on machine learning. PMLR, 2016.
Ozan Sener and Vladlen Koltun. Multi-task learning as multi-objective optimization. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pp. 525-536, 2018.
Wenjie Wang, Fuli Feng, Xiangnan He, Liqiang Nie, and Tat-Seng Chua. Denoising implicit feedback for recommendation. Proceedings of the 14th ACM international conference on web search and data mining, pp. 373-381, 2021.
Zihong Wang, Yingxia Shao, Jiyuan He, Jinbao Liu, Shitao Xiao, Tao Feng, and Ming Liu. Diversity- aware deep ranking network for recommendation. In Proceedings of the 32nd ACM International Conference on Information and Knowledge Management, pp. 2564-257, 2023.
Jacek Wasilewski and Neil Hurley, Incorporating diversity in a learning to rank recommender system. In FLAIRS, pp. 572-578, 2016.
Haolun Wu, Chen Ma, Bhaskar Mitra, Fernando Diaz, and Xue Liu. A multi-objecciive optimiza- tion framework for multi-stakeholder fairness-aware recommendation. ACM Transactions on Information Systems, 41(2):1-29, 2022.
Liangwei Yang. Shengjie Wang, Yunzhe Tao, Jiankai Sun, Xiaolong Liu, Philip S Yu, and Taiqing Wang. Dgrec: Graph neural network for recommendation with diversified embedding generation. In Proceedings of the sixteenth ACM international conference on web search and data mining, pp. 661-669, 2023.
Yuying Zhao, Yu Wang, Yunchao Liu, Xueqi Cheng, Charu C Aggarwal, and Tyler Derr. Fairness and diversity in recommender systems: survey. ACM Transactions on Intelligent Systems and Technology, pp. 1-28, 2025.
Yong Zheng and David Xuejun Wang. A survey of recommender systems with multi-objective optimization. Neurocomputing, 474:141-153, 2022.
Yu Zheng, Chen Gao, Liang Chen, Depeng Jin, and Yong Li. Dgen: Diversified recommendation with graph convolutional networks. In Proceedings of the web conference 2021, pp. 401-412. 2021.
13

---

Published as a conference paper at ICLR 2026
A PROOFS
In this appendix, we provide detailed proofs for the results on the trade-off parameter B.
A.1 PROOF OF LEMMA 3
In the joint loss, we minimize both
Crel == LMSE
Laiv = - -DDRO
so decreasing Laiv increases diversity. We denote the gradients
Drel V Lrel,
Odiv
V Ldiv,
g3 .grel + (1 - B)ddiv,
8 E [0,
so that direct evverstyy-guider tuning (DDT) performs
θ +
1993-
Let
(grel. ydiv) 1-,,
[ret > O, b == ||gav| > O,
ab
For an infinitesimal step along gg. the first-order changes in each objective are
d
dze = dt (re - -tgs) = -((grel, gB),
d (f - bgg)
daiv dt
-9ddiv.
1=
Substituting Os Bgret + (1 B)gdiv gives
8rel - -lgell 1|2 (1 (9 gdiv) -Ba² (1 -B)abp,
daiv - (1 - 1lll 11? 3 B(aavv Irel) = (1 - B)b2 - -Babp.
We require both decreases:
8ret < 0, daiv < 0.
Rearranging each inequality yields the following necessary and sufficient conditions:
Ba (1 B3bp 0,
(A)
(1 3)b + Bap 0.
(B)
We now analyze (A) and (B) systematically. Inequality (A). Ba + (1 (1-9)bp
> 3(a bp) > bp.
If bp > 0, then 3
-bp
we need further analyze p.
bp*
If bp 0, then
-bp we need further analyze p.
a-bp"
If a bp = 0, then the inequality holds for all B.
Inquulity (B).
(1 3-b3bb+- + Sap = b Blap b) 0.
If ap b > 0, then
b (RHS negative, vacuous for [0, 1).
If ap b < 0, then
< 8-op' we need further analyze p.
If ap b. then the inequality holds for all B.
14

---

Published as a conference paper at ICLR 2026
We specialize the above formulas to the three relevant regimes of p.
Case I: Aligned gradients (p > 0). Here the two gradients point in similar directions. We revisit inequalities (A) and (B). Condition (A).
3a + (1 3)bp 0
B(a-bp) bp.
If a bp O, then the right-hand side is nonpositive (-bp S 0 since P 0). Thus the inequality is automatically satisfied for all 3 E [0, 1]; no restriction is imposed.
If bp < O, then the inequality gives B b-bp But since bp and -bp < the right-hand side positive. In fact, one can check that _bp still satisfies the inequality.
Therefore any B E [0. 1]
a-
Condition (B).
(1 - --b Bap
b B(ap b) 0.
If ap b b20 0, then the right-hand side is increasing in , and at B 0 equals > 0, so the inequality holds for all 3 E [0,1] 1].
If ap b < 0, then the inequality becomes B < 5-ap Since ap b < 0, the denominator
b ap > 0 and thus the right-hand side exceeds 1. Hence the condition imposes restriction within 3 E [0, 1].
Both inequalities (A) and (B) are therefore automatically satisfied when > Thus
8ee < and
daiv <
for every [0. 1].
In other words, when gradients are aligned, any convex combination yields a valid common descent direction. At 3 0 or 3 1, one loss still strictly decreases while the other is nonincreasing; for B 1, both decrease strictly.
Case II: Orthogonal gradients (p- 0). Plugging p = 0 into (A) and (B) gives
Ba 00 + 8>0 0,
(1 B)b + B < 1.
8rel < 0, daiv
for all $ E (0, 1).
Case III: Opposing gradients (p < 0). Let p = -lpl 0. Then
blpl
From (A): 3
=: L,
From (B): 3 < b+ alp
= U.
a + blpl
Thus the feasible interval is
blpl
b
BE (L.U)
+ blpl' b alp
This interval nonempty whenever |p| < 1, since
ab(1 (pl3)
U-L
(b + alp)) a + blpl) > 0.
If |pl = 1 (exactly opposite gradients), then any 8 (0, 1) still makes both d daiv < 0.
We have characterized the feasible region of
If p 0: every 3 [0, 1] yields common descent.
If p 0: every 3 (0, 1) yields common descent.
If p 0: feasible region is (L, U) as defined above.
This completes the proof of Lemma 3.
15

---

Published as a conference paper at ICLR 2026
mace computation
Difirent able diveriity somputation
Diffo ondobo
Model updote
Approsimate top-
R: prodicted raling
2
topek marns
DDRRO Eq. (9)
approximation error
Fg(0.1)
DRRO Eq.
observed naing
Bxact Non-d
Luce
10 L Mau-
2
LASS
FOONT Pl (1 av
Padevancee-rrly training updates nool paramater gradient V. Vefuss
instadd we updates mode parometer grodiont Valyunr
Figure 3: DDT workflow
A.2
PROOF OF 3*
We seek 8 such that (8) (B). Expanding: 2a² + (1 - Baabp- = -1 -113 + Babp.
Rearranging,
(a² + + 2abp))3 622 -abp.
Thus
b(b ap)
8r a² + b² 2abp
When p O, substitution verifies L 8 < U. where (L, U) is the interval from Lemma 3. Hence 3* is feasible and maximizes the minimum first-order decrease.
B ALGORITHMS
Algorithm workflow diagram Figure and 4 illustrate the workflow of DDT and MDR.
DDT directly optimizes the joint loss LIOINT BLuSE (1 B) via standard gradient descent using innovative soft-ranking, top-k selection, pairwise diversity penalties, and adaptive diversity weights.
For MDR, nutshell, the meta learning procedure reweights using two different stages: the model stage and the design stage. While the design stage updates w using gradients V..LIOINT from loss + diversity; the model stage updates the model through gradients Ve of the w-weighted relevance objective. In detail, consider the outer loop to be the standard mini-batch training procedure that updates the model parameters for each mini-batch of data. For each data point in the mini-batch, we initialise the weight Wa,i and copy the current model (()) as meta-model. The inner loop is one-step optimisation of the V..LIO where LIOINT is evaluated using the meta-mos model. The value of weight is the gradient direction that maximises the LJOINT- Then we rescale and normalize the weight vector to obtain the weight distribution that implicitly encodes the contribution of the data point towards diversity gain. Then the outloop updates the model parameter by the weighted MSE loss (which is relevance-only).
16

---

Published as a conference paper at ICLR 2026
Approsimaat top-
& predicted rating
soft rank
Seft too matii
soflrosh(2))
LOORO- Eq.
inttaiinn Ortor men rasicatchee.
F.U.I)
Pa
L
LIOINT flay Bllas
wekgin updiate
Σ, F,
mant V.loney)
R: observed rating
Voliu Updano model parameter use reweighed rolevanee objective
Figure 4: MDR workflow
Direct diversity-guided tuning To obtain top-k relevant and diverse recommendations wittinennnnttioossitttttuttt without relying on post-hoc re-ranking or heuristic interventions-we propose an integrated approach. Our approach DDT integrates 3--balanced relevance-diversity loss into a standard end-to-end training pipeline. Specifically, we use gradient-based joint optimization, which promotes relevance and diversity. The overall procedure outlined in Algorithm This method preserves the simplicity and efficiency of conventional end-to-end training while explicitly aligning the optimization process with our dual objective. By embedding both goals within a single differentiable framework, DDT eliminates the need for separate re-ranking stages.
Meta diversity-guiders reweighting Our second approach is inspired by meta-learning and bilevel optimization, where hyperparameters per-example weights are optimized with respect to valida- tion objective (Ren et al., 2018). Here we ask "Which training examples, if included in the update, will reduce both training loss and improve diversity?" To this end, estimate and assign importance weights W.,i E [0, 1] to training samples (u, i, r) for each batch B S Ωr yielding
MSE
Σ (a.))
Wu,i RRusi Runs))
2
(10)
As shown in Algorithm 2, we begin each mini-bated update (line 4) by initializing the per-example weights W,,i = 0, effectively ignoring all samples. We then perform a one-step inner update to obtain temporary model @' using the weighted training loss EiSE (line 5-6), where w is currently all zero. Next, we re-evaluate predictions using the updated parameters (line 7) and compute the joint meta-loss LIONNT (line 8), combining relevance and diversity. For this, we compute the gradient of LJOINI with respect to producing a utility score for each sample. We rectify it via max( - V CJOINT 0) (line 9), and normalize (line 10) obtain the final per-example weights w, enforcing ( W = 1. Normalization ensures that the overall gradient magnitude- and thus the effective learning aitee-reming consistent across training steps, similar to standard Stochastic Gradient Descent (SGD), which averages over the batch (Ren et al., 2018). The final model update is then performed using this reweighted loss (line 11). This implicit diversity-guide optimization in MDR allows us to explore whether modifying the data sampling distribution alone can help achieve a decent eelevance-diversin trade-off. We provide empirical answer in Section 5, demonstrating that such implicit reweighting can be a practical and effective alternative to explicit multi mujti-objecttiv optimization.
17

---

Published as a conference paper at ICLR 2026
Algorithm Direct Diversity-Guidee Tuning (DDT)
Require:
Initialization model parameter θ training data Gt, item distance matrix D, trade-off B,
learning rate 1)
1: for each epoch from 1 to T do
2: for each mini-batch
S2p do Ru,i - (u, i), for all (u, i, RR,,)) B
3:
4:
EMSE t Compute LMSE (asseRRu,i Ru.t B) with (1)
5:
LDDRO +- compute DDRO with (5)
6:
LIOINT + B LMSE - (1 3) LDDRO
7: θ < θ 11 VeLJOINT
8: return
Algorithm 2 Meta Diversity Reweighting (MDR)
Require: Initialization model parameter O training data S2, item distance matrix D, trade-off (3,
learning rate
1: for each epoch from T do
2: for mini batches B E S2y do
3:
Ru, < Fe i), for all (u, Ru,i) B
4:
W ++0 0
5:
MSE Lu
MISE Ru,i Ru,,, B) with (10)
6:
O' +
-11 Ve VoLMES update meta model.
7:
R uit t F-rer(u, i). for all (u, i, R..i) B
8:
LIOINT + B MSE (1 5) LDDRO
9:
max w LJOINT. 0);
ΣS w- normalization according Ren al. (2018)
11:
θ - θ 11 VeLxi (Ry,s, (o Ru,i, w; B3)
return
+
C CONTROLLED TOy EXAMPLE ILLUSTRATING MDR
We present a controlled toy example to illustrate how Meta Diversity Reweighting (MDR) computes sample weights within mini-batei and how the resulting weights favor diversity.
In high-level, we consider mini-batch containing user-item interactions
{(U, ((U1, (b), (U,, (a), Ia)).
with learnable nonnegative weights
W = (wi, W2, W3),
Σ w -w==1 = 1.
Intuitively, MDR assigns larger weights items that contribute more diversity. This is achieved by performing a one-step update and computing the gradient of the joint relevance relevance-diversity objective with respect the weights. The resulting gradient encodes how increasing each sample's weight would influence diversity. After rectification and normalization, the weights form probability distribution that implicitly captures each sample's contribution to diversity improvement.
We consider a single mini-batch from the first training epoch containing five interactions from the same user:
Users: [5794, 5794, 5794, 5794, 5794],
Items: [1, 2, 3, 4, 5].
Ratings: [5.0, 5.0, 5.0, 5.0, 5.0].
We note that the items map [534, 38, 1604, 1958, 2064] in the MovieLens dataset. All ratings set equal to control for relevance effects, ensuring that weight differences arise primarily from the diversity objective.
18

---

Published as a conference paper at ICLR 2026
We define the pairwise item distance matrix
T0.0
1.0
0.5
1.0
0.0
1.0
0.0
1.0
0.0
1.0
D
0.5
1.0
0.0
1.0
0.5
1.0
0.0
1.0
0.0
1.0
0.0
1.0
0.5
1.0
0.0
This matrix reveals the following structure:
Cluster A: items 1 and 5 (distance 0).
Cluster items and (distance 0).
Item 3 lies between the clusters, with distance 0.5 to Cluster A and distance 1 to Cluster B.
Since diversity increases when selected items are far apart, we expect MDR to assign higher weight to representative items from different clusters, particularly item 3, which bridges both clusters.
Next, we give the step-by-step computation following Algorithm 2.
Step 1: Weight initialization.
Weights are initialized as
w (0)
[0, 0, 0, 0, 0].
Step Meta-model update.
We copy the current model parameters O' to obtain a meta-model
θ + Oe
and perform one gradient step on the weighted relevance loss:
O - O v/ (G')
Step 3: Compute weight gradient from joint objective. We evaluate the joint relevance-diversity objective the meta-model and compute
In this example, the gradient is
2-24499561, 1675.825, 60342.375, -39423.586, -35152.387].
negative gradient indicates that increasing the corresponding weight would reduce the joint loss (i.e., improve diversity and relevance).
Step Gradient rectification.
We convert the gradient into nonnegative importance scores:
@L
ú max
Ow,
yielding
22449.561, 0, 60342.375, 39423.586, 35152.387].
Step 5: Normalization.
We normalize to obtain the weight distribution:
" Σ W +
resulting in
w [0.143, 0.000, 0.383, 0.251, 0.223).
As expected, the toy example clearly shows that the MDR improve diversity by rise the weight of items from diverse clusters: the MDR give the highest weight to "middle" item 3, and relatively high weight for one (item 4) from cluster A and one (item 5) from cluster B, while the ather item from same cluster is low even 0. Interestingly, the weight of item 4 (B) is higher than item 5 (A), since item has the highest weight (0.383) and is closer to cluster A.
This controlled example demonstrates that MDR automatically assigns higher weights to samples that increase diversity, without explicitly encoding cluster information. The diversity signal emerges naturally from the gradient of the joint objective through the meta-learning step.
19

---

Published as a conference paper at ICLR 2026
D DETAILED EXPERIMENT SETTING.
Datasets. To evaluate our methods across different recommendation scenarios, we consider datasets from three domains: entertainment, product, and social recommendations, detailed in Tab. 1. To cover the entertainment domain, we use Netflix (Bennett & Lanning, 2007) and MovieLens (Harper & Konstan, 2015) for user-movie recommendations, user-music recommendations. These datasets contain user ratings on a 5-point scale [1, 5). along with genre or category annotations. For Netflix (respectively, Yahoo-R2), we randomly sample 000 items (respectively, 5 500) and retain users with at least 20 ratings (respectively. 100+ ratings). In the product recommendation setting, we use the Coat (Schnabel al., 2016), which captures user-coat interactions in e-commerce. It contains [1, 5] ratings and item 'meta' attributes. Finally, for the social recommendation scenario, we consider the KuaiRec (Gao al., 2022), which is collected from mobile video-sharing platform, which includes play duration, video length, and 'watch ratios' from 0 (never watched) to 2 (twice watched), which we linearly interpolate to 5-star ratings for consistency.
well as Yahoo-R2 (Dror et al., 2012) for
Baselines. We compare our algorithm against broad set of state-of-ther of-tte--rttttttt recommender-systemm methods, as well as diversification techniques covering greedy, probabilistic, and graph-based strategies. To study the impact of diversification, we employ classical Non-negative MF (NMF) (Lee & Seung. 1999) as relevance-only baseline that does not use any diversity mechanisms. We also include two baselines from the post-processing family: Maximal Marginal Relevance (MMR) (Carbonell & Goldstein, 1998) and Diversity-weighter Utility Maximization (DUM) ( (Ashkan et al., 2015), both greedy diversification techniques applied top of NMF the underlying model.
MMR greedily selects top-k items that maximizes a weighted combination of relevance and dissim- ilarity with previously selected items. DUM, on the other hand, uses submodular combination of relevance and category- diversity reward. Determinantal Point Processes (DPP) (Chen al., 2018) estimates the likelihood of item to be diverse and relevant the determinant of an item-item similarity kernel matrix, from which we select the top-k using a greedy selection. Finally, we include recent embedding-basee method, Diversified GNN Recommender (DGRec) (Yang et al., 2023), which introduces a diversity-aware aggregation mechanism into graph neural networks by selecting neighbors that maximize coverage over item categories.
Evaluation criteria. We evaluate model accuracy using metrics suitable for top-k recommendations, hit rate, precision, and recall metries. For each user u, let R., denote the top-k recommended items and To the set of ground -tutt relevant items (i.e., rated above 4 in the test set). Hit rate measures whether at least one relevant item appears in R.; precision is the fraction of items in R. that are in T,; and recall is the fraction of relevant items in Tu that are retrieved in R...
However, the above requires known ground-truth compute the potential user satisfaction as the relevance score
evaluate the relevance of unseen items, we
Relevance(u) u)
Ipp(Ra.)) r). RER.
Σ
(11)
using preference likelihoods p( (R...) E [0.
Experiment environment. We conduct all experiments on 2 AMD Epyc 7742 CPUs, TBB of RAM and I NVIDIA DGX-A100 GPU, Our code is written in Python v3.1 All results are averaged over 10 independent runs. We adopt the Adam optimizer during the optimization.
Detailed Parameter setting. In all experiment, we set = 10 for a exact rank approximation. During the optimization with adaptive B, we choose learning rate I = 0.01 for the matrix factoriza- tion model, and = 0.001 for the neural network model. We keep the Adam weight decay parameter as the default.
E ADDITIONAL EXPERIMENT
We report the additional experiment results that are omitted in the main content. Figure 5 shows the adaptive 3 based optimization of two datasets (MovieLens and Netflix) and two models (NMF and NCF). The patterns are consistently aligned with the analysis. Figure 6 shows the 4 accuracy metric
20

---

Published as a conference paper at ICLR 2026
D
Lower bound
Upper bound
Lower bound
Upper bound
Lacss
DDRO
DRO
Luss
DDRO
DRO
NCF Train
NCF Line-tune
NMF + Train
Fine-tune
1.0
1.0
0.5
00. 0.5
0.5
0.0
0.0
0.0
0.0
45
75
90
0
15
30
45
60
75
90
0
15
30
15
60
75
90
15
30
45
60
75
90
Diversity
1.0
1.0
Loss Diveritty
1.0
0.9
0.9
0.8
0.8
0.8
Epoch
Epnch
Epoch
Epoch
(a) MovieLens
(b) MovieLens
D
Upper
bound
Upper bound
DDRO
DRO
Lussr
DDRO
DRO
NCF Train
NCF
Fine-tune
NMF Train
NMF Fine-tune
1.0
1.0
1.0
1.0
0.5
2 0.5
0.5
0.0
0.0
0.0
15
30
15
60
75 90
15
30
45
60
75
90
0
15
30
15
60
75
90
15
30
45
60
75
90
Loss Diversity
Loss Diverityy
1.0
0.95
0.9
0.9
0.8
0.8
0.85
0.7
90
30
15
30
45 60 Epoch
75 90
30
Epoch
Epuch
Epoch
(c) Netflix
(d) Netflix
Figure 5: Optimization of relevance-diversity joint objective with adaptive B.
Table 5: Diversity gain achieved by Direct Diversity Tuning (DDT) top-1 i and k + 1 ~ 2k: recommendations across five datasets Coat, R2 Netflix, MovieLens, and KuaiRec. We vary the diversity reward parameter & {5, 10, 20, 30, 40} and apply MDR to tre-trained NMF and NCF, reporting diversity gain. We report the mean and standard deviation across 10 runs.
Diversity gain (%)
Dataset
1 A
I++1 2k
E=
k=
k=10
E=
Kuailtoe
815(1.6)
5.202
6,3 (0.6)
(1.3)
(0.)
(1.3)
10.1 (0.7)
299 (0.9)
10.3 (0.8)
(0.8)
NMP
Coat
5.2(3.4)
(4.0)
(131)
(3.7
(1.3)
(4(1.))
(..))
-0.02.0)
1.8 (0.7)
(1.2(1.1)
MovieLens
(614)11.1)
(2.9
(6.9)
12.1 (4.4)
24.1
(2.)
21.7
6.(1.7)
19.7 (2.6)
6.(144)
Netflix
(2.2
4.8 (3.2)
13.0(1.3)
2.11
(0.9
(..17
9.2
(2400.)
8.7 (0.5)
1.45
Yahoo-RL R2
49.8(19.11
(10.8)
(9.0412.4)
65.1 (7.7
(2.4)
(3.5
31,(113)
66.5 (2.3)
(9.6
KuaiRee
1.3(1.8)
(0.2
(.34)
(1.8
1.1 (0.9)
(.9(1.2)
(.3)
0.2
1.4 (0.6)
0.0
NOF
Coat
0.8.2..3)
0.9 (1.2)
0.8
0.3 (0.7)
0.7 (0.9)
(..
0.3 (0.6)
(0.3
(0.4)
(0.6)
Moviel
(1.2(11.8)
6.1
(5.9)
366 (2.3)
(3.)
00.5 (.9.).)))
(2.)
(1.3)
(1.7
0.6(1.0
Netflix
4.642.44
(.4()
-0.1 (..3)
1.1 (0.)
(0.4
-0.2 (0.3)
0.2 (0.3)
-0.3
Yahoo-R2
(150
(8.8
(10.2)
23 (8.3)
19.5
(7.1
16.7 (8.1)
(5.4).
(6.9
(.2))
results: recall, mean squared error, hit rate, and precision, along with the diversity reword objective score (DRO). Similar to Table 3, Table presents the in-objective and out-obeective diversity gain of NMF and NCF with MDR.
21

---

Published as a conference paper at ICLR 2026
NCF
Cost
Yahoo
Netflx
MovicLens
KuaiRee
0.8
0.8
0.2
Recall
0.6
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
2.0
2.00
2.0
2.0
MSS
1.75
1.5
1.5
1.50
1.0
0.0
0.5
0.5
1.0
0.0
0.5
1.0
0.0
1.0
0.5
HB
0.6
0.8
0.8
0.8
0.8
0.0
8
8
0.6
0.8
Predisuu
0.8
0.2
0.4
0.6
0.0
0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
0.85
1.0
0.975
Diverdty
0.95
0.8
0.950
0.50
0.90
0.925
0.6
0.75
0.0
0.85
0.500
0.0
0.5
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
Figure 6: Performance comparison of DDT and MDR applied to two recommender models (NMF and NCF) across five datasets: Coat, Yahoo-R2, Netflx, MovieLens, and KuaiRec, We vary the parameter E [0. 1] to control the trade-off between relevance and diversity, where 3 = 1 corresponds to optimizing only the relevance loss LMSE and
corresponds optimizing only the diversity loss LDDRO. The r-axis indicates the value of B. The y-axis shows recall (top row), mean squared error( second row), hit rate (third row), precision (fourth row), and diversity reword objective (DRO) score with & 10 (bottom row). In each setting, we initialize from pre-trained model (using LMSF only), then fine-tune with the joint loss LIONNT for 10 epochs, selecting the best result by diversity score. All experiments repeated 10 times, and we report the mean and standard deviation.
22

---

Published as a conference paper at ICLR 2026
Metric
Before
DDT (3 0.2)
DDT (3 0.5)
DDT 0.8)
BPR
0.3125
0.3008
0.3008
0.2973
Precision @ 10
0.0040
0.004)
0.0039
0.0040
Recall@ 10
0.0396
0.0404
0.0394
0.0404
HitRate @ 10
0.0396
0.0404
0.0394
0,0404
NDCG@ 10
0.0191
0.0200
0.0200
0.0203
MAP@ 10
0.0129
0.0139
0,0142
0,0143
DDRO
0.7776
0.8778
0.8813
0.8833
DRO
0.7803
0.8812
0.8815
0.8848
Table 6: Performance before and after DDT fine-tuning with BPR loss under different values of B Each model fine-tuned for 10 epochs.
23


Reviews:
[{"review_id": "DuW6eh7ABw", "reviewer": "Reviewer_LYyX", "summary": "The paper proposes a model-agnostic, differentiable top-k diversity objective for ranking. It relaxes the non-differentiable top-k set via soft ranking/permutahedron projection and couples it with a relevance loss, optimizing a joint objective either Direct Diversity Tuning and Meta-Diversity Reweighting. Experiments on five datasets (MovieLens, Netflix, Yahoo-R2, Coat, KuaiRec) with MF/NCF models show large diversity gains with minimal relevance drops.", "questions": "1. How sensitive are results to the choice of the item similarity matrix (e.g., Jaccard on genres vs. learned embedding cosine vs. taxonomy distance)?\n\n2. Any plans for counterfactual or bandit-style evaluation to account for feedback loops and exposure bias?\n\n3. Does optimizing DDRO measurably increase long-tail exposure?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper proposes a differentiable top-k diversity via soft rankings.\n2. The MGDA-based adaptive parameter has a transparent feasible region and a closed-form. It guarantees convergence to Pareto-stationary points under standard conditions.\n3. Two complementary routes (DDT vs. MDR) cover both objective-level and data-level interventions.", "weaknesses": "1. The DDRO approximation error depends on $\\epsilon$ and k. For larger k, the method needs a smaller $\\epsilon$ to be accurate, which may affect stability/cost. The paper notes this, but practical guidance/ablation across datasets is limited.\n\n2. While MDR reduces bias via reweighting, classic recommendation confounders (position bias, missing-not-at-random feedback) aren’t explicitly modeled.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BY1GvbHiUh", "reviewer": "Reviewer_B1Yz", "summary": "This paper addresses the issue that existing recommendation models often optimize solely for relevance and user engagement, which may amplify data bias and reduce the diversity of recommended results. To tackle this, the authors propose a general pairwise Top-K diversity optimization framework that formulates relevance and diversity as a joint optimization problem. The approach achieves diversity enhancement by modifying the learning objective function and introducing a reweighting mechanism.", "questions": "See weakness", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.The authors propose Direct Diversity-Guided Tuning (DDT), which augments the original loss function by incorporating a relevance–diversity joint term.\n\n2.The authors further introduce Meta-Diversity Reweighting (MDR), a mechanism that retains the conventional relevance-based training pipeline while adopting the joint loss as a meta-objective to reweight training samples dynamically.\n\n3.Rich experimental results", "weaknesses": "1.Readability needs improvement. Although the paper includes a series of lemmas and proofs to validate the proposed method, the workflow of DDT remains abstract and difficult to follow. It is recommended to include a model architecture or workflow diagram for DDT and MDR to make the methodology more comprehensible.\n\n2.The rationale of meta-learning-based reweighting requires further clarification. In Section 4.2, the authors employ meta-learning to adjust sample weights, yet it is not clear how this process ensures the preservation of relevance. Specifically, the computation of the weight w_{u,i} is not described in detail (it seems to be obtained via an MLP). If this is the case, additional explanation is needed to justify why using an MLP-based reweighting does not compromise the relevance term in Equation (8).\n\n3.Implementation details of DDT are insufficient. The loss formulation of DDT is not presented in Section 4.1 but rather briefly mentioned in Section 3, which is not reader-friendly. Furthermore, Algorithm 1 only states “compute DDRO with (5)” without further elaboration. I hope the author can describe the practical details of the DDT method in section 4.1.\n\nIf the author can resolve my doubts, I am willing to raise my score.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2HcKnKp6Nn", "reviewer": "Reviewer_n8VJ", "summary": "This paper focuses on relevance-diversity optimization in recommender systems. Specifically, it proposes a differentiable top-$k$ diversity reward objective (DDRO) that can be integrated into RS training in a model-agnostic and loss-agnostic manner. To achieve this, the authors introduce the softrank technique to approximate the non-differentiable sorting operation, and derive a soft relaxation of the top-$k$ diversity reward (DRO), which is defined as the average pairwise distance dissimilarity among the top-$k$ recommended items. Furthermore, they borrow the idea of MGDA for multi-objective optimization to balance relevance and diversity during training. The proposed method is evaluated in both training-from-scratch and fine-tuning settings on various datasets and two recommendation backbones, exhibiting improvements in both relevance and diversity.", "questions": "- **Evaluation Metrics.** In the main results (Tables 2, 3, and 5, Figures 1 and 3), the reported revelance metric is defined as the proportion of recommended items whose likelihood is greater than a certain threshold. However, this metric is not commonly used in the recommender systems literature --- it merely measures the sharpness of the predicted probability distribution, which may not directly reflect the recommendation quality. Although Figure 4 presents Recall, MSE, HR, and Precision results, the baseline methods are not compared. Thus, I suggest the authors include more standard recommendation metrics (e.g., Recall@$k$ and NDCG@$k$) in the main results for fair comparison.\n- **Recommendation Losses.** It seems that the proposed DDRO can be integrated with any recommendation loss. Nonetheless, the experiments only consider the MSE loss, which has been shown to be suboptimal for recommendation tasks. It would be more convincing if the authors could evaluate DDRO with other widely used recommendation losses, such as AUC-oriented losses (e.g., BPR [R1]), ranking-oriented losses (e.g., PSL [R2]), and top-$k$-oriented losses (e.g., LambdaLoss@$k$ [R3] and SL@$k$ [R4]), to see whether the improvements still hold on both accuracy and diversity compared to these SOTA baselines.\n- **Top-$k$ Approximation.** The authors employ the top-$k$ soft indicator based on the softrank technique to handle the non-differentiability. Recently, there are several works on differentiable top-$k$ operators that provide an alternative quantile-based approximation (e.g., SL@$k$ [R4] and SONG@$k$ [R5]). It would be beneficial to compare these approaches with the approximation used in this paper, both theoretically and empirically.\n- **Minor Comments.** The proof of Lemma 3 seems to be generated by LLMs (although no errors were found). The proof is in fact quite straightforward, so I suggest the authors simplify it to make the presentation more concise.\n\n**References:**\n- [R1] BPR: Bayesian Personalized Ranking from Implicit Feedback. UAI '09.\n- [R2] PSL: Rethinking and Improving Softmax Loss from Pairwise Perspective for Recommendation. NIPS '24.\n- [R3] On Optimizing Top-K Metrics for Neural Ranking Models. SIGIR '22.\n- [R4] Breaking the Top-K Barrier: Advancing Top-K Ranking Metrics Optimization in Recommender Systems. KDD '25.\n- [R5] Large-scale Stochastic Optimization of NDCG Surrogates for Deep Learning with Provable Convergence. ICML '22.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- This paper introduces a differentiable top-$k$ diversity objective by leveraging the softrank technique, facilitating end-to-end optimization.\n- This paper performs relevance-diversity optimization in a joint manner with theoretical guarantees.\n- The proposed method is evaluated with comprehensive experiments on multiple datasets and recommendation backbones, demonstrating its effectiveness in improving both relevance and diversity.", "weaknesses": "- The evaluation metrics used in the main results are not widely adopted in the previous work, making it difficult to assess the recommendation quality.\n- The experiments only consider the MSE loss, which may limit the generalizability of the proposed method.\n- The choice of top-$k$ approximation could be further justified by comparing with other differentiable top-$k$ operators.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)