ERM. The traditional Empirical Risk Minimization (ERM) algorithm optimizes over the loss on i.i.d. data, i.e.,
min β,Φ R(β · Φ), where R(β · Φ) = E (x,y)∼D (β Φ(x), y).
(1) In the OOD generalization problem, one faces a set of (training) environments E, where each environment e ∈ E corresponds to a unique **data** distribution D e . When facing multiple environments, the ERM objective simply mixes the **data** together and takes the form
(ERM) min β,Φ e R e (β · Φ),(2)
where R e (β · Φ) = E (x,y)∼De (β Φ(x), y).