Bucketed Ranking-Based Losses

• Bucketed ranking-based losses are loss functions that partition samples into groups (or buckets) to enforce relative ranking constraints, improving optimization in imbalanced settings.
• They enable flexible priority encoding, integrating hard negative mining and quantile focus by grouping predictions based on semantic similarity, confidence, or rank intervals.
• These techniques offer computational advantages and scalability, demonstrating robust performance across applications like recommendation systems, object detection, and image segmentation.

Bucketed ranking-based losses are a class of loss functions used in machine learning tasks where model performance depends on the relative ordering or ranking of instances, rather than simple pointwise prediction accuracy. These methods divide candidates, predictions, or samples into groups (“buckets”) according to either semantic similarity, quantized codes, confidence, rank intervals, or application-specific properties, with ranking or ordering constraints enforced at the bucket or group level. Such strategies yield tractable optimization, allow for flexible encoding of priorities (e.g., hard negative mining, quantile focus), and achieve significant computational savings in large-scale or imbalance-prone settings.

1. Formal Definitions and Generalization of Bucketed Ranking Losses

Bucketed ranking-based losses instantiate the general family of rank-based (or spectral) losses by partitioning either the set of samples or the sorted per-sample losses into disjoint “buckets,” where ranking, calibration, or sorting objectives are applied within or across these groups.

• General rank-based loss: Given a set $\{(x_i, y_i)\}_{i=1}^n$, model $f(x_i;\theta)$, and base loss $\ell$, define:

$L_i(\theta) = \ell(y_i f(x_i;\theta))$

Sorting these losses $\ell_{(1)} \leq \cdots \leq \ell_{(n)}$, rank-based or spectral loss takes the form:

$L(\theta) = \sum_{i=1}^n w_i \ell_{(i)}(\theta)$

where $w$ is non-negative and determines which quantiles or ranks are prioritized.

• Bucketed losses are specified by constraining $w$ to be piecewise constant over disjoint rank intervals (buckets). Canonical examples include top-$k$ average (one “bucket” for top $k$ ranks) and multi-quantile averaging, as well as multi-resolution or cluster-based buckets for structured user/item groups in recommendation and detection (Xiao et al., 2023).

2. Instantiations in Major Application Domains

2.1 Recommendation Systems: Group-wise Hierarchical Bucketing

In hierarchical group-wise ranking (Yan et al., 15 Jun 2025), buckets are defined via residual vector quantization (RVQ) of user embedding vectors. Each user $e_u \in \mathbb R^d$ is quantized via a cascade of $L$ codebooks $C^{(\ell)}$, yielding an $L$-stage code $c_u = (c_{u,1}, ..., c_{u,L})$ and inducing a prefix tree (trie) structure.

• At level $\ell$, users sharing the same code prefix $(c_{u,1}, ..., c_{u,\ell})$ belong to the same bucket. Within each bucket, listwise ranking losses (e.g., softmax cross-entropy, or binary-calibrated “ListCE” variants) are applied to user-item interactions.
• Loss formulation for bucket $G_m^{(\ell)}$ at level $\ell$:

$$L_m^{(\ell)} = -\sum_{i \in G_m^{(\ell)}} \tilde y_i \log \frac{\sigma_i}{\sum_{j \in G_m^{(\ell)}} \sigma_j}, \quad \tilde y_i = \frac{y_i}{\sum_{j \in G_m^{(\ell)}} y_j + \varepsilon}$$

where $\sigma_i = \sigma(s_i)$ is the sigmoid of the logit.

• The per-level loss $L^{(\ell)}$ is averaged across buckets, then linearly combined with uncertainty-based weights.

Key property: As $\ell$ increases, buckets become finer (users more similar), causing within-bucket negatives to become “harder” and better approximating hard-negative mining. This hierarchical organization allows scalable, informative contrastive signals without explicit nearest neighbor retrieval (Yan et al., 15 Jun 2025).

2.2 Object Detection: Negative Bucketing for Efficient Pairwise Comparisons

Standard ranking-based losses in detection (e.g., AP loss, Rank&Sort loss) require $O(PN)$ operations for $P$ positives and $N$ negatives (Yavuz et al., 19 Jul 2024). Bucketed ranking-based losses overcome this by grouping negatives into $B \leq P + 1$ contiguous buckets based on the ordering of prediction scores.

• Construction: After sorting predictions by score, contiguous runs of negatives between positives form buckets. Each bucket is represented by its mean score.
• Loss (Bucketed AP): For positive $i$,

$$N^b_{FP}(i) = \sum_{k: s^b_k > s_i} b_k, \quad \mathrm{rank}^b(i) = 1 + \sum_{j: s_j > s_i, \,y_j=1} 1 + N^b_{FP}(i)$$

$$\ell^b_R(i) = \frac{N^b_{FP}(i)}{\mathrm{rank}^b(i)}$$

• Computational advantage: Pairwise comparisons reduce to $O(P^2 + N \log N)$, matching the original gradient exactly when $B=P+1$.
• Sorting among positives (for RS loss) remains $O(P^2)$, orthogonal to negative bucketing (Yavuz et al., 19 Jul 2024).

2.3 Structured Ranking Data: Bucket Orders and Mass Transport

In ranking models without vector space structure (e.g., permutations), bucketed ranking loss quantifies distortion relative to an ideal “bucket order” (Achab et al., 2018):

• Given: Distribution $P$ on permutations $\Sigma$ of $n$ items.
• Bucket order: Partition $(B_1, ..., B_K)$. The expected cross-bucket inversion count (Kendall’s $\tau$) defines distortion:

$$D(P,B) = \sum_{k<\ell} \sum_{i \in B_k, j \in B_\ell} P\{\Sigma(j) < \Sigma(i)\}$$

• Empirically, model selection proceeds by minimization of empirical distortion, complexity penalization, or dynamic programming segmentation of the Kemeny-median (Achab et al., 2018).

2.4 Image Segmentation and Edge Detection: Certainty-based Bucketing

Here, bucketed losses group positive (edge or instance) pixels by derived confidence/uncertainty, as in high-certainty vs low-certainty buckets (certified by consensus across multiple annotators), and all negatives as another bucket (Cetinkaya et al., 4 Mar 2024). Rank-based and sort-based penalties are computed to both separate positives from negatives and favor high-certainty positives.

3. Algorithmic Implementation and Complexity

The core technical motivation for bucketed ranking-based loss is computational tractability and stabilization of the ranking signal. Generic computational steps include:

• Bucket assignment: Grouping samples based on code prefixes, rank intervals, semantic confidence, or geometric proximity. Complexity for quantization or sorting-based bucketing ranges from $O(LKd)$ per embedding (hierarchical) (Yan et al., 15 Jun 2025), to $O(N \log N)$ for sorting-based detection buckets (Yavuz et al., 19 Jul 2024).
• Within-bucket loss calculation: Application of listwise, softmax, cross-entropy, or AP/RS-type losses within each bucket; vectorized aggregation over groups.
• Optimization: Proximal ADMM is applied in settings where the bucketed loss is a weighted sum over sorted losses. PAVA enables $O(n)$ or $O(n^2)$ routines, e.g., for quantile/top-$k$ buckets (Xiao et al., 2023).

Caching quantized user codes and vectorized implementation of bucket operations are standard best practices, with memory and compute tradeoffs closely studied per application (Yan et al., 15 Jun 2025, Yavuz et al., 19 Jul 2024).

4. Empirical Performance and Practical Insights

Extensive experiments across modalities demonstrate both theoretical and practical merits.

• Recommender benchmarks: Hierarchical group-wise bucketed losses yield consistent GAUC/AUC improvements over flat listwise and joint calibration-ranking losses; best performance is observed when bucket resolution is moderate (e.g., $K=16$, $L=3$ in the hierarchical trie) (Yan et al., 15 Jun 2025).
• Object detection: Bucketed AP and RS loss match or slightly surpass unbucketed variants (AP or accuracy differences $<0.02$) at $2\times$–$6\times$ faster iteration times. First transformer-based detectors (CoDETR) are efficiently trained with ranking-based objectives via bucketing (Yavuz et al., 19 Jul 2024).
• Structured permutations: Bucket orders sharply reduce model dimension with mild loss in fidelity, as evidenced in full-ranking-to-bucket models for datasets like Sushi and Cars (Achab et al., 2018).
• Edge detection: Certainty-based bucketing in RankED outperforms state-of-the-art on NYUD-v2, BSDS500, Multi-cue (up to 5.7% AP improvement), with no loss of robustness in imbalanced or uncertain conditions (Cetinkaya et al., 4 Mar 2024).

Ablation studies confirm that bucketed ranking terms (especially at finer bucket levels) yield the strongest ranking gradient signals, and omitting hierarchical or quantized components leads to measurable degradation (Yan et al., 15 Jun 2025).

5. Theoretical Guarantees and Optimization Properties

• Convergence: For spectral and bucketed rank-based losses, convergence to $\varepsilon$-KKT points in $O(1/\varepsilon^2)$ (nonsmooth ADMM) and $O(1/\varepsilon^4)$ (with smoothed regularizer) is established under weakly convex regularization (Xiao et al., 2023).
• Generalization and consistency: In permutation-based settings, empirical minimization of bucket-induced distortion generalizes uniformly with rates governed by the Rademacher complexity of the family of bucketed orders; fast rates under strong stochastic transitivity are proven (Achab et al., 2018).
• Optimality and approximation: For detection and recommender systems, exact gradient matching is achieved if the number of buckets matches the theoretical maximum per configuration (e.g., $B=P+1$ in detection), and empirical studies validate negligible loss in task quality if further bucket coalescence is performed (Yavuz et al., 19 Jul 2024, Yan et al., 15 Jun 2025).

6. Variants, Best Practices, and Limitations

Design choices for bucketed ranking losses primarily concern bucket assignment schemes, within-bucket loss selection, and the aggregation weights across resolution or buckets. Recommended practices include:

• Detection: Implement exact bucketing ($B=P+1$), with small $H_\delta$ smoothing for stability, and vectorized operator execution for GPU efficiency (Yavuz et al., 19 Jul 2024).
• Recommendation: Use hierarchical codebook and bucket trie's intermediate depths for optimal tradeoff between bucket informativeness and training signal robustness (Yan et al., 15 Jun 2025).
• Sorting/quantile loss: Pool–adjacent–violators algorithm for efficient handling of multiple disjoint bucket constraints (Xiao et al., 2023).

Limitations include the necessity for appropriate bucket granularity (excessively fine buckets dilute ranking signal and harm generalization) and potential instability under severe imbalance if within-bucket sample sizes become too small (Yan et al., 15 Jun 2025). For detection, if positives are extremely dense, even $O(P^2)$ sorting among positives can become a bottleneck, requiring merged or subsampled buckets (Yavuz et al., 19 Jul 2024).

7. Summary Table: Bucketed Ranking Loss Variants

Domain/Task	Bucket Assignment	Intra-Bucket Loss	Core Benefit
Recommendation	Trie prefix on quantized embeddings	Sigmoid-ListCE, softmax	Hard negative mining, calibration
Object Detection	Run-based buckets in score order	AP, RS (bucketed)	$O(P^2+N\log N)$ time, accuracy
Permutation Models	Partition of items (Kemeny-median)	Mass-transport (distort)	Dimensionality reduction
Edge Detection	Certainty/disagreement threshold	AP, RS (cert.-aware)	Imbalance/uncertainty robustness

Each instantiation adapts bucket construction and intra-bucket loss to task structure, operational constraints, and metric alignment.

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Bucketed ranking-based losses provide a principled, versatile, and efficient mechanism for embedding ranking structure at scale, integrating listwise groupings, hard negative mining, quantile focus, and uncertainty handling, with strong empirical performance and rigorous optimization properties across major machine learning domains (Yan et al., 15 Jun 2025, Yavuz et al., 19 Jul 2024, Achab et al., 2018, Cetinkaya et al., 4 Mar 2024, Xiao et al., 2023).