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 $\ell_{\infty}$ benchmark from RobustBench by 1032.64\%/1494.46\%. Our code is available at https://github.com/Leon-Leyang/curvature-tuning.

The number of linear regions has been studied as a proxy of complexity for ReLU networks. However, the empirical success of network compression techniques like pruning and knowledge distillation, suggest that in the overparameterized setting, linear regions density might fail to capture the effective nonlinearity. In this work, we propose an efficient algorithm for discovering linear regions and use it to investigate the effectiveness of density in capturing the nonlinearity of trained VGGs and ResNets on CIFAR-10 and CIFAR-100. We contrast the results with a more principled nonlinearity measure based on function variation, highlighting the shortcomings of linear regions density. Furthermore, interestingly, our measure of nonlinearity clearly correlates with model-wise deep double descent, connecting reduced test error with reduced nonlinearity, and increased local similarity of linear regions.