# PiRank: Scalable Learning To Rank via Differentiable Sorting

arXiv: https://arxiv.org/abs/2012.06731 · HTML: https://mlsanity.com/paper/2012.06731/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Dec 2020.

## Built on by (most cited first, 27 of 27)

- [Lero: A Learning-to-Rank Query Optimizer](https://mlsanity.com/paper/2302.06873/) (2023)
- [Large-scale Stochastic Optimization of NDCG Surrogates for Deep Learning with Provable Convergence](https://mlsanity.com/paper/2202.12183/) (2022)
- [Listwise Reward Estimation for Offline Preference-based Reinforcement Learning](https://mlsanity.com/paper/2408.04190/) (2024)
- [Learning by Sorting: Self-supervised Learning with Group Ordering Constraints](https://mlsanity.com/paper/2301.02009/) (2023)
- [Improved Diversity-Promoting Collaborative Metric Learning for Recommendation](https://mlsanity.com/paper/2409.01012/) (2024)
- [Where-to-Unmask: Ground-Truth-Guided Unmasking Order Learning for Masked Diffusion Language Models](https://mlsanity.com/paper/2602.09501/) (2026)
- [Algorithmic Foundations of Empirical X-risk Minimization](https://mlsanity.com/paper/2206.00439/) (2022)
- [Adaptive Neural Ranking Framework: Toward Maximized Business Goal for Cascade Ranking Systems](https://mlsanity.com/paper/2310.10462/) (2023)
- [COS-DPO: Conditioned One-Shot Multi-Objective Fine-Tuning Framework](https://mlsanity.com/paper/2410.08316/) (2024)
- [Learning semantic image quality for fetal ultrasound from noisy ranking annotation](https://mlsanity.com/paper/2402.08294/) (2024)
- [Interpretable Tabular Foundation Models via In-Context Kernel Regression](https://mlsanity.com/paper/2602.02162/) (2026)
- [Finding Pareto Trade-offs in Fair and Accurate Detection of Toxic Speech](https://mlsanity.com/paper/2204.07661/) (2022)
- [Scaling Laws for Online Advertisement Retrieval](https://mlsanity.com/paper/2411.13322/) (2024)
- [QAMRO: Quality-aware Adaptive Margin Ranking Optimization for Human-aligned Assessment of Audio Generation Systems](https://mlsanity.com/paper/2508.08957/) (2025)
- [EZ-Sort: Efficient Pairwise Comparison via Zero-Shot CLIP-Based Pre-Ordering and Human-in-the-Loop Sorting](https://mlsanity.com/paper/2508.21550/) (2025)
- [The Evolution of Reranking Models in Information Retrieval: From Heuristic Methods to Large Language Models](https://mlsanity.com/paper/2512.16236/) (2025)
- [Holistic Utility Preference Learning for Listwise Alignment](https://mlsanity.com/paper/2410.18127/) (2024)
- [Learning Cascade Ranking as One Network](https://mlsanity.com/paper/2503.09492/) (2025)
- [RankList -- A Listwise Preference Learning Framework for Predicting Subjective Preferences](https://mlsanity.com/paper/2508.09826/) (2025)
- [Scalable Unidirectional Pareto Optimality for Multi-Task Learning with Constraints](https://mlsanity.com/paper/2110.15442/) (2021)
- [Pretrained deep models outperform GBDTs in Learning-To-Rank under label scarcity](https://mlsanity.com/paper/2308.00177/) (2023)
- [SpatialRank: Urban Event Ranking with NDCG Optimization on Spatiotemporal Data](https://mlsanity.com/paper/2310.00270/) (2023)
- [Efficient Pointwise-Pairwise Learning-to-Rank for News Recommendation](https://mlsanity.com/paper/2409.17711/) (2024)
- [Forecast Collapse in Time-Series Foundation Models](https://mlsanity.com/paper/2608.14106/) (2026)
- [RewardRank: Optimizing True Learning-to-Rank Utility](https://mlsanity.com/paper/2508.14180/) (2025)
- [Data Efficient Sample Selection for In-Context Learning](https://mlsanity.com/paper/2609.06670/) (2026)
- [Principled Top-k Selection for Language Models with Hybrid Gradients](https://mlsanity.com/paper/2610.04162/) (2026)

## Abstract (arXiv)

A key challenge with machine learning approaches for ranking is the gap between the performance metrics of interest and the surrogate loss functions that can be optimized with gradient-based methods. This gap arises because ranking metrics typically involve a sorting operation which is not differentiable w.r.t. the model parameters. Prior works have proposed surrogates that are loosely related to ranking metrics or simple smoothed versions thereof, and often fail to scale to real-world applications. We propose PiRank, a new class of differentiable surrogates for ranking, which employ a continuous, temperature-controlled relaxation to the sorting operator based on NeuralSort [1]. We show that PiRank exactly recovers the desired metrics in the limit of zero temperature and further propose a divide and-conquer extension that scales favorably to large list sizes, both in theory and practice. Empirically, we demonstrate the role of larger list sizes during training and show that PiRank significantly improves over comparable approaches on publicly available internet-scale learning-to-rank benchmarks.