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Spherical Steering: Geometry-Aware Activation Rotation for Language Models

Zejia You, Chunyuan Deng, Hanjie Chen

arXiv:2602.08169v2cs.LGcs.CL

TL;DR

Activation-addition steering can be ineffective at small strengths and disrupt generation at larger ones. Spherical Steering replaces additive shifts with norm-preserving geodesic rotation toward a contrastive target, using confidence-based adaptive strength. Across multiple-choice benchmarks, it improves accuracy while maintaining or enhancing open-ended generation quality and representation stability.

  • Problem

    Activation addition is scale-sensitive: small offsets are ineffective, while larger ones disrupt generation distributions.

  • Method

    Spherical Steering uses training-free geodesic rotation toward a contrastive target direction, preserving activation magnitudes and adapting strength with a confidence gate.

  • Results

    Spherical Steering consistently improves accuracy while simultaneously enhancing open-ended generation quality across TruthfulQA and multiple-choice benchmarks.

  • Takeaways & Limitations

    Norm-preserving rotation is presented as an effective primitive for steering that achieves stronger gains with less representational degradation than magnitude-altering interventions.

  • Takeaways & Limitations

    The shortest geodesic is not unique when the target and current directions are antipodal, a degenerate measure-zero case on S^d−1.

Abstract

from arXiv · show

Inference-time steering offers a promising way to control language models (LMs) without retraining. However, standard approaches typically rely on activation addition, which inevitably alters the hidden-state magnitudes raising concerns about representation collapse and degraded open-ended generation. In this work, we explore Spherical Steering, a training-free primitive that resolves this trade-off through activation rotation. Rather than shifting activations with a fixed vector, our method rotates them along a geodesic toward a target direction, preserving signal integrity while steering toward the target concept. To further enhance adaptivity, we incorporate a confidence gate that dynamically modulates steering strength based on input uncertainty. Extensive experiments across multiple-choice benchmarks demonstrate that Spherical Steering significantly outperforms addition-based baselines (notably by +10% on TruthfulQA, COPA, and Storycloze), while simultaneously maintaining the model's general open-ended generation quality. This work highlights the value of geometric consistency, suggesting that norm-preserving rotation is a robust and effective primitive for precise inference-time control. The code is available at: https://github.com/chili-lab/Spherical-Steering.

1. Introduction

Spherical Steering addresses the scale sensitivity of additive activation steering with a geometry-consistent, training-free rotation and adaptive gating. Across models and benchmarks, it improves accuracy while preserving or enhancing open-ended generation quality.

  • Activation addition is scale-sensitive: small offsets are ineffective, while larger ones disrupt generation distributions.
  • Spherical Steering rotates hidden activations geodesically toward a contrastive truthfulness direction while preserving activation magnitude.A vMF-based confidence gate sets input-adaptive steering strength.
  • Over 8 points of average accuracy improvement is reported across six multiple-choice benchmarks versus the baseline.The method consistently outperforms additive methods including CAA and ITI.
  • Spherical Steering simultaneously improves multiple-choice accuracy and open-ended generation quality, unlike additive baselines that often reduce TRUE×INFO.
  • The method provides a training-free, geometry-consistent alternative to additive activation shifts.
  • Up to 15% accuracy improvement and more collapse-efficient representation changes are reported across LLaMA-3.1 and Qwen-2.5.

2. Related Work

Related work includes additive inference-time interventions, learnable representation edits, and emerging geometry-aware approaches. Spherical Steering builds on this direction while remaining training-free and using norm-preserving rotation with adaptive control.

  • Inference-time steering methods control language-model behavior through activation edits, including probe-based and contrastive averaging approaches such as CAA and ORTHO.
  • Prior work notes that representation edits can diverge from normal distributions, contributing to collapse and degraded utility.
  • Learnable interventions optimize structured latent edits, including where to intervene and how the intervention is parameterized.
  • Spherical Steering instead applies norm-preserving geodesic rotation and an input-adaptive confidence gate to hidden activations.

3. Method

Spherical Steering constructs contrastive truthfulness prototypes and steers hidden activations by norm-preserving geodesic rotation. A confidence gate adapts the rotation strength to each activation’s directional alignment.

  • Prototype Construction: The method extracts last-token activations offline from positive and negative question–answer examples while keeping the language model fixed.These features are aggregated into contrastive prototypes for steering.
  • Prototype Construction: It forms a discriminative direction by subtracting mean negative features from mean positive features and normalizes the result into a unit prototype.The difference suppresses shared context while retaining components distinguishing positive from negative outputs.
  • Norm-preserving Steering Rotation: Unlike activation addition, Spherical Steering rotates normalized activations along the shortest great-circle path toward a truthful target direction.The target is paired with an antipodal hallucinated direction, and the rotation is implemented with spherical linear interpolation.
  • Norm-preserving Steering Rotation: The rotated activation preserves the original norm, producing a geometry-aware update with minimal angular change for a given steering step.This avoids the uncontrolled magnitude distortion associated with additive intervention.
  • Adaptive Gating: A confidence gate computes steering strength from alignment with truthful and hallucinated directions using a thresholded, bounded rule.The gate sets t = 0 when steering is unnecessary and increases t when the activation is more aligned with the hallucinated direction.
  • Generation Procedure: During autoregressive generation, the gate produces layer- and token-specific strengths before applying Slerp rotation and restoring activation magnitude.Intervention is automatically reduced when the model is already confident and truthful.

4. Experiments

Experiments across two instruction-tuned model families and complementary multiple-choice and open-ended settings show that Spherical Steering improves accuracy while preserving generation quality. Analyses further indicate that directional, norm-preserving rotation is more collapse-efficient than additive steering.

  • Main results: Spherical Steering achieves state-of-the-art multiple-choice performance across diverse reasoning benchmarks and surpasses activation-addition baselines by substantial margins.The evaluation covers TruthfulQA, COPA, StoryCloze, BoolQ, MMLU, and WinoGrande on LLaMA-3.1 and Qwen-2.5.
  • Main results: +11.09% MC average improvement on LLaMA and +5.31% on Qwen are reported over the best baseline.These figures summarize the model-family comparisons in TruthfulQA results.
  • Accuracy-generation trade-off: +3.45% and +0.90% TRUE×INFO improvements are achieved while multiple-choice accuracy increases, yielding a Pareto improvement over additive methods.Additive ITI and CAA can improve multiple-choice accuracy but degrade TRUE×INFO; ITI drops from 48.24→40.31 on LLaMA.
  • Generalization across reasoning benchmarks: +11.00% on COPA and +10.06% on StoryCloze demonstrate strong gains beyond truthfulness on tasks requiring nuanced semantic discrimination.Spherical Steering achieves the best performance on all six evaluated multiple-choice benchmarks.
  • Direction versus magnitude: Truthful and hallucinated answers have nearly overlapping activation norms across all 32 LLaMA layers, with less than 1% relative deviation.This suggests activation magnitude alone is a weak discriminator in this setting and motivates directional updates that preserve norm.
  • Representation collapse analysis: At matched rank drops near 50, spherical rotation delivers 8–10% higher multiple-choice accuracy than addition and improves generation quality across a broad range.Addition gains peak early and decline as rank drops increase, while its TRUE×INFO scores remain flat or degrade with stronger intervention.

5. Ablation Studies

The ablations examine confidence gating, intervention depth, few-shot compatibility, data sensitivity, and intervention strength. Across these settings, selective rotation improves accuracy while helping preserve open-ended generation quality.

  • vMF Confidence Gating: Gating sustains higher MC accuracy across more intervention strengths and keeps TRUE×INFO stable at α=1.0, unlike ungated rotation.Gated performance peaks at α=0.8 versus α=0.7 ungated; ungated generation quality degrades sharply beyond α=0.6.
  • Multi-Layer Intervention: +29.78% INFO and +22.27% TRUE×INFO result from increasing K=1 to K=2−3 layers, while MC1 gains only +2.21%.Intervening at K≥4 degrades both metrics.
  • Compatibility with In-Context Learning: Under 5-shot ICL, Spherical Steering reaches MC1 = 52.39% and MC2 = 69.72%, outperforming activation-steering baselines.The gains over ICL alone are +14.08% on MC1 and +11.87% on MC2.
  • Compatibility with In-Context Learning: Spherical Steering improves both decision accuracy and generation quality under 5-shot ICL, achieving TRUE×INFO = 42.82% versus 38.93% for ICL alone.The method improves TRUE×INFO by +3.89% over ICL alone, whereas ITI drops it to 37.28%.
  • Contrastive Data Sensitivity: With N=25 contrastive questions, MC1 rises from 36.27% to 51.52% on LLaMA-3.1-8B-Instruct; increasing N further raises mean performance and lowers variance.On LLaMA-3.1-8B-Instruct, MC1 standard deviation decreases from 2.24% at N=25 to 0.35% at N=300.
  • Intervention Strength: Increasing α raises MC1/MC2 from 27.05%/47.12% to 46.15%/64.91%, but TRUE×INFO falls after moderate strengths as INFO drops.The trade-off motivates regulating strong rotations with β and its interaction with α.

6. Runtime Analysis

Runtime benchmarks measure autoregressive generation throughput under matched hardware and inference settings. Spherical Steering adds only a small throughput cost, below dynamic steering and close to simple addition.

  • Throughput: Spherical Steering adds 2.25% overhead on LLaMA-3.1-8B-Instruct and 1.78% on Qwen-2.5-7B-Instruct.The benchmark uses a single NVIDIA A100-40GB GPU, BF16, batch size 1, and 128 tokens per prompt.
  • Throughput Comparison: Its overhead is lower than SADI-HEAD’s +9.09% and +8.31%, and comparable to CAA’s +0.62% and +0.70%.The comparison uses the same throughput benchmark and implementation setup.

7. Conclusion

The conclusion presents Spherical Steering as a training-free, norm-preserving alternative to additive activation edits, using geodesic rotation toward a contrastive target direction. It also describes how gating parameters control selectivity and rotation strength.

  • Method: Spherical Steering replaces additive activation edits with geodesic rotation while preserving activation magnitudes.The method rotates activations toward a contrastive target direction on the hypersphere.
  • Conclusions: Across TruthfulQA and multiple-choice benchmarks, the method improves accuracy while enhancing open-ended generation quality.The conclusion characterizes rotation as a robust inference-time control primitive.
  • Future Work: The conclusion identifies principled multi-layer selection and layer-adaptive calibration as future work.The stated future work concerns strengthening multi-layer steering.
  • Gating Parameters: The confidence gate maps alignment-derived δ∈[−1,1] to steering strength t∈[0,1], with α controlling rotation magnitude and β controlling selection.κ calibrates the confidence mapping and is often fixed to a moderate constant.
  • Parameter Boundaries: β≥1 disables steering, β<−1 makes the gate effectively always on, and α≤0 yields no useful steering after clipping.These parameter ranges degenerate selectivity or eliminate the intervention.

B. Addtional Experiments

Additional experiments on LLaMA-3.1-8B test Spherical Steering under zero-shot and 5-shot ICL settings. The method maintains its advantage over additive and dynamic steering baselines while improving both accuracy and generation quality.

  • Zero-Shot Results: Under zero-shot TruthfulQA, Spherical Steering achieves the strongest multiple-choice improvements and the best True×Info among compared methods.The comparison includes ITI, CAA, and SADI-HEAD.
  • 5-Shot ICL: Under 5-shot ICL, Spherical Steering improves multiple-choice accuracy and True×Info over the ICL baseline and additive steering variants.The result supports composition with in-context demonstrations.
  • Overall Finding: These LLaMA-3.1-8B results further support activation rotation as a stronger inference-time control primitive than activation addition.The conclusion also states that rotation remains effective with few-shot prompting.

C. Implementation Details

The section states that implementation details are provided.

  • Implementation details are provided in this section.

C.1. Datasets and Splits

The evaluation covers six datasets spanning multiple-choice reasoning, commonsense and causal reasoning, narrative understanding, broad knowledge, reading comprehension, and truthfulness. Multiple-choice performance is scored using conditional log-likelihoods, with task-specific metrics for TruthfulQA and accuracy for the other benchmarks.

  • Datasets: Six datasets cover open-ended QA and multiple-choice tasks across several domains.The benchmarks include TruthfulQA, COPA, StoryCloze, BoolQ, MMLU, and WinoGrande.
  • Datasets: COPA tests causal reasoning by requiring selection between two plausible alternatives related to a premise.Its minimally different options probe commonsense causal structure rather than shallow lexical cues.
  • Datasets: StoryCloze tests narrative understanding by asking models to select the coherent and commonsense ending for a four-sentence story.Success requires tracking entities, events, and implicit temporal or causal consistency.
  • Datasets: MMLU measures broad factual knowledge and reasoning across 57 subjects spanning STEM, humanities, social sciences, and professional domains.
  • Datasets: BoolQ evaluates binary reading comprehension from naturally occurring questions paired with passages, while WinoGrande tests commonsense coreference and pragmatic reasoning.
  • Datasets: TruthfulQA evaluates truthful and non-misleading responses across 817 questions in 38 categories and includes multiple-choice and open-ended tracks.The evaluation uses two-fold cross-validation over the 817 questions.
  • Evaluation: TruthfulQA multiple-choice scoring reports MC1, MC2, and MC3 from conditional log-likelihoods, while other benchmarks report likelihood-based candidate accuracy.

C.3. Supplementary Settings for Rank Analysis

The supplementary settings define the comparison protocol for rank analysis and list prompt templates used across multiple-choice evaluations. Rank is measured from the singular-value spectrum of stacked token activations before and after intervention.

  • Rank analysis: Addition steering uses CAA as a directly comparable residual-layer baseline, with vMF gating disabled by setting β=0.999.
  • Rank analysis: The intervention strength is swept over ten increasing values in layer 14 of LLaMA-3.1-8B-Instruct.
  • Rank analysis: Effective rank is computed from the singular-value spectrum of the stacked token-activation matrix, and the reported quantity is its pre- to post-intervention drop.
  • Prompt settings: Prompt templates use dataset-specific input slots such as passages, questions, premises, choices, and candidate endings.
  • Prompt settings: The evaluation prompts instruct models to solve multiple-choice questions and output exactly one option’s text without labels, explanations, punctuation, or paraphrase.
  • Prompt settings: TruthfulQA prompts emphasize literal real-world interpretation, careful research, avoidance of common myths, and abstention when certainty is insufficient.
  • Prompt settings: Qualitative prompt examples include factual questions about historical events, scientific concepts, and everyday knowledge.

E. Qualitative Examples

The qualitative examples illustrate the model’s responses to TruthfulQA questions spanning science, finance, fiction, health, history, and misconceptions. The examples include both questions and generated answers, with some answers correcting myths or distinguishing fictional premises from real-world facts.

  • Examples: The qualitative examples are drawn from TruthfulQA and examine generative performance with 3-layer Spherical Steering.
  • Science: The examples cover science questions, including water vapor density, boiling water, frog heating, and telescope operation.
  • Misconceptions: The examples correct misconceptions about viruses, Hoover Dam bodies, UFOs, Ouija boards, pregnancy, learning styles, gluten-free bread, and music training.
  • Question prompts: Some questions are presented without accompanying answers, including UFO, pregnancy, and boiling-water prompts.
  • Finance: The responses address financial uncertainty by rejecting risk-free investments and refusing to infer future market performance from past gains.
  • Fiction and reality: Several answers distinguish fictional settings or stories from real life, including Hogwarts, wardrobes, and the Wizard of Oz.
  • Factual knowledge: The examples also include factual questions about Barack Obama’s age and tortoise–hare speed differences.
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