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Model Predictive Control (MPC) is a well-established approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is generally infeasible, MPC determines a suboptimal feedback control by repeatedly solving finite time optimal control problems. Although MPC has been successfully used in many applications, applying MPC to large-scale systems -- arising, e.g., through discretization of partial differential equations -- requires the solution of high-dimensional optimal control problems and thus poses immense computational effort. We consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis (RB) method as a low-dimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as feedback control for the original large-scale system. We analyze the proposed RB-MPC approach by first developing a posteriori error bounds for the errors in the optimal control and associated cost functional. These bounds can be evaluated efficiently in an offline-online computational procedure and allow us to guarantee asymptotic stability of the closed-loop system using the RB-MPC approach in several practical scenarios. We also propose an adaptive strategy to choose the prediction horizon of the finite time optimal control problem. Numerical results are presented to illustrate the theoretical properties of our approach.

This paper is concerned with low-rank matrix optimization, which has found a wide range of applications in machine learning. This problem in the special case of matrix sensing has been studied extensively through the notion of Restricted Isometry Property (RIP), leading to a wealth of results on the geometric landscape of the problem and the convergence rate of common algorithms. However, the existing results can handle the problem in the case with a general objective function subject to noisy data only when the RIP constant is close to 0. In this paper, we develop a new mathematical framework to solve the above-mentioned problem with a far less restrictive RIP constant. We prove that as long as the RIP constant of the noiseless objective is less than $1/3$, any spurious local solution of the noisy optimization problem must be close to the ground truth solution. By working through the strict saddle property, we also show that an approximate solution can be found in polynomial time. We characterize the geometry of the spurious local minima of the problem in a local region around the ground truth in the case when the RIP constant is greater than $1/3$. Compared to the existing results in the literature, this paper offers the strongest RIP bound and provides a complete theoretical analysis on the global and local optimization landscapes of general low-rank optimization problems under random corruptions from any finite-variance family.

In the literature on Kleene algebra, a number of variants have been proposed which impose additional structure specified by a theory, such as Kleene algebra with tests (KAT) and the recent Kleene algebra with observations (KAO), or make specific assumptions about certain constants, as for instance in NetKAT. Many of these variants fit within the unifying perspective offered by Kleene algebra with hypotheses, which comes with a canonical language model constructed from a given set of hypotheses. For the case of KAT, this model corresponds to the familiar interpretation of expressions as languages of guarded strings. A relevant question therefore is whether Kleene algebra together with a given set of hypotheses is complete with respect to its canonical language model. In this paper, we revisit, combine and extend existing results on this question to obtain tools for proving completeness in a modular way. We showcase these tools by giving new and modular proofs of completeness for KAT, KAO and NetKAT, and we prove completeness for new variants of KAT: KAT extended with a constant for the full relation, KAT extended with a converse operation, and a version of KAT where the collection of tests only forms a distributive lattice.

There has been a recent surge in statistical methods for handling the lack of adequate positivity when using inverse probability weighting. Alongside these nascent developments, a number of questions have been posed about the goals and intent of these methods: to infer causality, what are they really estimating and what are their target populations? Because causal inference is inherently a missing data problem, the assignment mechanism -- how participants are represented in their respective treatment groups and how they receive their treatments -- plays an important role in assessing causality. In this paper, we contribute to the discussion by highlighting specific characteristics of the equipoise estimators, i.e., overlap weights (OW) matching weights (MW) and entropy weights (EW) methods, which help answer these questions and contrast them with the behavior of the inverse probability weights (IPW) method. We discuss three distinct potential motives for weighting under the lack of adequate positivity when estimating causal effects: (1) What separates OW, MW, and EW from IPW trimming or truncation? (2) What fundamentally distinguishes the estimand of the IPW, i.e., average treatment effect (ATE) from the OW, MW, and EW estimands (resp. average treatment effect on the overlap (ATO), the matching (ATM), and entropy (ATEN))? (3) When should we expect similar results for these estimands, even if the treatment effect is heterogeneous? Our findings are illustrated through a number of Monte-Carlo simulation studies and a data example on healthcare expenditure.

Understanding the impact of the most effective policies or treatments on a response variable of interest is desirable in many empirical works in economics, statistics and other disciplines. Due to the widespread winner's curse phenomenon, conventional statistical inference assuming that the top policies are chosen independent of the random sample may lead to overly optimistic evaluations of the best policies. In recent years, given the increased availability of large datasets, such an issue can be further complicated when researchers include many covariates to estimate the policy or treatment effects in an attempt to control for potential confounders. In this manuscript, to simultaneously address the above-mentioned issues, we propose a resampling-based procedure that not only lifts the winner's curse in evaluating the best policies observed in a random sample, but also is robust to the presence of many covariates. The proposed inference procedure yields accurate point estimates and valid frequentist confidence intervals that achieve the exact nominal level as the sample size goes to infinity for multiple best policy effect sizes. We illustrate the finite-sample performance of our approach through Monte Carlo experiments and two empirical studies, evaluating the most effective policies in charitable giving and the most beneficial group of workers in the National Supported Work program.

Edge inference is becoming ever prevalent through its applications from retail to wearable technology. Clusters of networked resource-constrained edge devices are becoming common, yet there is no production-ready orchestration system for deploying deep learning models over such edge networks which adopts the robustness and scalability of the cloud. We present SEIFER, a framework utilizing a standalone Kubernetes cluster to partition a given DNN and place these partitions in a distributed manner across an edge network, with the goal of maximizing inference throughput. The system is node fault-tolerant and automatically updates deployments based on updates to the model's version. We provide a preliminary evaluation of a partitioning and placement algorithm that works within this framework, and show that we can improve the inference pipeline throughput by 200% by utilizing sufficient numbers of resource-constrained nodes. We have implemented SEIFER in open-source software that is publicly available to the research community.

More than 100,000 children in the foster care system are currently waiting for an adoptive placement in the United States, where adoptions from foster care occur through a semi-decentralized search and matching process with the help of local agencies. Traditionally, most agencies have employed a family-driven search process, where prospective families respond to announcements made by the caseworker responsible for a child. However, recently some agencies switched to a caseworker-driven search process, where the caseworker conducts a targeted search of suitable families for the child. We introduce a novel search-and-matching model to capture essential aspects of adoption and compare these two search processes through a game-theoretical analysis. We show that the search equilibria induced by (novel) threshold strategies form a lattice structure under either approach. Our main theoretical result establishes that the equilibrium outcomes in family-driven search can never Pareto dominate the outcomes in caseworker-driven search, but there are instances where each caseworker-driven search outcome Pareto dominates all family-driven search outcomes. We also find that when families are sufficiently impatient, caseworker driven search is better for all children. We illustrate numerically that for a wide range of parameters, most agents are better off under caseworker-driven search.

In this paper, we address the dichotomy between heterogeneous models and simultaneous training in Federated Learning (FL) via a clustering framework. We define a new clustering model for FL based on the (optimal) local models of the users: two users belong to the same cluster if their local models are close; otherwise they belong to different clusters. A standard algorithm for clustered FL is proposed in \cite{ghosh_efficient_2021}, called \texttt{IFCA}, which requires \emph{suitable} initialization and the knowledge of hyper-parameters like the number of clusters (which is often quite difficult to obtain in practical applications) to converge. We propose an improved algorithm, \emph{Successive Refine Federated Clustering Algorithm} (\texttt{SR-FCA}), which removes such restrictive assumptions. \texttt{SR-FCA} treats each user as a singleton cluster as an initialization, and then successively refine the cluster estimation via exploiting similar users belonging to the same cluster. In any intermediate step, \texttt{SR-FCA} uses a robust federated learning algorithm within each cluster to exploit simultaneous training and to correct clustering errors. Furthermore, \texttt{SR-FCA} does not require any \emph{good} initialization (warm start), both in theory and practice. We show that with proper choice of learning rate, \texttt{SR-FCA} incurs arbitrarily small clustering error. Additionally, we validate the performance of our algorithm on standard FL datasets in non-convex problems like neural nets, and we show the benefits of \texttt{SR-FCA} over baselines.

In 1954, Alston S. Householder published Principles of Numerical Analysis, one of the first modern treatments on matrix decomposition that favored a (block) LU decomposition-the factorization of a matrix into the product of lower and upper triangular matrices. And now, matrix decomposition has become a core technology in machine learning, largely due to the development of the back propagation algorithm in fitting a neural network. The sole aim of this survey is to give a self-contained introduction to concepts and mathematical tools in numerical linear algebra and matrix analysis in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of the Euclidean space, Hermitian space, Hilbert space, and things in the complex domain. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.