Scale-Invariant Learning-to-Rank

• The paper introduces scale-invariant learning-to-rank, ensuring that ranking order remains constant under any positive rescaling of unstable features.
• It presents the SIR framework, which decomposes the scoring function into deep and wide components to mitigate scaling drift in production environments.
• Empirical results demonstrate that SIR models preserve NDCG and achieve significant speedups compared to traditional LTR methods in large-scale applications.

Scale-invariant learning-to-rank (LTR) refers to a class of ranking frameworks and objectives whose predictions remain invariant when certain input features are rescaled by arbitrary positive multiplicative constants at prediction time. This property directly addresses practical challenges in production LTR systems, where input scales may unpredictably drift between training and inference environments due to shifts in business rules, feature engineering policies, or distributed system idiosyncrasies. Scale-invariant LTR is especially relevant for large-scale e-commerce, online travel agencies, and content recommendation deployments, where ranking stability and operational robustness are paramount (Petrozziello et al., 2021, Petrozziello et al., 2024, Zhang et al., 11 Jun 2025).

1. Mathematical Formulation of Scale-Invariance in LTR

A learning-to-rank model is scale-invariant if, for any query $i$, and items $j$ and $k$, rescaling a subset of positive-valued item features (e.g., price, discount) by any positive $c$ preserves the order: $$f(\mathbf{x}_{ij}) > f(\mathbf{x}_{ik}) \implies f(\tilde{\mathbf{x}}_{ij}) > f(\tilde{\mathbf{x}}_{ik})$$ where $$\tilde{\mathbf{x}}_{ij}^{I,S} = c\,\mathbf{x}_{ij}^{I,S}$$. Here, the 'unstable' features $\mathbf{x}_{ij}^{I,S}$ are those susceptible to scale drift, whereas 'fixed-scale' features $\mathbf{x}_{ij}^{I,F}$ and shared query-level features $\mathbf{x}_i^Q$ are guaranteed stable (Petrozziello et al., 2024). This property must hold for all $c > 0$, for the full class of positive-valued rescalings encountered in practice.

2. Canonical Architectures: The SIR Framework

A prominent architectural realization is the Scale-Invariant Ranking (SIR) scoring function, which decomposes the overall score for an item into a deep and a wide component: $j$0

• $j$1 is a multi-layer perceptron over query and fixed-scale item features.
• $j$2 computes a (linear) Kronecker product of a low-dimensional query embedding $j$3 with the elementwise logarithm $j$4 of scale-unstable features, then performs an inner product with a learned $j$5.

This formulation guarantees scale-invariance because any positive rescaling $j$6 of $j$7 results in an additive constant $j$8 which cancels in pairwise score differences: $j$9 Thus, the induced orderings—and therefore the ranking—are invariant to feature scaling (Petrozziello et al., 2021, Petrozziello et al., 2024).

3. Integration into Standard LTR Algorithms

The SIR scoring architecture serves as a plug-and-play replacement for the base network $k$0 in classical LTR objectives. List-wise algorithms such as ListNet and ListMLE, as well as pair-wise or pair-rank methods like RankNet, LambdaRank, and SoftRank, remain unmodified apart from swapping in $k$1:

• ListNet: Softmax-normalized loss over all permutations, now using SIR scores.
• ListMLE: Negative log-likelihood of the ground-truth item order, parameterized by SIR.
• RankNet / LambdaRank: Pairwise preferences, with SIR scores plugged into the standard pairwise cross-entropy or weighted $k$2NDCG objective. No change to optimizer, loss formulation, regularization, or gradient flow is required (Petrozziello et al., 2021, Petrozziello et al., 2024).

4. Data Preprocessing, Simulation Protocols, and Empirical Results

In large e-commerce settings, features are categorized as either stable (e.g., categorical IDs, counts) or susceptible to scale drift (e.g., prices, discounts). At training, standardization may be applied for numerical stability, but SIR-based models require no runtime normalization of unstable features. Instead, the key requirement is strict positivity for features entering the log-transformed wide path (Petrozziello et al., 2024).

Empirical evaluations use large datasets such as ExpediaHotels (56 million sessions), RecTour, and MSLR Web30k. Production-scale scenarios are simulated by applying multiplicative perturbations (e.g., currency conversion, price aggregation) to subsets of the test set, leaving the training data unchanged. Results—measured primarily by NDCG—demonstrate that classical LTR algorithms are highly sensitive to such scaling, with observed NDCG drops up to 14.7% (e.g., ListMLE 0.668 → 0.570 under currency+aggregation shift), whereas SIR-augmented models preserve NDCG within ±0.001 of baseline across all scenarios (Petrozziello et al., 2021, Petrozziello et al., 2024).

Algorithm	Test NDCG	Perturbed (Case 4) NDCG	NDCG Drop/Preservation
ListMLE	0.668	0.570 (orig)/0.667 (SIR)	Drop 0.098 / Δ 0.001
ListNet	0.526	0.512 (orig)/0.522 (SIR)	Drop 0.014 / Δ 0
RankNet	0.663	0.577 (orig)/0.662 (SIR)	Drop 0.086 / Δ 0.001

This robustness is achieved with negligible runtime cost and no additional inference latency.

5. Algorithmic Scale-Invariance in Ranking Optimization

Beyond feature-based scale-invariance, scale-invariance is also conceptualized as invariance of optimization complexity to ranking cutoff size $k$3 and candidate pool size $k$4. The PL-Rank-3 algorithm for Plackett-Luce LTR models achieves this by reorganizing the policy-gradient estimator so that gradient computation per sample scales as $k$5—the same asymptotic as efficient sorting and a single dataset pass (Oosterhuis, 2022). Unlike prior methods scaling as $k$6, PL-Rank-3’s runtime is effectively independent of the top-K cutoff $k$7 for practical $k$8, and enables large-scale application of gradient-based stochastic ranking methods.

This form of scale-invariance applies to the computational dimension rather than the input space, but both are critical for industrial-scale deployment. Experimental results on Yahoo!, MSLR-Web30k, and Istella confirm 5–20$k$9 speedups to target NDCG relative to prior Plackett-Luce or StochasticRank methods, with no observed drop in ranking accuracy (Oosterhuis, 2022).

6. Metric Learning and Discrete Scale-Invariance

Scale-invariance also arises in collaborative filtering and metric learning settings, where absolute distance-based margins may fail on item sets with heterogeneous or imbalanced intra-class variation. Discrete scale-invariant metric learning (DSIML) implements angle-based (as opposed to absolute distance) margins in binary Hamming space: $c$0 with $c$1 hash codes (Zhang et al., 11 Jun 2025). The angle-based margin is strictly invariant to multiplicative rescalings of item representations, thus conferring stability even for imbalanced item groups.

Optimization is performed by smoothing the hinge via log-sum-exp, bounding it with a quadratic variational form, and alternating between binary quadratic programming for codes and closed-form updates of variational parameters. Experimentally, DSIML outperforms non-scale-invariant recommenders on Amazon CD, Movie, and Book datasets by 5–25% in NDCG@50, while yielding 5–18$c$2 speedups in online ranking due to the hashing structure (Zhang et al., 11 Jun 2025).

7. Practical Considerations, Limitations, and Open Directions

Practical deployment of scale-invariant LTR architectures eliminates the need for per-query feature normalization at inference—sidestepping the overhead and brittleness associated with real-time statistics, batch/layer normalization, and cross-shard synchronization. The only nonlinear operation on unstable features is an elementwise log, enabling low-latency, fully-distributed serving (Petrozziello et al., 2024).

However, current approaches handle only strictly positive multiplicative shifts in $c$3. Negative, additive, or more complex non-affine transformations remain open problems. The approach also presumes prior identification of which features may drift, though automatic detection of unstable features and extensions to categorical type drift are noted as prospective research directions (Petrozziello et al., 2024). In metric learning, scale-invariance currently applies via angle-margins in binary space; more general continuous-space or mixed-objective extensions are ongoing.

In summary, scale-invariant learning-to-rank frameworks provide a robust, mathematically-grounded safeguard against unpredictable scaling drift of input features or computational bottlenecks, and are increasingly viewed as essential for industrial LTR system reliability (Petrozziello et al., 2021, Oosterhuis, 2022, Petrozziello et al., 2024, Zhang et al., 11 Jun 2025).