Pretty Linear Liars: Unregularized Probes Favor Causally Unfaithful Features

Linear probing is a common technique in mechanistic interpretability despite the fact that the decodability of a given feature does not necessarily imply the network uses that feature. Causal faithfulness, not merely decodability, is the appropriate criterion for whether a feature serves a defined active role in the model. Meanwhile, probe regularization has been used for feature discovery, but the impact of regularization on feature causality remains unclear. We study probe regularization systematically, starting with unregularized ordinary least squares (OLS) regression and considering increasing amounts of regularization via ridge regression. We show theoretically that OLS probes amplify low-variance directions that have spurious correlations with the target feature, whereas regularization reduces influence from these directions. Empirically, we find in a toy transformer and Llama-2-7B that causal faithfulness increases up to a point, and then decreases again, and that the optimal regularization strength leads to features bearing striking similarity (in both direction and variance) to those learned by optimizing for intervention outcomes directly. Together, these results provide mechanistic justification for regularizing regression probes.