Abstract:
This paper studies the prediction of a target zz from a pair of random variables (x,y)(x,y), where the ground-truth predictor is additive E[z∣x,y]=f⋆(x)+g⋆(y)E[z∣x,y]=f⋆(x)+g⋆(y). We study the performance of empirical risk minimization (ERM) over functions f+gf+g, f∈Ff∈F and g∈Gg∈G, fit on a given training distribution, but evaluated on a test distribution which exhibits covariate shift. We show that, when the class FF is "simpler" than GG (measured, e.g., in terms of its metric entropy), our predictor is more resilient to *heterogenous covariate shifts* in which the shift in xx is much greater than that in yy. These results rely on a novel Hölder style inequality for the Dudley integral which may be of independent interest. Moreover, we corroborate our theoretical findings with experiments demonstrating improved resilience to shifts in "simpler" features across numerous domains.
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