  The asymptotic mean squared test error and sensitivity of the Random Features
Regression model (RFR) have been recently studied. We build on this work and
identify in closed-form the family of Activation Functions (AFs) that minimize
a combination of the test error and sensitivity of the RFR under different
notions of functional parsimony. We find scenarios under which the optimal AFs
are linear, saturated linear functions, or expressible in terms of Hermite
polynomials. Finally, we show how using optimal AFs impacts well-established
properties of the RFR model, such as its double descent curve, and the
dependency of its optimal regularization parameter on the observation noise
level.
