  Robust training methods against perturbations to the input data have received
great attention in the machine learning literature. A standard approach in this
direction is adversarial training which learns a model using
adversarially-perturbed training samples. However, adversarial training
performs suboptimally against perturbations structured across samples such as
universal and group-sparse shifts that are commonly present in biological data
such as gene expression levels of different tissues. In this work, we seek to
close this optimality gap and introduce Group-Structured Adversarial Training
(GSAT) which learns a model robust to perturbations structured across samples.
We formulate GSAT as a non-convex concave minimax optimization problem which
minimizes a group-structured optimal transport cost. Specifically, we focus on
the applications of GSAT for group-sparse and rank-constrained perturbations
modeled using group and nuclear norm penalties. In order to solve GSAT's
non-smooth optimization problem in those cases, we propose a new minimax
optimization algorithm called GDADMM by combining Gradient Descent Ascent (GDA)
and Alternating Direction Method of Multipliers (ADMM). We present several
applications of the GSAT framework to gain robustness against structured
perturbations for image recognition and computational biology datasets.
