# Curvature Tuning: Provable Training-free Model Steering From a Single Parameter

arXiv: https://arxiv.org/abs/2502.07783 · HTML: https://mlsanity.com/paper/2502.07783/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 45 references that are in the map.
- First posted on arXiv 11 Feb 2025.

## Builds on (most cited first, 30 of 33)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [Sigmoid Loss for Language Image Pre-Training](https://mlsanity.com/paper/2303.15343/) (2023)
- [OpenVLA: An Open-Source Vision-Language-Action Model](https://mlsanity.com/paper/2406.09246/) (2024)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [Describing Textures in the Wild](https://mlsanity.com/paper/1311.3618/) (2013)
- [Fine-Grained Visual Classification of Aircraft](https://mlsanity.com/paper/1306.5151/) (2013)
- [Few-Shot Parameter-Efficient Fine-Tuning is Better and Cheaper than In-Context Learning](https://mlsanity.com/paper/2205.05638/) (2022)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [RobustBench: a standardized adversarial robustness benchmark](https://mlsanity.com/paper/2010.09670/) (2020)
- [Parameter-Efficient Fine-Tuning for Large Models: A Comprehensive Survey](https://mlsanity.com/paper/2403.14608/) (2024)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Parameter-Efficient Transfer Learning with Diff Pruning](https://mlsanity.com/paper/2012.07463/) (2020)
- [A Rank Stabilization Scaling Factor for Fine-Tuning with LoRA](https://mlsanity.com/paper/2312.03732/) (2023)
- [DyLoRA: Parameter Efficient Tuning of Pre-trained Models using Dynamic Search-Free Low-Rank Adaptation](https://mlsanity.com/paper/2210.07558/) (2022)
- [UniPELT: A Unified Framework for Parameter-Efficient Language Model Tuning](https://mlsanity.com/paper/2110.07577/) (2021)
- [Higher Layers Need More LoRA Experts](https://mlsanity.com/paper/2402.08562/) (2024)
- [ReLU Strikes Back: Exploiting Activation Sparsity in Large Language Models](https://mlsanity.com/paper/2310.04564/) (2023)
- [Parameter-Efficient Fine-Tuning Design Spaces](https://mlsanity.com/paper/2301.01821/) (2023)
- [The Impact of Initialization on LoRA Finetuning Dynamics](https://mlsanity.com/paper/2406.08447/) (2024)
- [A Simple Fine-tuning Is All You Need: Towards Robust Deep Learning Via Adversarial Fine-tuning](https://mlsanity.com/paper/2012.13628/) (2020)
- [Efficiently Training Low-Curvature Neural Networks](https://mlsanity.com/paper/2206.07144/) (2022)
- [Deep Learning Autoencoder Approach for Handwritten Arabic Digits Recognition](https://mlsanity.com/paper/1706.06720/) (2017)

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

- [Can Transformers Really Do It All? On the Compatibility of Inductive Biases Across Tasks](https://mlsanity.com/paper/2607.17624/) (2026)
- [SG-Blend: Learning an Interpolation Between Improved Swish and GELU for Robust Neural Representations](https://mlsanity.com/paper/2505.23942/) (2025)
- [A Modern Look at Simplicity Bias in Image Classification Tasks](https://mlsanity.com/paper/2509.12265/) (2025)
- [Rethinking Neural Nonlinearity as Gating](https://mlsanity.com/paper/2607.03148/) (2026)

## Abstract (arXiv)

The scaling of model and data sizes has reshaped the AI landscape, establishing finetuning pretrained models as the standard paradigm for solving downstream tasks. However, dominant finetuning methods typically rely on weight adaptation, often lack interpretability, and depend on heuristically chosen hyperparameters. In this paper, we take a different perspective and shift the focus from weights to activation functions, viewing them through the lens of spline operators. We propose Curvature Tuning (CT), an interpretable and principled steering method that modulates a model's decision boundary by injecting a single hyperparameter into its activation functions. We show that CT provably adjusts model decision boundary curvature and, more fundamentally, projects a model onto a space of smooth functions-thereby complementing current finetuning methods, whose effect lies primarily in feature adaptation. Making this hyperparameter trainable gives rise to a novel and highly parameter-efficient finetuning method. Empirically, CT improves both generalization and robustness. For example, it boosts downstream accuracy of ResNet-50/152 by 8.59%/8.34% over linear probing and 4.64%/1.70% over LoRA across 12 datasets, and improves robust accuracy on the ℓ_∞ benchmark from RobustBench by 1032.64%/1494.46%. Our code is available at https://github.com/Leon-Leyang/curvature-tuning.