  Data used to train machine learning models can be adversarial--maliciously
constructed by adversaries to fool the model. Challenge also arises by privacy,
confidentiality, or due to legal constraints when data are geographically
gathered and stored across multiple learners, some of which may hold even an
"anonymized" or unreliable dataset. In this context, the distributionally
robust optimization framework is considered for training a parametric model,
both in centralized and federated learning settings. The objective is to endow
the trained model with robustness against adversarially manipulated input data,
or, distributional uncertainties, such as mismatches between training and
testing data distributions, or among datasets stored at different workers. To
this aim, the data distribution is assumed unknown, and lies within a
Wasserstein ball centered around the empirical data distribution. This robust
learning task entails an infinite-dimensional optimization problem, which is
challenging. Leveraging a strong duality result, a surrogate is obtained, for
which three stochastic primal-dual algorithms are developed: i) stochastic
proximal gradient descent with an $\epsilon$-accurate oracle, which invokes an
oracle to solve the convex sub-problems; ii) stochastic proximal gradient
descent-ascent, which approximates the solution of the convex sub-problems via
a single gradient ascent step; and, iii) a distributionally robust federated
learning algorithm, which solves the sub-problems locally at different workers
where data are stored. Compared to the empirical risk minimization and
federated learning methods, the proposed algorithms offer robustness with
little computation overhead. Numerical tests using image datasets showcase the
merits of the proposed algorithms under several existing adversarial attacks
and distributional uncertainties.
