Title: Stablizing Adversarial Invariance Induction by Discriminator Matching. Abstract: Incorporating the desired invariance into representation learning is a key challenge in many situations, e.g., for domain generalization and privacy/fairness constraints. An adversarial invariance induction (AII) shows its power on this purpose, which maximizes the proxy of the conditional entropy between representations and attributes by adversarial training between an attribute discriminator and feature extractor. However, the practical behavior of AII is still unclear as the previous analysis assumes the optimality of the attribute classifier, which is rarely held in practice. This paper first analyzes the practical behavior of AII both theoretically and empirically, indicating that AII has theoretical difficulty as it maximizes variational {\em upper} bound of the actual conditional entropy, and AII catastrophically fails to induce invariance even in simple cases as suggested by the above theoretical findings. We then argue that a simple modification to AII can significantly stabilize the adversarial induction framework and achieve better invariant representations. Our modification is based on the property of conditional entropy; it is maximized if and only if the divergence between all pairs of marginal distributions over $z$ between different attributes is minimized. The proposed method, {\em invariance induction by discriminator matching}, modify AII objective to explicitly consider the divergence minimization requirements by defining a proxy of the divergence by using the attribute discriminator. Empirical validations on both the toy dataset and four real-world datasets (related to applications of user anonymization and domain generalization) reveal that the proposed method provides superior performance when inducing invariance for nuisance factors. 