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Learning robust models that generalize well under changes in the data distribution is critical for real-world applications. To this end, there has been a growing surge of interest to learn simultaneously from multiple training domains - while enforcing different types of invariance across those domains. Yet, all existing approaches fail to show systematic benefits under fair evaluation protocols. In this paper, we propose a new learning scheme to enforce domain invariance in the space of the gradients of the loss function: specifically, we introduce a regularization term that matches the domain-level variances of gradients across training domains. Critically, our strategy, named Fishr, exhibits close relations with the Fisher Information and the Hessian of the loss. We show that forcing domain-level gradient covariances to be similar during the learning procedure eventually aligns the domain-level loss landscapes locally around the final weights. Extensive experiments demonstrate the effectiveness of Fishr for out-of-distribution generalization. In particular, Fishr improves the state of the art on the DomainBed benchmark and performs significantly better than Empirical Risk Minimization. The code is released at https://github.com/alexrame/fishr.
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