  Modern neural models trained on textual data rely on pre-trained
representations that emerge without direct supervision. As these
representations are increasingly being used in real-world applications, the
inability to \emph{control} their content becomes an increasingly important
problem.
  We formulate the problem of identifying and erasing a linear subspace that
corresponds to a given concept, in order to prevent linear predictors from
recovering the concept. We model this problem as a constrained, linear minimax
game, and show that existing solutions are generally not optimal for this task.
We derive a closed-form solution for certain objectives, and propose a convex
relaxation, R-LACE, that works well for others. When evaluated in the context
of binary gender removal, the method recovers a low-dimensional subspace whose
removal mitigates bias by intrinsic and extrinsic evaluation. We show that the
method -- despite being linear -- is highly expressive, effectively mitigating
bias in deep nonlinear classifiers while maintaining tractability and
interpretability.
