High-Precision Scoring (HPS) for Retrieval

• High-Precision Scoring (HPS) is an inference-time protocol that upcasts low-precision logits to FP32 to reduce spurious ties in retrieval systems.
• HPS enhances metric stability by computing the final scoring step in FP32, thereby preserving low-precision efficiency while improving measures such as nDCG, MRR, and Recall.
• The procedure is integrated post-neural forward pass without retraining, providing a targeted numerical fix to mitigate ranking uncertainty in diverse retrieval pipelines.

Searching arXiv for the specified paper and closely related retrieval-evaluation work. High-Precision Scoring (HPS) is an inference-time evaluation procedure for low-precision retrieval systems that upcasts the final scoring step to higher precision in order to reduce spurious ties among candidate scores and thereby stabilize ranking metrics such as Recall@k, nDCG@k, and MRR@k. In the formulation introduced in "Reliable Evaluation Protocol for Low-Precision Retrieval" (Yang et al., 5 Aug 2025), HPS leaves the neural forward pass in low-precision as usual, but casts the small tensor of logits to FP32 immediately before applying the scoring function and sorting. The same acronym appears in an unrelated binary-event forecasting context, where it denotes a construction of scoring rules that incentivize precision rather than a retrieval protocol (Neyman et al., 2020).

1. Definition and problem setting

In modern neural retrieval, it is common to quantize model weights and activations to low-precision, for example FP16 or BF16, to save memory and speed up inference (Yang et al., 5 Aug 2025). The retrieval-specific problem addressed by HPS is not quantization in general, but the instability introduced when relevance scores are computed in low-precision. Scoring functions such as softmax, sigmoid, and dot-product map real-valued logits into a narrow range, often $(0,1)$, and with only 7–10 bits of mantissa many distinct logits collapse onto exactly the same floating-point value. The resulting "spurious ties" among candidate scores make downstream ranking metrics highly sensitive to arbitrary tie resolution.

The protocol in (Yang et al., 5 Aug 2025) therefore targets evaluation reliability rather than model retraining or architectural redesign. Its purpose is to reduce score variation caused by tied candidates while preserving the efficiency benefits of low-precision inference. The paper presents HPS together with Tie-aware Retrieval Metrics (TRM): HPS reduces the number of spurious ties, and TRM reports expected scores, range, and bias to quantify the remaining order uncertainty.

A central practical feature is that HPS is described as a simple remedy. The entire neural forward pass remains in BF16 or FP16, and only the final scoring step is executed in FP32. This suggests that the protocol is best understood as a narrowly targeted numerical intervention at the ranking boundary rather than a general mixed-precision scheme.

2. Mathematical formulation

The paper defines $z_i \in \mathbb{R}^d$ as the final pre-score output, or logit, for a query–document pair $(q,d_i)$, originally stored in a $B$-bit mantissa format (Yang et al., 5 Aug 2025). Let $\phi$ denote the scoring function, which may be a softmax-based probability, $\phi(z)=\operatorname{sigmoid}(z)$, or $\phi(z)=\operatorname{dot}(h_q,h_d)$ for bi-encoder embeddings.

Under the low-precision baseline, all operations, including $\phi$, run in low-precision. The score is written as

$s_i^{\text{low}}=\phi_B(z_i).$

Sorting by $s_i^{\text{low}}$ induces tie groups

$z_i \in \mathbb{R}^d$0

and if $z_i \in \mathbb{R}^d$1 straddles the cutoff $z_i \in \mathbb{R}^d$2, random tie-breaking induces large variance in ranking metrics.

HPS changes only the precision of the final scoring step:

$z_i \in \mathbb{R}^d$3

The forward pass itself is unchanged. Because the tensor of $z_i \in \mathbb{R}^d$4 values is usually small, often one logit per candidate and often a few hundred candidates, the added FP32 work is limited to a small vector. The intended effect is to restore fine granularity in the score space and thereby collapse far fewer ties.

The paper’s framing emphasizes that the numerical pathology arises after the model has already produced its low-precision outputs. Accordingly, HPS does not alter learned representations, loss functions, or encoder computations. It intervenes at the stage where ranking becomes vulnerable to reduced floating-point granularity.

3. Algorithmic procedure and computational properties

For a single query with $z_i \in \mathbb{R}^d$5 candidates and pre-computed logits $z_i \in \mathbb{R}^d$6 in low-precision, the HPS procedure in (Yang et al., 5 Aug 2025) is:

1. Optionally encode query and documents entirely in low-precision, yielding logits $z_i \in \mathbb{R}^d$7.
2. Convert logits to FP32:

$z_i \in \mathbb{R}^d$8

with $z_i \in \mathbb{R}^d$9 time and negligible memory.

1. Compute high-precision scores:

$(q,d_i)$0

also in $(q,d_i)$1 time.

1. Sort candidates by $(q,d_i)$2 in descending order.
2. Compute ranking metrics such as MRR@k and Recall@k on the sorted list.

The reported complexity overhead is $(q,d_i)$3 extra work per query (Yang et al., 5 Aug 2025). Since $(q,d_i)$4 is typically a few hundred and $(q,d_i)$5 is described as a lightweight operation, the added wall-clock cost is characterized as almost no cost relative to a pure low-precision run. The practical guidelines are more explicit: runtime overhead is stated as under 1–2% of total inference time, and the memory overhead is the temporary FP32 copy of $(q,d_i)$6 floats, exemplified as $(q,d_i)$7.

These details define HPS as a "surgical" modification. There is no retraining, no weight conversion, and no mixed-precision scheduling. The only required patch is in the inference script, immediately after the vector of logits or pairwise scores is computed and before sorting or thresholding.

4. Placement in retrieval pipelines

HPS is described as model-agnostic and applicable to any scoring function and any architecture, including cross-encoder, bi-encoder, and hybrid systems (Yang et al., 5 Aug 2025). The common pattern is that expensive representation learning remains low-precision, while only the terminal score computation is upcast.

In a bi-encoder retrieval pipeline, the query and documents are embedded in low-precision and dot products or cosine scores are calculated in low-precision. The vector of $(q,d_i)$8 dot-products is then upcast to FP32 and rescored or sorted. In reranking setups using cross-encoders, the full forward pass runs in BF16 or FP16, producing logits $(q,d_i)$9; these logits are then upcast, passed through sigmoid or softmax in FP32, and sorted.

The protocol is explicitly inference-time only. There is no need to re-train or change any weights (Yang et al., 5 Aug 2025). This matters operationally because HPS can be inserted into existing retrieval stacks without modifying model checkpoints or optimizer configurations. A plausible implication is that HPS is particularly attractive in evaluation settings where reproducibility and fair model comparison are more important than marginal throughput differences.

The paper also frames HPS as complementary to TRM rather than a replacement for robust reporting. HPS reduces the incidence of tie-induced instability, while TRM makes any residual uncertainty transparent by reporting expected score, range, and bias.

5. Empirical behavior and metric stability

The empirical motivation for HPS is that pure low-precision scoring can produce large uncertainty in standard IR metrics when tied candidates straddle a cutoff (Yang et al., 5 Aug 2025). The paper reports experiments on multiple models with three scoring functions and two retrieval datasets, and states that HPS dramatically reduces tie-induced instability while TRM accurately recovers expected metric values.

A concrete example is given for BF16 reranker outputs on MIRACLReranking using Qwen3-Reranker-0.6B:

Precision	Metric	Tie-oblivious Mobl / Mobl with HPS
BF16	nDCG@10	75.04 / 73.35
BF16	MRR@10	78.45 / 77.22

Metric	Range (no HPS)	Range (HPS)
nDCG@10	25.59 %p	1.13 %p
MRR@10	38.03 %p	1.21 %p

The interpretation given in the paper is direct. Pure BF16 scoring yields uncertainty of up to 38 percentage points in MRR@10, while upcasting only the final scoring step slashes the range by more than 95% and preserves almost identical mean performance (Yang et al., 5 Aug 2025). The same section further states that, under HPS, the reranker’s nDCG@10 and MRR@10 sit within 1% of the FP32 baseline, effectively eliminating spurious ranking reversals.

The broader experimental claim is that the same pattern appears for sigmoid-based and product-based scoring. HPS reduces metric variability to near zero at trivial cost. This suggests that the instability is a numerical artifact of low-precision scoring granularity rather than an intrinsic deficiency of the retrieval models themselves.

6. Reporting, interpretation, and terminological ambiguity

The evaluation protocol in (Yang et al., 5 Aug 2025) includes both HPS and Tie-aware Retrieval Metrics. TRM is designed to report the expected score $B$0, the range, and the bias so that the order uncertainty of tied candidates is explicit. The paper’s recommended practice is therefore not only to reduce ties through FP32 final scoring, but also to publish tie-aware statistics when residual ties remain.

One common misconception is to treat a single tie-oblivious metric value as definitive even when low-precision scoring creates large tie groups. The paper’s examples show that such scores can differ substantially from the expectation under tie permutations, and that the range itself can be large enough to affect conclusions about comparative system quality (Yang et al., 5 Aug 2025). HPS addresses this instability, but TRM is the mechanism that quantifies the uncertainty that remains.

A second source of confusion is terminological. The phrase "High-Precision Scoring" is also used in "Binary Scoring Rules that Incentivize Precision" (Neyman et al., 2020), but there it refers to a proper-scoring-rule construction in a biased-coin model, not to a retrieval evaluation protocol. In that setting, the object of study is a scoring rule $B$1, with an incentivization index

$B$2

used to characterize how a proper scoring rule encourages an expert to pay a cost to refine a belief about a binary event. The two usages share an emphasis on "precision," but they concern different problems: one is low-precision numerical scoring in retrieval, the other is incentive design for probabilistic forecasting.

For retrieval research, the relevant meaning of HPS is therefore the FP32 upcasting of the final scoring step. Its significance lies in making low-precision evaluation more consistent and reliable while leaving the efficiency-oriented low-precision forward pass intact (Yang et al., 5 Aug 2025).