When causal quantities cannot be point identified, researchers often pursue partial identification to quantify the range of possible values. However, the peculiarities of applied research conditions can make this analytically intractable. We present a general and automated approach to causal inference in discrete settings. We show causal questions with discrete data reduce to polynomial programming problems, and we present an algorithm to automatically bound causal effects using efficient dual relaxation and spatial branch-and-bound techniques. The user declares an estimand, states assumptions, and provides data (however incomplete or mismeasured). The algorithm then searches over admissible data-generating processes and outputs the most precise possible range consistent with available information -- i.e., sharp bounds -- including a point-identified solution if one exists. Because this search can be computationally intensive, our procedure reports and continually refines non-sharp ranges that are guaranteed to contain the truth at all times, even when the algorithm is not run to completion. Moreover, it offers an additional guarantee we refer to as $\epsilon$-sharpness, characterizing the worst-case looseness of the incomplete bounds. Analytically validated simulations show the algorithm accommodates classic obstacles, including confounding, selection, measurement error, noncompliance, and nonresponse.

This paper considers a large class of problems where we seek to recover a low rank matrix and/or sparse vector from some set of measurements. While methods based on convex relaxations suffer from a (possibly large) estimator bias, and other nonconvex methods require the rank or sparsity to be known a priori, we use nonconvex regularizers to minimize the rank and $l_0$ norm without the estimator bias from the convex relaxation. We present a novel analysis of the alternating proximal gradient descent algorithm applied to such problems, and bound the error between the iterates and the ground truth sparse and low rank matrices. The algorithm and error bound can be applied to sparse optimization, matrix completion, and robust principal component analysis as special cases of our results.

We study the Sparse Plus Low Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix $\mathbf{D}$ into a sparse matrix $\mathbf{Y}$ containing the perturbations plus a low rank matrix $\mathbf{X}$. SLR is a fundamental problem in Operations Research and Machine Learning arising in many applications such as data compression, latent semantic indexing, collaborative filtering and medical imaging. We introduce a novel formulation for SLR that directly models the underlying discreteness of the problem. For this formulation, we develop an alternating minimization heuristic to compute high quality solutions and a novel semidefinite relaxation that provides meaningful bounds for the solutions returned by our heuristic. We further develop a custom branch and bound routine that leverages our heuristic and convex relaxation that solves small instances of SLR to certifiable near-optimality. Our heuristic can scale to $n=10000$ in hours, our relaxation can scale to $n=200$ in hours, and our branch and bound algorithm can scale to $n=25$ in minutes. Our numerical results demonstrate that our approach outperforms existing state-of-the-art approaches in terms of the MSE of the low rank matrix and that of the sparse matrix.

The increasing availability of massive data sets poses a series of challenges for machine learning. Prominent among these is the need to learn models under hardware or human resource constraints. In such resource-constrained settings, a simple yet powerful approach is to operate on small subsets of the data. Coresets are weighted subsets of the data that provide approximation guarantees for the optimization objective. However, existing coreset constructions are highly model-specific and are limited to simple models such as linear regression, logistic regression, and $k$-means. In this work, we propose a generic coreset construction framework that formulates the coreset selection as a cardinality-constrained bilevel optimization problem. In contrast to existing approaches, our framework does not require model-specific adaptations and applies to any twice differentiable model, including neural networks. We show the effectiveness of our framework for a wide range of models in various settings, including training non-convex models online and batch active learning.

A transition to a low-carbon electricity supply is crucial to limit the impacts of climate change. Reducing carbon emissions could help prevent the world from reaching a tipping point, where runaway emissions are likely. Runaway emissions could lead to extremes in weather conditions around the world -- especially in problematic regions unable to cope with these conditions. However, the movement to a low-carbon energy supply can not happen instantaneously due to the existing fossil-fuel infrastructure and the requirement to maintain a reliable energy supply. Therefore, a low-carbon transition is required, however, the decisions various stakeholders should make over the coming decades to reduce these carbon emissions are not obvious. This is due to many long-term uncertainties, such as electricity, fuel and generation costs, human behaviour and the size of electricity demand. A well choreographed low-carbon transition is, therefore, required between all of the heterogenous actors in the system, as opposed to changing the behaviour of a single, centralised actor. The objective of this thesis is to create a novel, open-source agent-based model to better understand the manner in which the whole electricity market reacts to different factors using state-of-the-art machine learning and artificial intelligence methods. In contrast to other works, this thesis looks at both the long-term and short-term impact that different behaviours have on the electricity market by using these state-of-the-art methods.

In today's digital economy, there are numerous situations where many players have to compete for limited resources. For instance, on ride-hailing platforms such as Uber and Lyft, drivers pick an area to go and then compete with other drivers for riding requests at that area; on content platforms such as Youtube and Tiktok, content providers choose a style/theme for their contents and then compete with other providers of the same theme for Internet traffic interested in that theme; on digital markets such as Amazon and Wayfair, retailers choose a particular product category (e.g., pet supplies or home&kitchen, etc.) to focus on and compete with other retailers for sale demands on that category. All these problems share the following similarity: (1) many players make a choice (e.g., a ride-sharing area or a content theme) from multiple options and their payoffs has negative externalities with other players of the same choice due to competition; (2) players have high uncertainty about the payoffs of their choices since the entire system's demand of riding requests or Internet traffic are unknown to an individual player, whereas the system usually has much fined-grained information about these uncertainties. An important operational task common in all these applications is the following: how can the system (the sender) strategically reveal her privileged information to influence the decisions of so many players (the receivers) in order to steer their collective decisions towards a desirable social outcome? This task, also known as information design or persuasion [1, 2, 3, 4], has attracted extensive recent interests.

This paper presents a new optimization approach to causal estimation. Given data that contains covariates and an outcome, which covariates are causes of the outcome, and what is the strength of the causality? In classical machine learning (ML), the goal of optimization is to maximize predictive accuracy. However, some covariates might exhibit a non-causal association to the outcome. Such spurious associations provide predictive power for classical ML, but they prevent us from causally interpreting the result. This paper proposes CoCo, an optimization algorithm that bridges the gap between pure prediction and causal inference. CoCo leverages the recently-proposed idea of environments, datasets of covariates/response where the causal relationships remain invariant but where the distribution of the covariates changes from environment to environment. Given datasets from multiple environments -- and ones that exhibit sufficient heterogeneity -- CoCo maximizes an objective for which the only solution is the causal solution. We describe the theoretical foundations of this approach and demonstrate its effectiveness on simulated and real datasets. Compared to classical ML and existing methods, CoCo provides more accurate estimates of the causal model.

Decision-making problems under uncertainty have broad applications in operations research, machine learning, engineering, and economics. When the data involves uncertainty due to measurement error, insufficient sample size, contamination, and anomalies, or model misspecification, distributionally robust optimization (DRO) is a promising approach to data-driven optimization, by seeking a minimax robust optimal decision that minimizes the expected loss under the most adverse distribution within a given set of relevant distributions, called ambiguity set. It provides a principled framework to produce a solution with more promising out-of-sample performance than the traditional sample average approximation (SAA) method for stochastic programming [86]. We refer to [81] for a recent survey on DRO. At the core of DRO is the choice of the ambiguity set. Ideally, a good ambiguity set should take account of the properties of practical applications while maintaining the computational tractability of resulted DRO formulation; and it should be rich enough to contain all distributions relevant to the decision-making but, at the same time, should not include unnecessary distributions that lead to overly conservative decisions. Various DRO formulations have been proposed in the literature. Among them, the ambiguity set based on Wasserstein distance has recently received much attention [104, 67, 17, 46]. The Wasserstein distance incorporates the geometry of sample space, and thereby is suitable for comparing distributions with non-overlapping supports and hedging against data perturbations [46].

Several methods have been proposed to solve such derivative-free stochastic optimization problems, and we refer the reader to [3, 38] for surveys of these methods. A popular class of these methods estimate the gradients using function values and employ standard gradient-based optimization methods using these estimators. Quasi-Newton methods are recognized as one of the most powerful methods for solving deterministic optimization problems. These methods build quadratic models of the objective information using only gradient information. Recently, researchers have been adapting these methods for stochastic settings when the gradient information is available. The empirical results in [15] indicate that a careful implementation of these methods can be efficient compared with the popular stochastic gradient methods. We adapt these methods to make them suitable for situations where the gradients are estimated using function values. We propose finite-difference derivative-free stochastic quasi-Newton methods for solving (1) by exploiting common random number (CRN) evaluations of f.

The Conditional Preference Network (CP-net) graphically represents user's qualitative and conditional preference statements under the ceteris paribus interpretation. The constrained CP-net is an extension of the CP-net, to a set of constraints. The existing algorithms for solving the constrained CP-net require the expensive dominance testing operation. We propose three approaches to tackle this challenge. In our first solution, we alter the constrained CP-net by eliciting additional relative importance statements between variables, in order to have a total order over the outcomes. We call this new model, the constrained Relative Importance Network (constrained CPR-net). Consequently, We show that the Constrained CPR-net has one single optimal outcome (assuming the constrained CPR-net is consistent) that we can obtain without dominance testing. In our second solution, we extend the Lexicographic Preference Tree (LP-tree) to a set of constraints. Then, we propose a recursive backtrack search algorithm, that we call Search-LP, to find the most preferable outcome. We prove that the first feasible outcome returned by Search-LP (without dominance testing) is also preferable to any other feasible outcome. Finally, in our third solution, we preserve the semantics of the CP-net and propose a divide and conquer algorithm that compares outcomes according to dominance testing.