Why Subliminal Learning Needs So Much Data: A Noisy Inverse View through Steering Vector Recovery

Luoyu Chen, Xiaoyu Ding, Weiqi Wang, Chenhan Zhang, Zhiyi Tian +2

Subliminal learning lets a student inherit a teacher's behavioral trait from semantically unrelated data, yet published demonstrations typically require tens of thousands of carrier examples. We ask where this data requirement comes from. Our testbed is subliminal steering: the teacher trait is a known residual-stream vector $Δ_T$, so transfer can be measured directly as parameter recovery. On identical carrier prefixes, we compare token-level (hard) NLL supervision with full-distribution (soft) KL supervision. At initialization the two objectives give nearly collinear gradients, and both align poorly with $Δ_T$. Under iterative optimization, however, they diverge: soft supervision recovers $Δ_T$ almost exactly from a few hundred carriers, while hard supervision stays well below it even with tens of thousands. We explain this gap by casting steering-vector distillation as a noisy linear inverse problem. Locally, the carrier task maps the trait through its Fisher matrix $F$, so gradients point toward $FΔ_T$ rather than $Δ_T$. Gradient descent then acts as a progressively less-damped inverse of $F$. With soft targets, this inverse restores low-curvature directions. With hard…