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Beyond Semantic IDs: Finite-Beam Reachability and Continuous-Query Generation in Generative Recommendation

Yu Sun, Renkai Xiang, Zuodong Xiang, Junwei Zhou, Yike Zhang, Jianxin Zhang, Zhihao Zhu, Xiaoyu Cao, Hailu Xu

arXiv:2610.12977v1cs.AI

TL;DR

Finite-beam Semantic-ID retrieval can leave held-out targets outside returned lists, motivating a closer study of output support. The paper compares single-step and autoregressive continuous-query methods with SID retrieval. CQG-Single shows a stable aggregate R@10 gain across seven seeds, while retrieval crossovers vary with checkpoint, popularity, and depth.

  • Problem

    Finite-beam SID decoding can leave valid target codes outside the explored paths, while SID support and coverage can be incomplete and popularity-skewed.

    Source: §1 · p. 2

  • Method

    The paper compares CQG-Single, which generates one continuous query, and CQG-AR, which generates continuous residuals autoregressively, with SID retrieval.

    Source: §1 · p. 2, §8 · p. 10

  • Results

    CQG-Single achieves a stable aggregate R@10 gain across seven training seeds and is numerically stronger than TIGER-style SID in 38 of 48 popularity–depth comparisons.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The observed retrieval crossover depends on checkpoint, item popularity, and depth, so neither interface is shown to dominate universally.

    Source: §7 · p. 10

    Evidence is specific to the evaluated model–search–compute configurations; billion-item training, ANN latency, memory, and dynamic-catalog accuracy remain untested.

    Source: §7 · p. 10

Abstract

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Generative recommendation (GR) of social media platforms commonly retrieves items by autoregressively decoding discrete Semantic IDs (SIDs) with finite-width beam search. This work examines how this output interface constrains whether held-out targets enter finite returned lists, while distinguishing catalog-wide scoreability from empirical top-K support. We introduce two continuous-query paradigms: CQG-Single generates one retrieval query in a single pass, and CQG-AR autoregressively generates continuous residual updates, retaining step-wise decoding while removing the SID item-token vocabulary, code lookup, and discrete beam expansion. These designs thereby eliminate the SID-specific reachability constraint induced by finite-beam decoding and avoid the technical challenge of aligning a discrete SID vocabulary with the pretrained backbone's vector space. Both methods make every encoded item directly scoreable through dense or approximate-nearest-neighbor retrieval. Under a unified protocol with valid-prefix constrained decoding, CQG-Single exceeds the TIGER-style baseline in all seven runs while using 55.3% as many trainable parameters. CQG-AR remains comparable on ML-1M and is numerically stronger on Beauty and Sports datasets. A controlled depth ablation further shows that additional continuous refinement does not monotonically improve shallow accuracy. These findings motivate decoupling output representation, autoregressive modeling, and retrieval budget when evaluating generative recommendation systems.

1 Introduction

The introduction examines how finite-beam SID decoding limits target support and presents CQG-Single and CQG-AR as continuous-query alternatives. Its reported comparisons show seed-stable gains alongside a crossover that depends on checkpoint, popularity, and retrieval depth.

1 Introduction

  • Generative recommendation retrieves items by generating symbolic identifiers; Semantic-ID systems quantize items into discrete code sequences that autoregressive models decode under constrained generation.The introduction describes SID pipelines as requiring code construction, vocabulary alignment or a separately trained decoder, and mapping generated codes back to items.Source: §1 · pp. 1–2
  • Finite-beam decoding explores only a prefix-limited subset of code paths, creating a checkpoint- and budget-specific target-support ceiling even for held-out items with valid codes.The introduction also identifies longtail items as having worse SID reachability because items are not distributed uniformly across SID leaves.Source: §1 · p. 2
  • CQG-Single achieves a stable aggregate R@10 gain across seven training seeds and is numerically stronger than TIGER-style SID in 38 of 48 popularity–depth comparisons.The reported crossover varies with checkpoint, item popularity, and retrieval depth rather than establishing a universal ranking.Source: §1 · p. 2
  • CQG-Single generates one continuous query, while CQG-AR generates continuous residual updates autoregressively, separating the effect of autoregression from a discrete SID vocabulary.Both use dense or approximate-nearest-neighbor retrieval without code-to-item lookup.Source: §1 · p. 2
  • The study distinguishes unique-code assignment, target-conditioned beam support, catalog-wide scoreability, and empirical union-of-top-K coverage, with output-based measures stratified by popularity.It evaluates retrieval quality, finite-output coverage, objective controls, and LLM-backbone diagnostics across ML-1M, Amazon Beauty, and Amazon Sports, limiting scalability claims to tested catalog sizes.Source: §1 · p. 2

2 Related Work

Related work establishes discrete identifier decoding and catalog scoring as existing retrieval interfaces. This paper instead studies finite-beam support and uses paired continuous-query methods to compare output representations while controlling for autoregressive structure.

2 Related Work

  • Prior Semantic-ID systems differ in code construction but generally retain structured outputs and constrained autoregressive decoding; this work studies finite-width search support rather than proposing another discrete tokenizer.The cited discussion distinguishes successful code assignment from whether a target is retrieved at inference time.Source: §2 · p. 2
  • CQG-Single removes both the item-token vocabulary and serial decoding, whereas CQG-AR retains step-wise generation through continuous residual updates.The paired design tests whether step-wise generation requires discrete identifier tokens; it does not claim dense retrieval itself as novel.Source: §2 · p. 3
  • Prior continuous or textual-query methods use diffusion, continuous latent generation, or textual retrieval queries, while adjacent multimodal and efficient-model studies do not establish results on SID reachability or these CQG methods.The related-work section positions CQG-Single as a minimal continuous-query control and CQG-AR as its autoregressive counterpart.Source: §2 · p. 3
  • Dense catalog scoring and identifier decoding are both established retrieval interfaces, so CQG-Single and CQG-AR provide a controlled comparison in a shared frozen item space.The comparison does not attribute its results to a larger item encoder or an additional reranker.Source: §2 · p. 3

3 Method

The method compares discrete SID retrieval with two continuous-query designs using the same user context and frozen item embedding space. It defines alternative training objectives and retrieves continuous outputs directly against the item index.

3.1 Problem Formulation

  • The task is next-item retrieval from a user's chronological interaction sequence; SID-based GR maps items to discrete code sequences and decodes valid code prefixes with beam search.Both continuous methods use a frozen matrix of L2-normalized item embeddings and the same user context available to the SID decoder.Source: §3.1 · p. 3

3.2 CQG-Single: Single-Step Continuous Query Generation

  • CQG-Single maps the final valid history state into the shared item space with a causal Transformer and produces a retrieval query in one forward pass without an item-token vocabulary.The query is ranked by cosine similarity, and retrieval uses exact dense scoring or ANN; every indexed item is scoreable, though not guaranteed to appear in a finite top-K list.Source: §3.2 · pp. 3–4, Figure 1 · p. 4

3.3 CQG-AR: Autoregressive Continuous Query Generation

  • CQG-AR retains an autoregressive decoder but replaces discrete SID-token prediction with continuous residual generation conditioned on user context and previous outputs.Residuals are fed back as subsequent inputs; after L steps, the refined query retrieves from the same frozen embedding matrix without codebook lookup or discrete beam expansion.Source: §3.3 · pp. 4–5

3.4 Training Objectives

  • The study evaluates full-catalog CE, sampled CE, in-batch InfoNCE, and MSE objectives for the continuous-query methods.Full-catalog CE provides complete catalog visibility at training time but costs O(N) per step; MSE preserves geometric alignment but yields lower retrieval accuracy than CE.Source: §3.4 · p. 5

3.5 Retrieval and Contrast with SID

  • CQG scores items directly in the embedding index, using exact catalog scoring or ANN, whereas SID retrieval requires codebook learning, discrete beam search, and code-to-item lookup.The reported CQG-AR retrieval uses only the final query; merging candidates from intermediate or interest-specific queries is untested.Source: §3.5 · p. 5

4 Experiments

Experiments use ML-1M for exact full-catalog diagnostics and Beauty and Sports for cross-domain evaluation. The setup specifies a leave-one-out protocol, comparison baselines, and continuous-query model configurations.

4.1 Experimental Setup

  • ML-1M is the primary controlled diagnostic, while Beauty and Sports test cross-domain behavior; dataset density measures interactions relative to the user-item catalog.Higher density indicates users interact with a larger fraction of the catalog; lower density indicates sparser interactions and a more pronounced long tail.Source: §4.1 · p. 5, Table 1 · p. 6
  • The processed Sports split differs from the commonly reported 5-core split, so its values are not directly compared with published results.The paper uses “TIGER” for its hierarchical implementation of k means in TIGER-style.Source: §4.1 · p. 6
  • Evaluation holds out the last item for testing and the second-to-last for validation, excludes historical items from ranking, and reports Recall@K and NDCG@K.Reachability analysis primarily uses K ∈{10, 30, 50, 70, 90}; cross-dataset evaluation uses K ∈{5, 10, 20}.Source: §4.1 · p. 6
  • Comparisons include SASRec, TIGER-style SID retrieval, published SID results where protocols allow, and random, last-item kNN, and mean-of-5 kNN retrieval.The cited prior methods are SASRec, TIGER, LETTER, and EAGER.Source: §4.1 · p. 6
  • A two-layer SASRec trained with BPR supplies a frozen 64-dimensional item-embedding table shared by both continuous methods; the main experiments use no textual or language-model item encoder.The checkpoint is selected by full-catalog validation NDCG@10.Source: §4.1 · p. 6
  • CQG-Single generates one normalized query, whereas CQG-AR autoregressively generates three residual updates and supervises only its final query with full-catalog cross-entropy.The Sports multimodal diagnostic instead uses a frozen fusion of collaborative and ImageNet-pretrained ResNet-18 features.Source: §4.1 · p. 6

5 Results

The results examine how finite returned lists support held-out targets and compare continuous-query retrieval with SID decoding across seeds, popularity strata, and datasets. CQG-Single’s gains are stable across ML-1M runs, while CQG-AR is comparable on ML-1M and stronger on the Amazon datasets.

5.1 Budget-Specific Finite-Beam Reachability

  • 66.7%, 67.3%, and 66.4% are the seed-42 TargetInOutput@100 values for TIGER, CQG-Single, and CQG-AR, respectively.Target support is the fraction of test users whose held-out target appears anywhere in the returned list; for the fixed model–search pair, it is a ceiling for reranking restricted to the decoded beam.Source: §5.1 · pp. 6–7
  • CQG-Single retains an advantage in overall user-weighted TargetInOutput@100 and all four item-balanced popularity quartiles; Figure 2 shows it matching or exceeding TIGER in each quartile.Figure 2 evaluates the first 100 positions of matched Top-200 lists; its bars are user weighted, unlike the item-balanced quartile values in Table 2.Source: Figure 2 · p. 7, §5.1 · p. 8
  • CQG achieves broader union-of-output catalog coverage than TIGER on Beauty and, more modestly, at Top-100 on ML-1M; 33.3% of held-out user targets miss the corrected Top-100 candidate list.That 33.3% is a user-target result, not the fraction of catalog items never retrieved; Table 4 reports the separate catalog-coverage diagnostic.Source: §5.1 · p. 8, Table 4 · p. 9

5.2 Seed-Stable Gains and Popularity-Dependent Crossover

  • 0.0107 mean paired R@10 gain: CQG-Single outperforms TIGER-style SID in all seven ML-1M runs (95% paired t interval [0.0021, 0.0193]; exact sign-flip p = .0156).The reported comparison is in Table 3.Source: Table 3 · p. 7, §5.2 · p. 8
  • CQG-AR remains comparable to TIGER-style SID on ML-1M and has higher mean R@10 on Beauty and Sports; on Sports, R@10 rises from 0.0236 to 0.0355 (50.4% relative).The seed-42 paired difference on Sports is 0.0146 (95% bootstrap CI [0.0119, 0.0173], p < .0002).Source: §5.2 · p. 8

6 Analysis

The analysis tests how continuous refinement, training-time catalog visibility, SID construction, and returned-list size affect the observed results. It finds no monotonic shallow-accuracy gain from added refinement and treats the robustness and list-size findings as bounded diagnostics.

Effect of Autoregressive Refinement

  • R@10 varies only from 0.2909 to 0.2944 as CQG-AR depth increases from one to four steps in the controlled ablation; additional refinements do not monotonically improve shallow retrieval.R@100 rises from 0.6550 at L = 1 to 0.6697 at L = 4, though intermediate depths are not monotonic; the ablation uses intermediate-loss weight 0.3, unlike the final primary recipe, which uses 0.Source: §6 · p. 9

Objective visibility

  • Sampled-negative CE reduces CQG-AR R@10 from 0.2982 to 0.2895 while retaining performance close to TIGER-style SID.The comparison indicates that training-time catalog visibility explains part, but not all, of the observed behavior; the architecture-level claim is limited to eliminating discrete code lookup and finite-beam exclusion.Source: §6 · p. 9

Robustness across SID constructions

  • On the closer T5/RQ-VAE Beauty reproduction, valid SIDs cover 44.0%±13.7% of the catalog in the union of Top-100 outputs, with coverage rising from 14.0%±7.6% for Q1 to 88.7% ± 13.0% for Q4.Across decoder seeds 42–44, 99.48% of generated sequences are valid SIDs; because ranking accuracy remains below the reference implementation, the authors treat this as a robustness diagnostic, not a reproduced TIGER benchmark.Source: §6 · p. 9, Table 6 · p. 10

Returned-list-size sensitivity

  • Increasing the seed-42 ML-1M TIGER returned-list size from 100 to 200 raises TargetInOutput from 66.7% to 76.3%.Across seven seeds, the means are 0.6667 ± 0.0018 and 0.7593 ± 0.0025, respectively; larger lists relax the observed support limitation, rather than establishing an architecture-invariant catalog ceiling.Source: §6 · p. 10

7 Discussion and Limitations

The observed tradeoff between finite-beam decoding and continuous retrieval depends on checkpoint, item popularity, and retrieval depth. The evidence is limited to evaluated configurations, with production-scale costs and cold-item accuracy still unmeasured.

7 Discussion and Limitations

  • Continuous retrieval removes discrete code lookup and finite-beam exclusion, but does not guarantee superior ranking accuracy: full-rank SASRec remains strongest on Sports.CQG-Single is suggested when shallow accuracy and inference efficiency are primary; CQG-AR preserves step-wise autoregressive conditioning when desired.Source: §7 · p. 10
  • Crossover patterns depend on checkpoint, popularity, and retrieval depth, so neither finite-beam decoding nor continuous retrieval is universally dominant.On ML-1M, TIGER is numerically stronger for rare targets at some shallow and intermediate cutoffs, while CQG-Single is generally stronger for popular targets and several deeper cutoffs; Beauty does not reproduce the rare-item shallow advantage.Source: §7 · p. 10
  • Evidence is specific to evaluated model–search–compute configurations; wider beams raise TIGER target support, and the hierarchical-k-means baseline is weaker than published SID systems.Full-catalog CE costs O(N) per update and was evaluated only at reported catalog sizes; billion-item training, ANN latency, memory, dynamic-catalog accuracy, and cold-item accuracy remain unmeasured.Source: §7 · p. 10

8 Conclusion

The study finds incomplete, popularity-skewed target support and catalog coverage, with crossover patterns depending on checkpoint, popularity, and retrieval depth. Its continuous-query comparison separates autoregression from discrete SID output constraints.

8 Conclusion

  • Target support and catalog coverage can be incomplete and popularity-skewed, with crossover patterns depending on checkpoint, popularity, and retrieval depth.The conclusion characterizes the crossover as conditional rather than universal.Source: §8 · p. 10
  • CQG-Single generates one continuous query, while CQG-AR replaces SID tokens with autoregressive residual vectors; a controlled ablation finds no consistent shallow benefit from refinement.CQG-Single is presented as a minimal vocabulary-free interface for controlled comparison with SID under a shared item space, not as the invention of dense retrieval.Source: §8 · p. 10
  • The results decouple autoregression from discrete output constraints and motivate matched evaluation of accuracy, target support, and catalog coverage.Source: §8 · p. 11

A Additional Cross-Dataset Diagnostics

Under the matched Amazon protocol, CQG-Single leads Beauty runs on R@10 and NDCG@10 and leads two of three Sports runs on R@10. Training-run intervals separate the continuous methods from TIGER on Sports, while Beauty intervals overlap.

A Additional Cross-Dataset Diagnostics

  • CQG-Single achieves the best R@10 and NDCG@10 in all Beauty runs and the best R@10 in two of three Sports runs.The comparison is against the study's TIGER-style baseline, not published TIGER.Source: A Additional Cross-Dataset Diagnostics · p. 12, Table 7 · p. 12
  • Table 7 reports complete seed-level Beauty and Sports results for 22,363 and 21,519 aligned test users, respectively.All methods use the same processed split and history filtering within each dataset; bold marks the best value for each dataset–seed pair and metric.Source: Table 7 · p. 12
  • On Sports, both continuous methods' 95% R@10 intervals lie above TIGER's; the Beauty intervals overlap.Beauty intervals: TIGER [0.0372, 0.0583], CQG-Single [0.0490, 0.0720], CQG-AR [0.0465, 0.0676]. Sports intervals: TIGER [0.0165, 0.0307], CQG-Single [0.0327, 0.0399], CQG-AR [0.0344, 0.0365]. These are training-run, not user-level intervals.Source: A Additional Cross-Dataset Diagnostics · p. 12

B Training-Seed Stability

CQG-Single exceeds the TIGER-style baseline in R@10 across all seven ML-1M runs. The seed pairing matches replicate indices, not identical random draws across architectures.

B Training-Seed Stability

  • CQG-Single exceeds TIGER in R@10 in all seven ML-1M runs; the mean paired difference is 0.0107 (95% t interval [0.0021, 0.0193]; exact sign-flip p = .0156).TIGER TargetInOutput is 0.6667 ± 0.0018 at output size 100 and 0.7593 ± 0.0025 at size 200.Source: B Training-Seed Stability · p. 12, Table 8 · p. 13
  • Paired seed means matched replicate index, not identical random draws across architectures; TIGER seeds also change hierarchical k-means.All methods share the frozen item table, split, and evaluation history.Source: B Training-Seed Stability · p. 12
  • Table 8 reports seven-seed ML-1M results under valid-prefix constrained TIGER decoding across all 6,040 test users.TIGER checkpoints are reselected by constrained validation; bold marks the best result for each seed/metric and the mean row.Source: Table 8 · p. 13

C Experimental Protocol and Reproducibility

The protocol evaluates fixed-checkpoint candidate generation and SASRec reranking alongside paired crossover diagnostics, using chronological full-catalog splits. CQG-Single reranking approaches full-rank SASRec Recall@10 across three datasets, while paired CQG-Single–TIGER differences remain statistically uncertain at the reported depths.

C.1 MSE-trained candidate generation and reranking.

  • 97.7–102% of full-rank SASRec Recall@10: CQG-Single followed by SASRec reranking reaches this range across ML-1M, Beauty, and Sports.These are fixed-checkpoint results, not multi-seed estimates.Source: Table 9 · p. 13, §C.1 · p. 13

C.2 Paired Crossover and Candidate-Generation Diagnostics

  • 0.0071, 0.0093, and 0.0040: CQG-Single–TIGER differences at K = 10, 50, and 90, with all paired-bootstrap 95% CIs including zero.The comparison uses paired seed-42 checkpoints and matched Top-200 outputs on the same 6,040 users.Source: Table 10 · p. 14

C.3 Computing Infrastructure

  • Experiments used one NVIDIA A800 80GB PCIe GPU, 14 allocated CPU cores, and approximately 120GB of host-visible memory.The software environment includes Python 3.12.3, PyTorch 2.8.0+cu128, and FAISS-CPU 1.14.3.Source: §C.3 · p. 14

C.4 Dataset Splits

  • Chronological user splits assign the last interaction to test, the penultimate to validation, and earlier interactions to training; evaluation ranks against the full catalog unless stated otherwise.Users with insufficient history after filtering are removed.Source: §C.4 · p. 14

C.5 Evaluation Metrics

This appendix specifies the support metrics, frozen item-space pipeline, continuous-query training and SID decoding, then reports objective, temperature, and language-backbone diagnostics. Full-catalog CE leads sampled-negative training for CQG-AR, while the best tested Qwen hybrid trails the scratch Transformer in its single-run diagnostic.

C.5 Evaluation Metrics

  • SID target support is per-user and target-conditioned, whereas union coverage counts distinct retrieved items; catalog-wide scoreability does not imply full finite-list coverage.The primary diagnostics separately address item-balanced support, matched union coverage, and training-seed variation.Source: §C.5 · pp. 14–15

D.1 Frozen Collaborative Item Space

  • The shared frozen item table supplies global collaborative item vectors for history inputs, objectives or catalog classification, retrieval, and TIGER-style clustering, isolating output-interface comparisons from item-encoder differences.The table is exported from a checkpoint-selected SASRec model trained without text or language-model features.Source: §D.1 · p. 16

D.2 CQG-Single Architecture and Optimization; D.3 CQG-AR Architecture and Optimization; D.4 Hyperparameters

  • Both methods output 64-dimensional queries: CQG-Single generates one, while CQG-AR appends three continuous residuals for the same next-item target and directly supervises only the final query.The depth ablation gives non-final queries intermediate weight 0.3; the final recipe uses intermediate_weight=0.Source: §D.2 · p. 16, §D.3 · pp. 16–17, Table 11 · p. 17

D.5 Continuous-Query Dataloader; D.6 Continuous Query Training Objectives

  • Training pairs padded histories with next-item targets mapped to frozen embeddings or full-catalog labels; MSE, InfoNCE, and CE variants share that sequential next-item protocol.All CQG-AR steps use the same next-item target, and inference can score every encoded item even though finite top-K coverage is empirical.Source: §D.5 · p. 17, §D.6 · p. 18

E.1 Semantic-ID Construction

  • The TIGER-style baseline uses depth-3 hierarchical paths, resolving remaining ML-1M code collisions with deterministic residual identifiers before decoder training.On ML-1M, 99.4% of items are unique after three cluster levels; tie-breaking makes the codes one-to-one.Source: §E.1 · p. 18

E.2 TIGER T5/RQ-VAE Reproduction Recipe; E.3 Valid-Prefix Decoding

  • 99.48% valid SID sequences: the Beauty T5/RQ-VAE reproduction's Top-100 union covers 44.0% ± 13.7% of the catalog, confirming popularity-skewed coverage but not published TIGER accuracy.Coverage ranges from 14.0% ± 7.6% in Q1 to 88.7% ± 13.0% in Q4; decoding uses a catalog trie to mask invalid prefixes and exact code lookup.Source: §E.2 · p. 18, §E.3 · p. 18

F.1 Objective, Catalog-Visibility, and Temperature Diagnostics

  • 0.0087: replacing full-catalog CE with 256 sampled negatives lowers CQG-AR mean R@10 from 0.2982 to 0.2895; lower temperatures improve shallow retrieval in the tested range.CQG-Single R@10 falls from 0.3114 at τ = 0.07 to 0.2846 at 0.20; aligned seed-42 CQG-AR falls from 0.2972 to 0.2725.Source: Figure 5 · p. 19, §F.1 · p. 19
  • 0.2118 versus 0.248: the best Qwen hybrid trails the single-run scratch model in R@10 at every cutoff, while joint training yields item–item cosine 0.9936.The figure and table describe single-run diagnostics, distinct from the three-seed CQG-Single MSE result.Source: G LLM Backbone Diagnostic · p. 19, Table 12 · p. 19, Figure 6 · p. 20

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