# Don't Forget the Nonlinearity: Unlocking Activation Functions in Efficient Fine-Tuning

arXiv: https://arxiv.org/abs/2509.13240 · HTML: https://mlsanity.com/paper/2509.13240/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 28 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 20 of 20)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [Visual Prompt Tuning](https://mlsanity.com/paper/2203.12119/) (2022)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [Vision Transformer Adapter for Dense Predictions](https://mlsanity.com/paper/2205.08534/) (2022)
- [Searching for Activation Functions](https://mlsanity.com/paper/1710.05941/) (2017)
- [LoRA-FA: Efficient and Effective Low Rank Representation Fine-tuning](https://mlsanity.com/paper/2308.03303/) (2023)
- [Exploring the Limits of Large Scale Pre-training](https://mlsanity.com/paper/2110.02095/) (2021)
- [Learning Activation Functions to Improve Deep Neural Networks](https://mlsanity.com/paper/1412.6830/) (2014)
- [Kolmogorov-Arnold Transformer](https://mlsanity.com/paper/2409.10594/) (2024)
- [Neural networks and rational functions](https://mlsanity.com/paper/1706.03301/) (2017)
- [AReLU: Attention-based Rectified Linear Unit](https://mlsanity.com/paper/2006.13858/) (2020)
- [LoLDU: Low-Rank Adaptation via Lower-Diag-Upper Decomposition for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2410.13618/) (2024)
- [Precise Apple Detection and Localization in Orchards using YOLOv5 for Robotic Harvesting Systems](https://mlsanity.com/paper/2405.06260/) (2024)

## Built on by (most cited first, 4 of 4)

- [Learning Rate Matters: Vanilla LoRA May Suffice for LLM Fine-tuning](https://mlsanity.com/paper/2602.04998/) (2026)
- [FeRA: Frequency-Energy Constrained Routing for Effective Diffusion Adaptation Fine-Tuning](https://mlsanity.com/paper/2511.17979/) (2025)
- [Nonlinearity as Rank: Generative Low-Rank Adapter with Radial Basis Functions](https://mlsanity.com/paper/2602.05709/) (2026)
- [SPOT-E: Test-Time Entropy Shaping with Visual Spotlights for Frozen VLMs](https://mlsanity.com/paper/2606.20244/) (2026)

## Abstract (arXiv)

Existing parameter-efficient fine-tuning (PEFT) methods primarily adapt weight matrices while keeping activation functions fixed. We introduce NoRA, the first PEFT framework that directly adapts nonlinear activation functions in pretrained transformer-based models. NoRA replaces fixed activations with learnable rational functions and applies structured low-rank updates to numerator and denominator coefficients, with a group-wise design that localizes adaptation and improves stability at minimal cost. On vision transformers trained on CIFAR-10 and CIFAR-100, NoRA matches or exceeds full fine-tuning while updating only 0.4% of parameters (0.02M), achieving accuracy gains of +0.17% and +0.27%. When combined with LoRA (NoRA++), it outperforms LoRA and DoRA under matched training budgets by adding fewer trainable parameters. On LLaMA3-8B instruction tuning, NoRA++ consistently improves generation quality, yielding average MMLU gains of +0.3%--0.8%, including +1.6% on STEM (Alpaca) and +1.3% on OpenOrca. We further show that NoRA constrains adaptation to a low-dimensional functional subspace, implicitly regularizing update magnitude and direction. These results establish activation-space tuning as a complementary and highly parameter-efficient alternative to weight-based PEFT, positioning activation functions as first-class objects for model adaptation.