  A pervasive phenomenon in machine learning applications is distribution
shift, where training and deployment conditions for a machine learning model
differ. As distribution shift typically results in a degradation in
performance, much attention has been devoted to algorithmic interventions that
mitigate these detrimental effects. In this paper, we study the effect of
distribution shift in the presence of model misspecification, specifically
focusing on $L_{\infty}$-misspecified regression and adversarial covariate
shift, where the regression target remains fixed while the covariate
distribution changes arbitrarily. We show that empirical risk minimization, or
standard least squares regression, can result in undesirable misspecification
amplification where the error due to misspecification is amplified by the
density ratio between the training and testing distributions. As our main
result, we develop a new algorithm -- inspired by robust optimization
techniques -- that avoids this undesirable behavior, resulting in no
misspecification amplification while still obtaining optimal statistical rates.
As applications, we use this regression procedure to obtain new guarantees in
offline and online reinforcement learning with misspecification and establish
new separations between previously studied structural conditions and notions of
coverage.
