PreferGrow: Discrete Diffusion Recommender

• PreferGrow is a discrete diffusion-based recommender that models user preference ratios via fading and growing processes over item indices to address sparse feedback.
• It replaces continuous noise injection with a discrete replacement approach that guarantees Markovianity and reversibility, directly aligning with ranking tasks.
• Empirical results on benchmarks like MovieLens and Steam show that PreferGrow variants significantly outperform traditional methods in top-K recommendation accuracy.

Searching arXiv for PreferGrow and closely related papers to ground the article. PreferGrow is a discrete diffusion-based recommender system that models preference ratios by fading and growing user preferences over the discrete item corpus. In the formulation reported for recommendation, it departs from continuous diffusion on item embeddings or score vectors and instead operates directly on item indices, with a forward “preference fading” process that replaces a user’s preferred item with alternatives and a reverse “preference growing” process that reconstructs user-specific preferences from estimated ratios. The method is presented as aligning the diffusion process with the discrete and ranking-oriented nature of recommendation tasks, and it is described as offering a matrix-based formulation with theoretical guarantees on Markovianity and reversibility (Hu et al., 30 Sep 2025).

1. Definition and problem setting

PreferGrow is introduced for standard top-$K$ recommendation and sequential recommendation over a discrete item corpus $\mathcal{X}$ of size $N$. The user preference data is a pair $(u,i)$, where $u$ is either a user ID or, in the sequential case, the sequence of items the user has interacted with, and $i$ is the preferred item. The goal is to rank all items in $\mathcal{X}$ for each user $u$ so that the positive item or items appear in the top-$K$ positions (Hu et al., 30 Sep 2025).

The motivating difficulty is extreme sparsity in implicit-feedback recommendation. The reported datasets have user–item matrices that are more than 99.9% empty for most datasets, which makes it difficult to learn reliable scores $p(i\mid u)$, avoid overfitting to observed positives, and exploit negative signal that is not explicitly labeled (Hu et al., 30 Sep 2025).

Within that setting, PreferGrow is characterized by three core aspects: discrete modeling of preference ratios, perturbing via preference fading, and preference reconstruction via growing. It models relative preference ratios between item pairs rather than operating in the item representation or raw score simplex; it fades user preferences by replacing the preferred item with alternatives; and it reconstructs user preferences by iteratively growing the preference signals from the estimated ratios (Hu et al., 30 Sep 2025).

2. Relation to diffusion-based recommendation

The paper situates PreferGrow against two families of diffusion recommenders. The first is item-level diffusion, where a positive item is encoded as an embedding $\mathcal{X}$0 and perturbed with Gaussian noise. The second is preference-score-level diffusion, where a preferred item is represented as a one-hot vector over items and perturbed within the probability simplex, using Gaussian, Bernoulli, or categorical constructions in prior work. The reported critique is that these methods either inject noise in a continuous embedding space although items are discrete indices, or impose simplex constraints and prior noise distributions that are not grounded in user behavior (Hu et al., 30 Sep 2025).

PreferGrow replaces those constructions with a discrete diffusion process on item indices. It models pairwise preference ratios rather than raw scores, and it treats perturbation as discrete replacement of the preferred item. The paper states that this is physically akin to negative sampling and eliminates the need for any prior noise assumption (Hu et al., 30 Sep 2025).

This positioning connects PreferGrow to a broader shift from absolute scoring to relative comparison. A related but distinct line in generative modeling, exemplified by preference-guided image generation with paired comparisons in “PrefGen” (Helbling et al., 2023), also treats relative judgments as easier to elicit and more natural than explicit scalar control. In recommendation, PreferGrow instantiates that relative-preference perspective through discrete pairwise ratios rather than human-in-the-loop comparison queries.

3. Preference ratios and the forward fading process

The central modeling object is a pairwise preference ratio. For items $\mathcal{X}$1 and $\mathcal{X}$2, the Bradley–Terry form is written as

$\mathcal{X}$3

PreferGrow leverages preference ratios induced by the diffusion process rather than directly parameterizing the unknown $\mathcal{X}$4 (Hu et al., 30 Sep 2025).

The forward diffusion is defined as a preference fading process. For each time $\mathcal{X}$5,

$\mathcal{X}$6

where $\mathcal{X}$7 is a Dirac delta, $\mathcal{X}$8 is a decreasing retention probability with $\mathcal{X}$9, and $N$0 is the fading matrix. In matrix form,

$N$1

Operationally, with probability $N$2 the process keeps the original preferred item $N$3, and with probability $N$4 it swaps that item for an alternative sampled from $N$5 (Hu et al., 30 Sep 2025).

The structural assumption is that $N$6 is idempotent, $N$7. Under that condition, the fading process is stated to be Markov and invertible, with

$N$8

and inverse

$N$9

The paper presents this as the basis for a reversible discrete diffusion with a closed-form transition structure (Hu et al., 30 Sep 2025).

A further proposition states that if the chain converges to a unified non-preference state $(u,i)$0, then

$(u,i)$1

so that the fading matrix is rank-1 and every column corresponds to sampling from the same non-preference distribution $(u,i)$2 (Hu et al., 30 Sep 2025).

4. Preference reconstruction through growing

The reverse process is the preference growing stage. The paper states that if the forward process with idempotent $(u,i)$3 converges to $(u,i)$4, then the reverse process is also Markovian and satisfies a Kolmogorov backward equation. The reverse transition is given in matrix form by

$(u,i)$5

The rate matrix is likewise expressed in terms of the forward rate and a matrix of preference ratios (Hu et al., 30 Sep 2025).

For training and inference, the unknown marginal ratios are approximated with a neural model. The paper gives the practical reverse-step form

$(u,i)$6

where $(u,i)$7 approximates $(u,i)$8 (Hu et al., 30 Sep 2025).

At inference time, generation starts from the non-preference state. One samples $(u,i)$9, then applies a sequence of reverse steps from $u$0 down to 0. The guided ratio estimate is

$u$1

where $u$2 is a learned non-preference user and $u$3 controls personalization strength (Hu et al., 30 Sep 2025). This guidance mechanism resembles classifier-free guidance in the sense used by the paper; a plausible implication is that PreferGrow separates generic and personalized preference components and recombines them only during reconstruction.

5. Fading mechanisms, negative sampling, and learning objective

The non-preference state $u$4 determines how fading replaces positives. The paper describes four settings (Hu et al., 30 Sep 2025):

Mechanism	Definition	Interpretation
Point-wise fading	$u$5	Replace any item with a general hard negative item
Pair-wise fading	$u$6	Uniform negative sampling over real items
Hybrid fading	$u$7	Mix point-wise and pair-wise preferences
Adaptive fading	$u$8	Learn the negative sampling distribution

The paper identifies pair-wise fading with classic uniform negative sampling, and it describes hybrid fading as combining “positive vs mask” and “positive vs random item” preferences (Hu et al., 30 Sep 2025).

The model architecture uses item embeddings, a user embedding from a sequential recommendation backbone instantiated as SASRec with RoPE positional encoding, and a time embedding. The preference ratio network is

$u$9

and the intended target is

$i$0

Thus the network predicts itemwise log-ratio scores conditioned on the current faded item, timestep, and user context (Hu et al., 30 Sep 2025).

For learning, PreferGrow adopts the Score Entropy loss from DDSE: $i$1 The loss uses the exact reference ratio induced by the forward kernel. The paper further states a proposition connecting Score Entropy to soft-label BCE: $i$2 and concludes that the two objectives share the same descent direction and the same optima (Hu et al., 30 Sep 2025). This places PreferGrow in direct continuity with pairwise ranking methods such as Bradley–Terry-style comparison and BCE with negatives.

6. Theoretical properties, empirical evaluation, and scope

The theoretical results emphasized for PreferGrow are Markovianity and reversibility of the fading process, existence of a unified non-preference state, validity of the reverse preference-growing process as a discrete diffusion, and applicability of the Score Entropy loss under the derived forward rate matrix $i$3 (Hu et al., 30 Sep 2025).

On complexity, the paper reports that for PreferGrow the model parameters are $i$4, loss computation is $i$5 per sample, the conceptual modeling target is $i$6 possible ratios but is implemented as $i$7, and inference is $i$8. The paper attributes the practical reduction to the rank-1 or rank-$i$9 structure of the fading matrix $\mathcal{X}$0 (Hu et al., 30 Sep 2025).

The reported implementation uses PyTorch on a single NVIDIA RTX 3090, SASRec with RoPE, batch size 256, Adam, early stopping on validation metrics, and diffusion steps $\mathcal{X}$1, with performance often saturating around 20 steps (Hu et al., 30 Sep 2025).

The empirical evaluation uses five benchmarks: MovieLens 1M, Steam, Amazon Beauty, Amazon Toys & Games, and Amazon Sports & Outdoors, with chronological user-based 8:1:1 splits and all-ranking evaluation using HR@K and NDCG@K for $\mathcal{X}$2. Baselines include SASRec, Caser, GRURec, DreamRec, PreferDiff, DiffRec, and DDSR (Hu et al., 30 Sep 2025).

The reported result is that PreferGrow, particularly the Hybrid and Adaptive variants, consistently outperforms all baselines across all datasets and $\mathcal{X}$3. On Steam, for example, the paper reports NDCG@10 values of approximately 0.0272 for SASRec, approximately 0.0268 for DiffRec, approximately 0.0216 for DDSR, and approximately 0.0508 for PreferGrow-Hybrid (Hu et al., 30 Sep 2025). In a multi-seed re-evaluation on NDCG@5, the paper reports 0.0193 ± 0.0002 for SASRec, 0.0104 ± 0.0004 for PreferDiff, and 0.0399 ± 0.0004 for PreferGrow on Steam, and 0.0507 ± 0.0004 for SASRec, 0.0348 ± 0.0005 for PreferDiff, and 0.0913 ± 0.0003 for PreferGrow on MovieLens (Hu et al., 30 Sep 2025).

Ablation results on Steam indicate that pairwise preference ratios are crucial, non-preference user modeling significantly boosts personalized guidance, and Hybrid and Adaptive fading improve over pure point-wise or pure pair-wise designs (Hu et al., 30 Sep 2025). The paper also notes limitations: higher modeling complexity, slower training than some baselines, continued $\mathcal{X}$4 dependence that remains challenging for very large item corpora, and the need for scalable Semantic IDs pipelines for industrial-scale deployment (Hu et al., 30 Sep 2025).

In a broader research context, the term “PreferGrow” appears in other domains as an organizing idea rather than this specific recommender architecture. In multi-objective quality-diversity, preference-conditioned policy-gradient mutations guide the growth of solutions toward desired trade-offs (Janmohamed et al., 2024). In LLM pretraining, a “preference curriculum” schedules data so that a model is always trained on data preferred by a stronger version of itself (Zhang et al., 21 Jan 2025). In both cases, the shared pattern is preference-guided growth, but PreferGrow in the strict technical sense denotes the discrete diffusion recommender built around preference fading and growing over item corpora (Hu et al., 30 Sep 2025).