aiqbench analysis
This research introduces Hyperbolic Entropy Steering (HEST), a novel activation steering technique that leverages hyperbolic geometry to more effectively guide large language models during complex mathematical reasoning tasks. By targeting high-entropy tokens where the model's decision path branches, HEST improves accuracy on benchmarks like MATH-500 and GSM8K without the performance degradation often seen with standard Euclidean vector additions. The method demonstrates that modeling the hierarchical nature of reasoning in non-Euclidean spaces offers a distinct advantage for enhancing model reliability in sparse activation scenarios.
Hesitation Has a Geometry: Entropy-Trained Hyperbolic Probes for Sparse Activation Steering
arXiv:2610.02391v1 Announce Type: cross Abstract: When a large language model solves a mathematical problem, its reasoning is largely hierarchical, and the solution often branches at a few tokens where the next-token entropy is high. Such tree-like structure embeds in hyperbolic space with far lower distortion than in Euclidean space. Activation steering, however, usually edits the hidden states of a pretrained model by adding one fixed Euclidean vector at every token, even though most tokens of a solution are already determined by the context. We propose Hyperbolic Entropy Steering (HEST), which embeds the hidden states in the Poincar\'e ball with a lightweight probe whose only label is the model's own next-token entropy. Where this entropy exceeds a threshold, HEST moves the embedded state along the geodesic of steepest descent of a readout of the probe and maps the change back to the hidden state. For the Busemann readout of a learned ideal point, we prove that a step of fixed length lowers it by the same amount at every state. On three instruction-tuned models from the Qwen2.5-Math and Llama-3.1 families, HEST with the Busemann readout improves greedy accuracy on MATH-500 and GSM8K in five of six settings, by up to 1.8 points, whereas a contrastive steering vector added at every token lowers accuracy. With a Euclidean probe trained in the same way, this gain disappears on Qwen2.5-Math-1.5B-Instruct. The gains are largest on problems where the model hesitates often, and accuracy on the remaining problems is almost unchanged.
Abstract:When a large language model solves a mathematical problem, its reasoning is largely hierarchical, and the solution often branches at a few tokens where the next-token entropy is high. Such tree-like structure embeds in hyperbolic space with far lower distortion than in Euclidean space. Activation steering, however, usually edits the hidden states of a pretrained model by adding one fixed Euclidean vector at every token, even though most tokens of a solution are already determined by the context. We propose Hyperbolic Entropy Steering (HEST), which embeds the hidden states in the Poincaré ball with a lightweight probe whose only label is the model's own next-token entropy. Where this entropy exceeds a threshold, HEST moves the embedded state along the geodesic of steepest descent of a readout of the probe and maps the change back to the hidden state. For the Busemann readout of a learned ideal point, we prove that a step of fixed length lowers it by the same amount at every state. On three instruction-tuned models from the Qwen2.5-Math and Llama-3.1 families, HEST with the Busemann readout improves greedy accuracy on MATH-500 and GSM8K in five of six settings, by up to 1.8 points, whereas a contrastive steering vector added at every token lowers accuracy. With a Euclidean probe trained in the same way, this gain disappears on Qwen2.5-Math-1.5B-Instruct. The gains are largest on problems where the model hesitates often, and accuracy on the remaining problems is almost unchanged.
Machine Learning (cs.LG); Computation and Language (cs.CL)
arXiv:2610.02391 [cs.LG]
(or arXiv:2610.02391v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.02391 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 339 words
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