# Differentiable Fast Top-K Selection for Large-Scale Recommendation

arXiv: https://arxiv.org/abs/2510.11472 · HTML: https://mlsanity.com/paper/2510.11472/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 31 references that are in the map.
- First posted on arXiv 13 Oct 2025.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Stochastic Optimization of Sorting Networks via Continuous Relaxations](https://mlsanity.com/paper/1903.08850/) (2019)
- [SoftSort: A Continuous Relaxation for the argsort Operator](https://mlsanity.com/paper/2006.16038/) (2020)
- [Smooth Loss Functions for Deep Top-k Classification](https://mlsanity.com/paper/1802.07595/) (2018)
- [Cascade Ranking for Operational E-commerce Search](https://mlsanity.com/paper/1706.02093/) (2017)
- [Fast, Differentiable and Sparse Top-k: a Convex Analysis Perspective](https://mlsanity.com/paper/2302.01425/) (2023)
- [An Alternative Cross Entropy Loss for Learning-to-Rank](https://mlsanity.com/paper/1911.09798/) (2019)
- [LapSum -- One Method to Differentiate Them All: Ranking, Sorting and Top-k Selection](https://mlsanity.com/paper/2503.06242/) (2025)
- [Stochastic smoothing of the top-K calibrated hinge loss for deep imbalanced classification](https://mlsanity.com/paper/2202.02193/) (2022)
- [Full Stage Learning to Rank: A Unified Framework for Multi-Stage Systems](https://mlsanity.com/paper/2405.04844/) (2024)
- [RecFlow: An Industrial Full Flow Recommendation Dataset](https://mlsanity.com/paper/2410.20868/) (2024)
- [Adaptive Neural Ranking Framework: Toward Maximized Business Goal for Cascade Ranking Systems](https://mlsanity.com/paper/2310.10462/) (2023)
- [Learning Cascade Ranking as One Network](https://mlsanity.com/paper/2503.09492/) (2025)

## Built on by (most cited first, 3 of 3)

- [Flow-Based Generative Modeling for Optimizing Sampling Policies in Compressed Sensing Applications](https://mlsanity.com/paper/2606.00078/) (2026)
- [Fast LapSum: Exact Differentiable Top-k at Million Scale](https://mlsanity.com/paper/2608.06912/) (2026)
- [Principled Top-k Selection for Language Models with Hybrid Gradients](https://mlsanity.com/paper/2610.04162/) (2026)

## Abstract (arXiv)

Cascade ranking is a widely adopted paradigm in large-scale information retrieval systems for Top-K item selection. However, the Top-K operator is non-differentiable, hindering end-to-end training. Existing methods include Learning-to-Rank approaches (e.g., LambdaLoss), which optimize ranking metrics like NDCG and suffer from objective misalignment, and differentiable sorting-based methods (e.g., ARF, LCRON), which relax permutation matrices for direct Top-K optimization but introduce gradient conflicts through matrix aggregation. A promising alternative is to directly construct a differentiable approximation of the Top-K selection operator, bypassing the use of soft permutation matrices. However, even state-of-the-art differentiable Top-K operator (e.g., LapSum) require O(n log n) complexity due to their dependence on sorting for solving the threshold. Thus, we propose DFTopK, a novel differentiable Top-K operator achieving optimal O(n) time complexity. By relaxing normalization constraints, DFTopK admits a closed-form solution and avoids sorting. DFTopK also avoids the gradient conflicts inherent in differentiable sorting-based methods. We evaluate DFTopK on both the public benchmark RecFLow and an industrial system. Experimental results show that DFTopK significantly improves training efficiency while achieving superior performance, which enables us to scale up training samples more efficiently. In the online A/B test, DFTopK yielded a +1.77% revenue lift with the same computational budget compared to the baseline. To the best of our knowledge, this work is the first to introduce differentiable Top-K operators into recommendation systems and the first to achieve theoretically optimal linear-time complexity for Top-K selection. We have open-sourced our implementation to facilitate future research in both academia and industry.