  Recent studies show that transformer-based architectures emulate gradient
descent during a forward pass, contributing to in-context learning capabilities
- an ability where the model adapts to new tasks based on a sequence of prompt
examples without being explicitly trained or fine tuned to do so. This work
investigates the generalization properties of a single step of gradient descent
in the context of linear regression with well-specified models. A random design
setting is considered and analytical expressions are derived for the
statistical properties and bounds of generalization error in a non-asymptotic
(finite sample) setting. These expressions are notable for avoiding arbitrary
constants, and thus offer robust quantitative information and scaling
relationships. These results are contrasted with those from classical least
squares regression (for which analogous finite sample bounds are also derived),
shedding light on systematic and noise components, as well as optimal step
sizes. Additionally, identities involving high-order products of Gaussian
random matrices are presented as a byproduct of the analysis.
