# The Gradient That Top-k Distillation Never Computes

arXiv: https://arxiv.org/abs/2610.15589 · HTML: https://mlsanity.com/paper/2610.15589/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in General knowledge distillation methods, one of 12 directions in Knowledge distillation.
- Builds on 36 of its 38 references that are in the map.
- First posted on arXiv 11 Oct 2026.

## Placed in

- Knowledge distillation / [General knowledge distillation methods](https://mlsanity.com/distill/general-knowledge-distillation-methods/)

## Builds on (most cited first, 30 of 36)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [FitNets: Hints for Thin Deep Nets](https://mlsanity.com/paper/1412.6550/) (2014)
- [MiniLM: Deep Self-Attention Distillation for Task-Agnostic Compression of Pre-Trained Transformers](https://mlsanity.com/paper/2002.10957/) (2020)
- [DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence](https://mlsanity.com/paper/2606.19348/) (2026)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [Sequence-Level Knowledge Distillation](https://mlsanity.com/paper/1606.07947/) (2016)
- [Rethinking On-Policy Distillation of Large Language Models: Phenomenology, Mechanism, and Recipe](https://mlsanity.com/paper/2604.13016/) (2026)
- [Representation Degeneration Problem in Training Natural Language Generation Models](https://mlsanity.com/paper/1907.12009/) (2019)
- [DistiLLM: Towards Streamlined Distillation for Large Language Models](https://mlsanity.com/paper/2402.03898/) (2024)
- [A Survey of On-Policy Distillation for Large Language Models](https://mlsanity.com/paper/2604.00626/) (2026)
- [Stealing Part of a Production Language Model](https://mlsanity.com/paper/2403.06634/) (2024)
- [Breaking the Softmax Bottleneck: A High-Rank RNN Language Model](https://mlsanity.com/paper/1711.03953/) (2017)
- [Liger Kernel: Efficient Triton Kernels for LLM Training](https://mlsanity.com/paper/2410.10989/) (2024)
- [f-Divergence Minimization for Sequence-Level Knowledge Distillation](https://mlsanity.com/paper/2307.15190/) (2023)
- [DistiLLM-2: A Contrastive Approach Boosts the Distillation of LLMs](https://mlsanity.com/paper/2503.07067/) (2025)
- [Autoregressive Knowledge Distillation through Imitation Learning](https://mlsanity.com/paper/2009.07253/) (2020)
- [Distillation Scaling Laws](https://mlsanity.com/paper/2502.08606/) (2025)
- [Logits of API-Protected LLMs Leak Proprietary Information](https://mlsanity.com/paper/2403.09539/) (2024)
- [Cut Your Losses in Large-Vocabulary Language Models](https://mlsanity.com/paper/2411.09009/) (2024)
- [Pre-training Distillation for Large Language Models: A Design Space Exploration](https://mlsanity.com/paper/2410.16215/) (2024)
- [Why do small language models underperform? Studying Language Model Saturation via the Softmax Bottleneck](https://mlsanity.com/paper/2404.07647/) (2024)
- [Decoupling KL and Trajectories: A Unified Perspective for SFT, DAgger, Offline RL, and OPD in LLM Distillation](https://mlsanity.com/paper/2605.16826/) (2026)
- [BiLD: Bi-directional Logits Difference Loss for Large Language Model Distillation](https://mlsanity.com/paper/2406.13555/) (2024)
- [Sparse Logit Sampling: Accelerating Knowledge Distillation in LLMs](https://mlsanity.com/paper/2503.16870/) (2025)
- [Breaking the Softmax Bottleneck via Learnable Monotonic Pointwise Non-linearities](https://mlsanity.com/paper/1902.08077/) (2019)
- [Your Teacher Can't Help You Here: Combating Supervision Fidelity Decay in On-Policy Distillation](https://mlsanity.com/paper/2605.30833/) (2026)
- [Sequences of Logits Reveal the Low Rank Structure of Language Models](https://mlsanity.com/paper/2510.24966/) (2025)
- [The Illusion of Certainty: Decoupling Capability and Calibration in On-Policy Distillation](https://mlsanity.com/paper/2604.16830/) (2026)
- [A Dual-Space Framework for General Knowledge Distillation of Large Language Models](https://mlsanity.com/paper/2504.11426/) (2025)
- [KDFlow: A User-Friendly and Efficient Knowledge Distillation Framework for Large Language Models](https://mlsanity.com/paper/2603.01875/) (2026)

## Abstract (arXiv)

Many deployed distillation pipelines show the student a truncated view of the teacher: keep the top-k probabilities, renormalize, and minimize a divergence on what remains. Renormalization is assumed to account for the missing mass. We prove that it removes the visibility of tail mass, not the freedom to move it. The renormalized objective depends only on log-odds among the retained tokens, so at each position its gradient along the directions that control tail mass is exactly zero; on a real teacher that component measures 2-3×10-15 of the gradient. The full-vocabulary gradient equals the truncated gradient plus a residual weighted by the tail masses, and training acts only on the first term. Without the cross-entropy term, a top-k student's tail mass drifts away from the teacher's while a full-vocabulary control moves slightly toward it; with it, the gap shrinks about fourfold but persists. On seven open-weight teachers, the retained set determines the teacher's distribution only from k=d+1 (d the hidden dimension): recovery is exact from there on all seven, and commonly cached sizes sit below it. In our off-policy runs the practical effect is real but smaller than the choice of teacher-student pair. Renormalization is a normalization, not a safeguard. The code is available at https://github.com/KurbanIntelligenceLab/top-k-distillation.