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We consider the problem of causal structure learning in the setting of heterogeneous populations, i.e., populations in which a single causal structure does not adequately represent all population members, as is common in biological and social sciences. To this end, we introduce a distance covariance-based kernel designed specifically to measure the similarity between the underlying nonlinear causal structures of different samples. Indeed, we prove that the corresponding feature map is a statistically consistent estimator of nonlinear independence structure, rendering the kernel itself a statistical test for the hypothesis that sets of samples come from different generating causal structures. Even stronger, we prove that the kernel space is isometric to the space of causal ancestral graphs, so that distance between samples in the kernel space is guaranteed to correspond to distance between their generating causal structures. This kernel thus enables us to perform clustering to identify the homogeneous subpopulations, for which we can then learn causal structures using existing methods. Though we focus on the theoretical aspects of the kernel, we also evaluate its performance on synthetic data and demonstrate its use on a real gene expression data set.

Parameter estimation in the empirical fields is usually undertaken using parametric models, and such models are convenient because they readily facilitate statistical inference. Unfortunately, they are unlikely to have a sufficiently flexible functional form to be able to adequately model real-world phenomena, and their usage may therefore result in biased estimates and invalid inference. Unfortunately, whilst non-parametric machine learning models may provide the needed flexibility to adapt to the complexity of real-world phenomena, they do not readily facilitate statistical inference, and may still exhibit residual bias. We explore the potential for semiparametric theory (in particular, the Influence Function) to be used to improve neural networks and machine learning algorithms in terms of (a) improving initial estimates without needing more data (b) increasing the robustness of our models, and (c) yielding confidence intervals for statistical inference. We propose a new neural network method MultiNet, which seeks the flexibility and diversity of an ensemble using a single architecture. Results on causal inference tasks indicate that MultiNet yields better performance than other approaches, and that all considered methods are amenable to improvement from semiparametric techniques under certain conditions. In other words, with these techniques we show that we can improve existing neural networks for `free', without needing more data, and without needing to retrain them. Finally, we provide the expression for deriving influence functions for estimands from a general graph, and the code to do so automatically.

Ensemble methods based on subsampling, such as random forests, are popular in applications due to their high predictive accuracy. Existing literature views a random forest prediction as an infinite-order incomplete U-statistic to quantify its uncertainty. However, these methods focus on a small subsampling size of each tree, which is theoretically valid but practically limited. This paper develops an unbiased variance estimator based on incomplete U-statistics, which allows the tree size to be comparable with the overall sample size, making statistical inference possible in a broader range of real applications. Simulation results demonstrate that our estimators enjoy lower bias and more accurate confidence interval coverage without additional computational costs. We also propose a local smoothing procedure to reduce the variation of our estimator, which shows improved numerical performance when the number of trees is relatively small. Further, we investigate the ratio consistency of our proposed variance estimator under specific scenarios. In particular, we develop a new "double U-statistic" formulation to analyze the Hoeffding decomposition of the estimator's variance.

Consider the problem of nonparametric estimation of an unknown $\beta$-H\"older smooth density $p_{XY}$ at a given point, where $X$ and $Y$ are both $d$ dimensional. An infinite sequence of i.i.d.\ samples $(X_i,Y_i)$ are generated according to this distribution, and two terminals observe $(X_i)$ and $(Y_i)$, respectively. They are allowed to exchange $k$ bits either in oneway or interactively in order for Bob to estimate the unknown density. We show that the minimax mean square risk is order $\left(\frac{k}{\log k} \right)^{-\frac{2\beta}{d+2\beta}}$ for one-way protocols and $k^{-\frac{2\beta}{d+2\beta}}$ for interactive protocols. The logarithmic improvement is nonexistent in the parametric counterparts, and therefore can be regarded as a consequence of nonparametric nature of the problem. Moreover, a few rounds of interactions achieve the interactive minimax rate: the number of rounds can grow as slowly as the super-logarithm (i.e., inverse tetration) of $k$. The proof of the upper bound is based on a novel multi-round scheme for estimating the joint distribution of a pair of biased Bernoulli variables.

This work derives methods for performing nonparametric, nonasymptotic statistical inference for population parameters under the constraint of local differential privacy (LDP). Given observations $(X_1, \dots, X_n)$ with mean $\mu^\star$ that are privatized into $(Z_1, \dots, Z_n)$, we introduce confidence intervals (CI) and time-uniform confidence sequences (CS) for $\mu^\star \in \mathbb R$ when only given access to the privatized data. We introduce a nonparametric and sequentially interactive generalization of Warner's famous "randomized response" mechanism, satisfying LDP for arbitrary bounded random variables, and then provide CIs and CSs for their means given access to the resulting privatized observations. We extend these CSs to capture time-varying (non-stationary) means, and conclude by illustrating how these methods can be used to conduct private online A/B tests.

Center-based clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is undoubtedly the k-means problem, which, given a set $P$ of points from a metric space and a parameter $k<|P|$, requires to determine a subset $S$ of $k$ centers minimizing the sum of all squared distances of points in $P$ from their closest center. A more general formulation, known as k-means with $z$ outliers, introduced to deal with noisy datasets, features a further parameter $z$ and allows up to $z$ points of $P$ (outliers) to be disregarded when computing the aforementioned sum. We present a distributed coreset-based 3-round approximation algorithm for k-means with $z$ outliers for general metric spaces, using MapReduce as a computational model. Our distributed algorithm requires sublinear local memory per reducer, and yields a solution whose approximation ratio is an additive term $O(\gamma)$ away from the one achievable by the best known sequential (possibly bicriteria) algorithm, where $\gamma$ can be made arbitrarily small. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by the doubling dimension $D$ of the metric space. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for general metrics.

Generalizing knowledge beyond source domains is a crucial prerequisite for many biomedical applications such as drug design and molecular property prediction. To meet this challenge, researchers have used optimal transport (OT) to perform representation alignment between the source and target domains. Yet existing OT algorithms are mainly designed for classification tasks. Accordingly, we consider regression tasks in the unsupervised and semi-supervised settings in this paper. To exploit continuous labels, we propose novel metrics to measure domain distances and introduce a posterior variance regularizer on the transport plan. Further, while computationally appealing, OT suffers from ambiguous decision boundaries and biased local data distributions brought by the mini-batch training. To address those issues, we propose to couple OT with metric learning to yield more robust boundaries and reduce bias. Specifically, we present a dynamic hierarchical triplet loss to describe the global data distribution, where the cluster centroids are progressively adjusted among consecutive iterations. We evaluate our method on both unsupervised and semi-supervised learning tasks in biochemistry. Experiments show the proposed method significantly outperforms state-of-the-art baselines across various benchmark datasets of small molecules and material crystals.

We consider universal approximations of symmetric and anti-symmetric functions, which are important for applications in quantum physics, as well as other scientific and engineering computations. We give constructive approximations with explicit bounds on the number of parameters with respect to the dimension and the target accuracy $\epsilon$. While the approximation still suffers from the curse of dimensionality, to the best of our knowledge, these are the first results in the literature with explicit error bounds for functions with symmetry or anti-symmetry constraints.

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.

Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalising to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop an Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by product. The results and their inferential implications are showcased on synthetic and real data.