MComparison of concept vector norms¶

All of our steering interventions add a scalar multiple of a concept vector to the residual stream. The effective strength of a given $\alpha$ therefore depends on the $L^{2}$ norm of the vector being added, which varies. Table 27 reports those norms for the reward vectors.

M.1Norm-matching control-vector steering¶

The steering evaluations in §4 compare the trained reward vectors $\mathbf{v}_{c}$ against the maze-naive control concept vectors $\mathbf{u}_{c}$ extracted by the same pipeline (§2.3). Because $\mathbf{u}_{c}$ and $\mathbf{v}_{c}$ are extracted by the same pipeline from checkpoints that differ only in whether maze training has occurred, the controls isolate the contribution of maze training to the downstream steering effects. A random-direction baseline could rule out “any vector steers,” but $\mathbf{u}_{c}$ is a more targeted control: it additionally rules out artifacts of the extraction pipeline itself (e.g. the trajectory construction or the difference-in-means procedure) and of the emoji tokens’ pre-existing representations. Any steering effect present for $\mathbf{v}_{c}$ but absent for $\mathbf{u}_{c}$ is therefore attributable to how maze training reshaped the model’s representations of the rewarded tiles.

A concern is that the trained vectors and the controls generally have different $L^{2}$ norms (Table 27). We therefore norm-match: control vectors are steered with scaled factors $\beta$ chosen so that $\beta \lVert \mathbf{u}_{c} \rVert = \alpha \lVert \mathbf{v}_{c} \rVert$⁠, so the residual-stream perturbation has the same magnitude at every nominal $\alpha$⁠. In figures we plot the control vectors at their nominal $\alpha$ rather than the scaled $\beta$⁠, so trained and control curves can be read against the same horizontal axis. This is the convention used wherever a trained reward vector is compared against a control in the body steering evaluations (§4); it is also reused for the maze-naive control comparisons in Appendix C, for the VAA comparison in Appendix H, and for the sentiment-subspace decomposition in Appendix F.5.

We do not renormalize the trained reward vectors themselves; their norms could be signals of representation strength.