  This paper considers the Pointer Value Retrieval (PVR) benchmark introduced
in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce
the label. More generally, the paper considers the learning of logical
functions with gradient descent (GD) on neural networks. It is first shown that
in order to learn logical functions with gradient descent on symmetric neural
networks, the generalization error can be lower-bounded in terms of the
noise-stability of the target function, supporting a conjecture made in
[ZRKB21]. It is then shown that in the distribution shift setting, when the
data withholding corresponds to freezing a single feature (referred to as
canonical holdout), the generalization error of gradient descent admits a tight
characterization in terms of the Boolean influence for several relevant
architectures. This is shown on linear models and supported experimentally on
other models such as MLPs and Transformers. In particular, this puts forward
the hypothesis that for such architectures and for learning logical functions
such as PVR functions, GD tends to have an implicit bias towards low-degree
representations, which in turn gives the Boolean influence for the
generalization error under quadratic loss.
