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Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings. In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios.

We consider the fundamental problem of sampling the optimal transport coupling between given source and target distributions. In certain cases, the optimal transport plan takes the form of a one-to-one mapping from the source support to the target support, but learning or even approximating such a map is computationally challenging for large and high-dimensional datasets due to the high cost of linear programming routines and an intrinsic curse of dimensionality. We study instead the Sinkhorn problem, a regularized form of optimal transport whose solutions are couplings between the source and the target distribution. We introduce a novel framework for learning the Sinkhorn coupling between two distributions in the form of a score-based generative model. Conditioned on source data, our procedure iterates Langevin Dynamics to sample target data according to the regularized optimal coupling. Key to this approach is a neural network parametrization of the Sinkhorn problem, and we prove convergence of gradient descent with respect to network parameters in this formulation. We demonstrate its empirical success on a variety of large scale optimal transport tasks.

Generalized linear mixed models are useful in studying hierarchical data with possibly non-Gaussian responses. However, the intractability of likelihood functions poses challenges for estimation. We develop a new method suitable for this problem, called imputation maximization stochastic approximation (IMSA). For each iteration, IMSA first imputes latent variables/random effects, then maximizes over the complete data likelihood, and finally moves the estimate towards the new maximizer while preserving a proportion of the previous value. The limiting point of IMSA satisfies a self-consistency property and can be less biased in finite samples than the maximum likelihood estimator solved by score-equation based stochastic approximation (ScoreSA). Numerically, IMSA can also be advantageous over ScoreSA in achieving more stable convergence and respecting the parameter ranges under various transformations such as nonnegative variance components. This is corroborated through our simulation studies where IMSA consistently outperforms ScoreSA.

We seek an entropy estimator for discrete distributions with fully empirical accuracy bounds. As stated, this goal is infeasible without some prior assumptions on the distribution. We discover that a certain information moment assumption renders the problem feasible. We argue that the moment assumption is natural and, in some sense, {\em minimalistic} -- weaker than finite support or tail decay conditions. Under the moment assumption, we provide the first finite-sample entropy estimates for infinite alphabets, nearly recovering the known minimax rates. Moreover, we demonstrate that our empirical bounds are significantly sharper than the state-of-the-art bounds, for various natural distributions and non-trivial sample regimes. Along the way, we give a dimension-free analogue of the Cover-Thomas result on entropy continuity (with respect to total variation distance) for finite alphabets, which may be of independent interest. Additionally, we resolve all of the open problems posed by J\"urgensen and Matthews, 2010.

In this article, we aim to provide a general and complete understanding of semi-supervised (SS) causal inference for treatment effects. Specifically, we consider two such estimands: (a) the average treatment effect and (b) the quantile treatment effect, as prototype cases, in an SS setting, characterized by two available data sets: (i) a labeled data set of size $n$, providing observations for a response and a set of high dimensional covariates, as well as a binary treatment indicator; and (ii) an unlabeled data set of size $N$, much larger than $n$, but without the response observed. Using these two data sets, we develop a family of SS estimators which are ensured to be: (1) more robust and (2) more efficient than their supervised counterparts based on the labeled data set only. Beyond the 'standard' double robustness results (in terms of consistency) that can be achieved by supervised methods as well, we further establish root-n consistency and asymptotic normality of our SS estimators whenever the propensity score in the model is correctly specified, without requiring specific forms of the nuisance functions involved. Such an improvement of robustness arises from the use of the massive unlabeled data, so it is generally not attainable in a purely supervised setting. In addition, our estimators are shown to be semi-parametrically efficient as long as all the nuisance functions are correctly specified. Moreover, as an illustration of the nuisance estimators, we consider inverse-probability-weighting type kernel smoothing estimators involving unknown covariate transformation mechanisms, and establish in high dimensional scenarios novel results on their uniform convergence rates, which should be of independent interest. Numerical results on both simulated and real data validate the advantage of our methods over their supervised counterparts with respect to both robustness and efficiency.

We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is exactly $2$. Furthermore, we prove that if the ratio is less than $2$, say $\alpha$, then a lexicographically maximal solution achieves $(\alpha/2)$-approximation, and this bound is tight.

We study the off-policy evaluation (OPE) problem in an infinite-horizon Markov decision process with continuous states and actions. We recast the $Q$-function estimation into a special form of the nonparametric instrumental variables (NPIV) estimation problem. We first show that under one mild condition the NPIV formulation of $Q$-function estimation is well-posed in the sense of $L^2$-measure of ill-posedness with respect to the data generating distribution, bypassing a strong assumption on the discount factor $\gamma$ imposed in the recent literature for obtaining the $L^2$ convergence rates of various $Q$-function estimators. Thanks to this new well-posed property, we derive the first minimax lower bounds for the convergence rates of nonparametric estimation of $Q$-function and its derivatives in both sup-norm and $L^2$-norm, which are shown to be the same as those for the classical nonparametric regression (Stone, 1982). We then propose a sieve two-stage least squares estimator and establish its rate-optimality in both norms under some mild conditions. Our general results on the well-posedness and the minimax lower bounds are of independent interest to study not only other nonparametric estimators for $Q$-function but also efficient estimation on the value of any target policy in off-policy settings.

We consider the problem of finding tuned regularized parameter estimators for linear models. We start by showing that three known optimal linear estimators belong to a wider class of estimators that can be formulated as a solution to a weighted and constrained minimization problem. The optimal weights, however, are typically unknown in many applications. This begs the question, how should we choose the weights using only the data? We propose using the covariance fitting SPICE-methodology to obtain data-adaptive weights and show that the resulting class of estimators yields tuned versions of known regularized estimators - such as ridge regression, LASSO, and regularized least absolute deviation. These theoretical results unify several important estimators under a common umbrella. The resulting tuned estimators are also shown to be practically relevant by means of a number of numerical examples.

Continuous determinantal point processes (DPPs) are a class of repulsive point processes on $\mathbb{R}^d$ with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behaviour when the observation window grows towards $\mathbb{R}^d$. This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on $\mathbb{R}^d$ and compare favourably to common alternative estimation methods for continuous DPPs.

Intersection over Union (IoU) is the most popular evaluation metric used in the object detection benchmarks. However, there is a gap between optimizing the commonly used distance losses for regressing the parameters of a bounding box and maximizing this metric value. The optimal objective for a metric is the metric itself. In the case of axis-aligned 2D bounding boxes, it can be shown that $IoU$ can be directly used as a regression loss. However, $IoU$ has a plateau making it infeasible to optimize in the case of non-overlapping bounding boxes. In this paper, we address the weaknesses of $IoU$ by introducing a generalized version as both a new loss and a new metric. By incorporating this generalized $IoU$ ($GIoU$) as a loss into the state-of-the art object detection frameworks, we show a consistent improvement on their performance using both the standard, $IoU$ based, and new, $GIoU$ based, performance measures on popular object detection benchmarks such as PASCAL VOC and MS COCO.