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Distributionally robust optimization (DRO) is a powerful tool for decision making under uncertainty. It is particularly appealing because of its ability to leverage existing data. However, many practical problems call for decision- making with some auxiliary information, and DRO in the context of conditional distributions is not straightforward. We propose a conditional kernel distributionally robust optimiza- tion (CKDRO) method that enables robust decision making under conditional distributions through kernel DRO and the conditional mean operator in the reproducing kernel Hilbert space (RKHS). In particular, we consider problems where there is a correlation between the unknown variable y and an auxiliary observable variable x. Given past data of the two variables and a queried auxiliary variable, CKDRO represents the conditional distribution P(y|x) as the conditional mean operator in the RKHS space and quantifies the ambiguity set in the RKHS as well, which depends on the size of the dataset as well as the query point. To justify the use of RKHS, we demonstrate that the ambiguity set defined in RKHS can be viewed as a ball under a metric that is similar to the Wasserstein metric. The DRO is then dualized and solved via a finite dimensional convex program. The proposed CKDRO approach is applied to a generation scheduling problem and shows that the result of CKDRO is superior to common benchmarks in terms of quality and robustness. 

Federated learning is a framework for distributed optimization that places emphasis on communication efficiency. In particular, it follows a client-server broadcast model and is appealing because of its ability to accommodate heterogene- ity in client compute and storage resources, non-i.i.d. data assumptions, and data privacy. Our contribution is to offer a new federated learning algorithm, FedADMM, for solving non-convex composite optimization problems with non-smooth regularizers. We prove the convergence of FedADMM for the case when not all clients are able to participate in a given communication round under a very general sampling model. 

This work considers the problem of learning the Markov parameters of a linear system from ob- served data. Recent non-asymptotic system identification results have characterized the sample complexity of this problem in the single and multi-rollout setting. In both instances, the number of samples required in order to obtain acceptable estimates can produce optimization problems with an intractably large number of decision variables for a second-order algorithm. We show that a randomized and distributed Newton algorithm based on Hessian-sketching can produce ε-optimal solutions and converges geometrically. Moreover, the algorithm is trivially parallelizable. Our re- sults hold for a variety of sketching matrices and we illustrate the theory with numerical examples. 

Learning a dynamical system from input/output data is a fundamental task in the control design pipeline. In the partially observed setting there are two components to identification: parameter estimation to learn the Markov parameters, and system realization to obtain a state space model. In both sub-problems it is implicitly assumed that standard numerical algorithms such as the singular value decomposition (SVD) can be easily and reliably computed. When trying to fit a high-dimensional model to data, even computing an SVD may be intractable. In this work we show that an approximate matrix factorization obtained using randomized methods can replace the standard SVD in the realization algorithm while maintaining the finite-sample performance and robustness guarantees of classical methods.