Fishr: Invariant Gradient Variances for Out-of-distribution Generalization

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 controlled evaluation protocols. In this paper, we introduce a new regularization -- named Fishr -- that enforces domain invariance in the space of the gradients of the loss: specifically, the domain-level variances of gradients are matched across training domains. Our approach is based on the close relations between the gradient covariance, the Fisher Information and the Hessian of the loss: in particular, we show that Fishr eventually aligns the domain-level loss landscapes locally around the final weights. Extensive experiments demonstrate the effectiveness of Fishr for out-of-distribution generalization. Notably, Fishr improves the state of the art on the DomainBed benchmark and performs consistently better than Empirical Risk Minimization. The code is released at https://github.com/alexrame/fishr.