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Domain generalization aims at performing well on unseen test environmentswith data from a limited number of training environments. Despite aproliferation of proposal algorithms for this task, assessing their performanceboth theoretically and empirically is still very challenging. Distributionalmatching algorithms such as (Conditional) Domain Adversarial Networks [Ganin etal., 2016, Long et al., 2018] are popular and enjoy empirical success, but theylack formal guarantees. Other approaches such as Invariant Risk Minimization(IRM) require a prohibitively large number of training environments -- linearin the dimension of the spurious feature space $d_s$ -- even on simple datamodels like the one proposed by [Rosenfeld et al., 2021]. Under a variant ofthis model, we show that both ERM and IRM cannot generalize with $o(d_s)$environments. We then present an iterative feature matching algorithm that isguaranteed with high probability to yield a predictor that generalizes afterseeing only $O(\log d_s)$ environments. Our results provide the firsttheoretical justification for a family of distribution-matching algorithmswidely used in practice under a concrete nontrivial data model.

Iterative Feature Matching: Toward Provable Domain Generalization with Logarithmic Environments