WAQS: Weight Absorption for Quadratic Probing and Affine Steering

Overview & Synopsis

Activation steering provides a lightweight alternative to fine-tuning for controlling large language model behavior. Existing methods, however, often face a trade-off between efficiency and expressiveness: fixed steering vectors incur negligible inference overhead but cannot adapt to the current model input, whereas input-dependent nonlinear interventions provide greater flexibility at additional runtime cost. We introduce \textbf{WAQS} (\textbf{W}eight \textbf{A}bsorption for \textbf{Q}uadratic probing and affine \textbf{S}teering), a framework that obtains state-dependent steering, without additional inference-time computation. WAQS models a concept with a quadratic probe, whose gradient yields an affine, activation-dependent steering direction. We show that this affine intervention can be folded exactly into adjacent linear transformations through an offline weight update. Moreover, we prove that quadratic functions form the maximal class of smooth concept scores whose gradient-based interventions are affine and therefore exactly absorbable into linear layers. Under a Gaussian representation model, our formulation admits an interpretation as the difference of class-conditional Mahalanobis pulls and naturally generalizes linear discriminant and difference-in-means steering. To scale to high-dimensional representations, we introduce a low-rank parameterization of the quadratic term. Across truthfulness, refusal, and sycophancy steering tasks, quadratic steering matches or improves upon constant linear steering while preserving model utility, with the absorbed implementation introducing no additional inference-time operations. Our results suggest that quadratic probes provide a practical middle ground between static linear steering and computationally expensive dynamic nonlinear interventions.