We revisit the fundamental problem of learning with distribution shift, in which a learner is given labeled samples from training distribution $D$, unlabeled samples from test distribution $D'$ and is asked to output a classifier with low test error. The standard approach in this setting is to bound the loss of a classifier in terms of some notion of distance between $D$ and $D'$. These distances, however, seem difficult to compute and do not lead to efficient algorithms. We depart from this paradigm and define a new model called testable learning with distribution shift, where we can obtain provably efficient algorithms for certifying the performance of a classifier on a test distribution. In this model, a learner outputs a classifier with low test error whenever samples from $D$ and $D'$ pass an associated test; moreover, the test must accept if the marginal of $D$ equals the marginal of $D'$. We give several positive results for learning well-studied concept classes such as halfspaces, intersections of halfspaces, and decision trees when the marginal of $D$ is Gaussian or uniform on $\{\pm 1\}^d$. Prior to our work, no efficient algorithms for these basic cases were known without strong assumptions on $D'$. For halfspaces in the realizable case (where there exists a halfspace consistent with both $D$ and $D'$), we combine a moment-matching approach with ideas from active learning to simulate an efficient oracle for estimating disagreement regions. To extend to the non-realizable setting, we apply recent work from testable (agnostic) learning. More generally, we prove that any function class with low-degree $L_2$-sandwiching polynomial approximators can be learned in our model. We apply constructions from the pseudorandomness literature to obtain the required approximators.

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