This monograph develops a comprehensive statistical learning framework that is robust to (distributional) perturbations in the data using Distributionally Robust Optimization (DRO) under the Wasserstein metric. Beginning with fundamental properties of the Wasserstein metric and the DRO formulation, we explore duality to arrive at tractable formulations and develop finite-sample, as well as asymptotic, performance guarantees. We consider a series of learning problems, including (i) distributionally robust linear regression; (ii) distributionally robust regression with group structure in the predictors; (iii) distributionally robust multi-output regression and multiclass classification, (iv) optimal decision making that combines distributionally robust regression with nearest-neighbor estimation; (v) distributionally robust semi-supervised learning, and (vi) distributionally robust reinforcement learning. A tractable DRO relaxation for each problem is being derived, establishing a connection between robustness and regularization, and obtaining bounds on the prediction and estimation errors of the solution. Beyond theory, we include numerical experiments and case studies using synthetic and real data. The real data experiments are all associated with various health informatics problems, an application area which provided the initial impetus for this work.

When I joined Toyota Research Institute (TRI) more than five years ago, I believed that an industrial research lab like TRI could make fundamental contributions to robotics that would be hard to make in an academic lab or a startup. And I joined with a commitment that we would share our best tools and results with the world through open-source software. Just before joining TRI, I competed in the DARPA Robotics Challenge to program a humanoid robot for nearly-autonomous operation in a disaster response scenario. This experience gave me a deep appreciation for the value of software engineering and helped me realize the world has never seen truly mature implementations of the best ideas from control theory, machine learning, mathematical optimization, and verification applied to robots at scale. The introduction of scale and real-world testing poses a host of basic research challenges that simply aren't visible in simpler prototypes.

Model predictive control (MPC) is a method to formulate the optimal scheduling problem for grid flexibilities in a mathematical manner. The resulting time-constrained optimization problem can be re-solved in each optimization time step using classical optimization methods such as Second Order Cone Programming (SOCP) or Interior Point Methods (IPOPT). When applying MPC in a rolling horizon scheme, the impact of uncertainty in forecasts on the optimal schedule is reduced. While MPC methods promise accurate results for time-constrained grid optimization they are inherently limited by the calculation time needed for large and complex power system models. Learning the optimal control behaviour using function approximation offers the possibility to determine near-optimal control actions with short calculation time. A Neural Predictive Control (NPC) scheme is proposed to learn optimal control policies for linear and nonlinear power systems through imitation. It is demonstrated that this procedure can find near-optimal solutions, while reducing the calculation time by an order of magnitude. The learned controllers are validated using a benchmark smart grid.

Optimizing the reliability and the robustness of a design is important but often unaffordable due to high sample requirements. Surrogate models based on statistical and machine learning methods are used to increase the sample efficiency. However, for higher dimensional or multi-modal systems, surrogate models may also require a large amount of samples to achieve good results. We propose a sequential sampling strategy for the surrogate based solution of multi-objective reliability based robust design optimization problems. Proposed local Latin hypercube refinement (LoLHR) strategy is model-agnostic and can be combined with any surrogate model because there is no free lunch but possibly a budget one. The proposed method is compared to stationary sampling as well as other proposed strategies from the literature. Gaussian process and support vector regression are both used as surrogate models. Empirical evidence is presented, showing that LoLHR achieves on average better results compared to other surrogate based strategies on the tested examples.

The predict-then-optimize framework is fundamental in practical stochastic decision-making problems: first predict unknown parameters of an optimization model, then solve the problem using the predicted values. A natural loss function in this setting is defined by measuring the decision error induced by the predicted parameters, which was named the Smart Predict-then-Optimize (SPO) loss by Elmachtoub and Grigas [arXiv:1710.08005]. Since the SPO loss is typically nonconvex and possibly discontinuous, Elmachtoub and Grigas [arXiv:1710.08005] introduced a convex surrogate, called the SPO+ loss, that importantly accounts for the underlying structure of the optimization model. In this paper, we greatly expand upon the consistency results for the SPO+ loss provided by Elmachtoub and Grigas [arXiv:1710.08005]. We develop risk bounds and uniform calibration results for the SPO+ loss relative to the SPO loss, which provide a quantitative way to transfer the excess surrogate risk to excess true risk. By combining our risk bounds with generalization bounds, we show that the empirical minimizer of the SPO+ loss achieves low excess true risk with high probability. We first demonstrate these results in the case when the feasible region of the underlying optimization problem is a polyhedron, and then we show that the results can be strengthened substantially when the feasible region is a level set of a strongly convex function. We perform experiments to empirically demonstrate the strength of the SPO+ surrogate, as compared to standard $\ell_1$ and squared $\ell_2$ prediction error losses, on portfolio allocation and cost-sensitive multi-class classification problems.

Knowing the causal structure of a system is of fundamental interest in many areas of science and can aid the design of prediction algorithms that work well under manipulations to the system. The causal structure becomes identifiable from the observational distribution under certain restrictions. To learn the structure from data, score-based methods evaluate different graphs according to the quality of their fits. However, for large nonlinear models, these rely on heuristic optimization approaches with no general guarantees of recovering the true causal structure. In this paper, we consider structure learning of directed trees. We propose a fast and scalable method based on Chu-Liu-Edmonds' algorithm we call causal additive trees (CAT). For the case of Gaussian errors, we prove consistency in an asymptotic regime with a vanishing identifiability gap. We also introduce a method for testing substructure hypotheses with asymptotic family-wise error rate control that is valid post-selection and in unidentified settings. Furthermore, we study the identifiability gap, which quantifies how much better the true causal model fits the observational distribution, and prove that it is lower bounded by local properties of the causal model. Simulation studies demonstrate the favorable performance of CAT compared to competing structure learning methods.

Classes of set functions along with a choice of ground set are a bedrock to determine and develop corresponding variants of greedy algorithms to obtain efficient solutions for combinatorial optimization problems. The class of approximate constrained submodular optimization has seen huge advances at the intersection of good computational efficiency, versatility and approximation guarantees while exact solutions for unconstrained submodular optimization are NP-hard. What is an alternative to situations when submodularity does not hold? Can efficient and globally exact solutions be obtained? We introduce one such new frontier: The class of quasi-concave set functions induced as a dual class to monotone linkage functions. We provide a parallel algorithm with a time complexity over $n$ processors of $\mathcal{O}(n^2g) +\mathcal{O}(\log{\log{n}})$ where $n$ is the cardinality of the ground set and $g$ is the complexity to compute the monotone linkage function that induces a corresponding quasi-concave set function via a duality. The complexity reduces to $\mathcal{O}(gn\log(n))$ on $n^2$ processors and to $\mathcal{O}(gn)$ on $n^3$ processors. Our algorithm provides a globally optimal solution to a maxi-min problem as opposed to submodular optimization which is approximate. We show a potential for widespread applications via an example of diverse feature subset selection with exact global maxi-min guarantees upon showing that a statistical dependency measure called distance correlation can be used to induce a quasi-concave set function.

In generative adversarial imitation learning (GAIL), the agent aims to learn a policy from an expert demonstration so that its performance cannot be discriminated from the expert policy on a certain predefined reward set. In this paper, we study GAIL in both online and offline settings with linear function approximation, where both the transition and reward function are linear in the feature maps. Besides the expert demonstration, in the online setting the agent can interact with the environment, while in the offline setting the agent only accesses an additional dataset collected by a prior. For online GAIL, we propose an optimistic generative adversarial policy optimization algorithm (OGAP) and prove that OGAP achieves $\widetilde{\mathcal{O}}(H^2 d^{3/2}K^{1/2}+KH^{3/2}dN_1^{-1/2})$ regret. Here $N_1$ represents the number of trajectories of the expert demonstration, $d$ is the feature dimension, and $K$ is the number of episodes. For offline GAIL, we propose a pessimistic generative adversarial policy optimization algorithm (PGAP). For an arbitrary additional dataset, we obtain the optimality gap of PGAP, achieving the minimax lower bound in the utilization of the additional dataset. Assuming sufficient coverage on the additional dataset, we show that PGAP achieves $\widetilde{\mathcal{O}}(H^{2}dK^{-1/2} +H^2d^{3/2}N_2^{-1/2}+H^{3/2}dN_1^{-1/2} \ )$ optimality gap. Here $N_2$ represents the number of trajectories of the additional dataset with sufficient coverage.

Counterfactual explanation methods interpret the outputs of a machine learning model in the form of "what-if scenarios" without compromising the fidelity-interpretability trade-off. They explain how to obtain a desired prediction from the model by recommending small changes to the input features, aka recourse. We believe an actionable recourse should be created based on sound counterfactual explanations originating from the distribution of the ground-truth data and linked to the domain knowledge. Moreover, it needs to preserve the coherency between changed/unchanged features while satisfying user/domain-specified constraints. This paper introduces CARE, a modular explanation framework that addresses the model- and user-level desiderata in a consecutive and structured manner. We tackle the existing requirements by proposing novel and efficient solutions that are formulated in a multi-objective optimization framework. The designed framework enables including arbitrary requirements and generating counterfactual explanations and actionable recourse by choice. As a model-agnostic approach, CARE generates multiple, diverse explanations for any black-box model in tabular classification and regression settings. Several experiments on standard data sets and black-box models demonstrate the effectiveness of our modular framework and its superior performance compared to the baselines.

Archetypal analysis is an unsupervised learning method for exploratory data analysis. One major challenge that limits the applicability of archetypal analysis in practice is the inherent computational complexity of the existing algorithms. In this paper, we provide a novel approximation approach to partially address this issue. Utilizing probabilistic ideas from high-dimensional geometry, we introduce two preprocessing techniques to reduce the dimension and representation cardinality of the data, respectively. We prove that, provided the data is approximately embedded in a low-dimensional linear subspace and the convex hull of the corresponding representations is well approximated by a polytope with a few vertices, our method can effectively reduce the scaling of archetypal analysis. Moreover, the solution of the reduced problem is near-optimal in terms of prediction errors. Our approach can be combined with other acceleration techniques to further mitigate the intrinsic complexity of archetypal analysis. We demonstrate the usefulness of our results by applying our method to summarize several moderately large-scale datasets.