FishBack: Pullback Fisher Geometry for Optimal Activation Steering in Transformers

• URL: http://arxiv.org/abs/2605.17231v1
• Date: Sun, 17 May 2026 03:00:16 GMT
• Title: FishBack: Pullback Fisher Geometry for Optimal Activation Steering in Transformers
• Abstract summary: Activation steering methods modify intermediate representations of language models to control output behavior, but universally assume the activation space is Euclidean.<n>We show this assumption fails drastically: the local geometry induced by the model's own output behavior deviates from the Euclidean metric by over 97% in relative spectral norm on GPT-2.
• Score: 7.108048394322025
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Activation steering methods modify intermediate representations of language models to control output behavior, but universally assume the activation space is Euclidean. We show this assumption fails drastically: the local geometry induced by the model's own output behavior -- the Fisher information metric of the softmax layer, pulled back through the Jacobian of subsequent layers -- deviates from the Euclidean metric by over 97% in relative spectral norm on GPT-2, with an effective dimensionality of only 2--17% of the ambient space. From this pullback Fisher metric, we derive a closed-form steering equation that identifies the minimum-distortion direction for any target concept, yielding a closed-form optimal direction at each point that can be applied iteratively without manifold fitting or data-driven geometry estimation. We call the resulting framework FishBack. The metric admits a layer-wise recursive decomposition, which reveals that existing methods -- CAA, ActAdd, ITI, and others -- each implicitly adopt a particular approximate metric, and that their performance gaps are quantitatively predicted by a single spectral diagnostic: the ratio of their implicit metric's cost to the Fisher-optimal cost. On GPT-2, iterative pullback steering consistently outperforms all Euclidean baselines across three verb-morphology concepts and four layers, with off-target KL reductions of $1.3\times$--$2.5\times$ relative to Euclidean gradient ascent and $1.5\times$ relative to CAA at matched concept probability.

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