  In this article, we propose a new variational approach to learn private
and/or fair representations. This approach is based on the Lagrangians of a new
formulation of the privacy and fairness optimization problems that we propose.
In this formulation, we aim to generate representations of the data that keep a
prescribed level of the relevant information that is not shared by the private
or sensitive data, while minimizing the remaining information they keep. The
proposed approach (i) exhibits the similarities of the privacy and fairness
problems, (ii) allows us to control the trade-off between utility and privacy
or fairness through the Lagrange multiplier parameter, and (iii) can be
comfortably incorporated to common representation learning algorithms such as
the VAE, the $\beta$-VAE, the VIB, or the nonlinear IB.
