  Overparameterized models that achieve zero training error are observed to
generalize well on average, but degrade in performance when faced with data
that is under-represented in the training sample. In this work, we study an
overparameterized Gaussian mixture model imbued with a spurious feature, and
sharply analyze the in-distribution and out-of-distribution test error of a
cost-sensitive interpolating solution that incorporates "importance weights".
Compared to recent work Wang et al. (2021), Behnia et al. (2022), our analysis
is sharp with matching upper and lower bounds, and significantly weakens
required assumptions on data dimensionality. Our error characterizations also
apply to any choice of importance weights and unveil a novel tradeoff between
worst-case robustness to distribution shift and average accuracy as a function
of the importance weight magnitude.
