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This paper explores connections between margin-based loss functions and consistency in binary classification and regression applications. It is shown that a large class of margin-based loss functions for binary classification/regression result in estimating scores equivalent to log-likelihood scores weighted by an even function. A simple characterization for conformable (consistent) loss functions is given, which allows for straightforward comparison of different losses, including exponential loss, logistic loss, and others. The characterization is used to construct a new Huber-type loss function for the logistic model. A simple relation between the margin and standardized logistic regression residuals is derived, demonstrating that all margin-based loss can be viewed as loss functions of squared standardized logistic regression residuals. The relation provides new, straightforward interpretations for exponential and logistic loss, and aids in understanding why exponential loss is sensitive to outliers. In particular, it is shown that minimizing empirical exponential loss is equivalent to minimizing the sum of squared standardized logistic regression residuals. The relation also provides new insight into the AdaBoost algorithm.

Consensus and leader election are fundamental problems in distributed systems. Consensus is the problem in which all processes in a distributed computation must agree on some value. Average consensus is a popular form of consensus, where the agreed upon value is the average of the initial values of all the processes. In a typical solution for consensus, each process learns the value of others' to determine the final decision. However, this is undesirable if processes want to keep their values secret from others. With this motivation, we present a solution to privacy-preserving average consensus, where no process can learn the initial value of any other process. Additionally, we augment our approach to provide outlier resistance, where extreme values are not included in the average calculation. Privacy is fully preserved at every stage, including preventing any process from learning the identities of processes that hold outlier values. To our knowledge, this is the first privacy-preserving average consensus algorithm featuring outlier resistance. In the context of leader election, each process votes for the one that it wants to be the leader. The goal is to ensure that the leader is elected in such a way that each vote remains secret and the sum of votes remain secret during the election. Only the final vote tally is available to all processes. This ensures that processes that vote early are not able to influence the votes of other processes. We augment our approach with shallow ranked voting by allowing processes to not only vote for a single process, but to designate a secondary process to vote towards in the event that their primary vote's candidate does not win the election.

Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-$m$ rate, where $m$ is the number of simulated samples. This can lead to significant computational challenges since a large $m$ is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.

We study deviation of U-statistics when samples have heavy-tailed distribution so the kernel of the U-statistic does not have bounded exponential moments at any positive point. We obtain an exponential upper bound for the tail of the U-statistics which clearly denotes two regions of tail decay, the first is a Gaussian decay and the second behaves like the tail of the kernel. For several common U-statistics, we also show the upper bound has the right rate of decay as well as sharp constants by obtaining rough logarithmic limits which in turn can be used to develop LDP for U-statistics. In spite of usual LDP results in the literature, processes we consider in this work have LDP speed slower than their sample size $n$.

Vaccination is widely acknowledged as one of the most effective tools for preventing disease. However, there has been a rise in parental refusal and delay of childhood vaccination in recent years in the United States. This trend undermines the maintenance of herd immunity and elevates the likelihood of outbreaks of vaccine-preventable diseases. Our aim is to identify demographic or socioeconomic characteristics associated with vaccine refusal, which could help public health professionals and medical providers develop interventions targeted to concerned parents. We examine US county-level vaccine refusal data for patients under five years of age collected on a monthly basis during the period 2012--2015. These data exhibit challenging features: zero inflation, spatial dependence, seasonal variation, spatially-varying dispersion, and a large sample size (approximately 3,000 counties per month). We propose a flexible zero-inflated Conway--Maxwell--Poisson (ZICOMP) regression model that addresses these challenges. Because ZICOMP models have an intractable normalizing function, it is challenging to do Bayesian inference for these models. We propose a new hybrid Monte Carlo algorithm that permits efficient sampling and provides asymptotically exact estimates of model parameters.

Over the past two decades, we have seen an exponentially increased amount of point clouds collected with irregular shapes in various areas. Motivated by the importance of solid modeling for point clouds, we develop a novel and efficient smoothing tool based on multivariate splines over the tetrahedral partitions to extract the underlying signal and build up a 3D solid model from the point cloud. The proposed smoothing method can denoise or deblur the point cloud effectively and provide a multi-resolution reconstruction of the actual signal. In addition, it can handle sparse and irregularly distributed point clouds and recover the underlying trajectory. The proposed smoothing and interpolation method also provides a natural way of numerosity data reduction. Furthermore, we establish the theoretical guarantees of the proposed method. Specifically, we derive the convergence rate and asymptotic normality of the proposed estimator and illustrate that the convergence rate achieves the optimal nonparametric convergence rate. Through extensive simulation studies and a real data example, we demonstrate the superiority of the proposed method over traditional smoothing methods in terms of estimation accuracy and efficiency of data reduction.

Conventional rule learning algorithms aim at finding a set of simple rules, where each rule covers as many examples as possible. In this paper, we argue that the rules found in this way may not be the optimal explanations for each of the examples they cover. Instead, we propose an efficient algorithm that aims at finding the best rule covering each training example in a greedy optimization consisting of one specialization and one generalization loop. These locally optimal rules are collected and then filtered for a final rule set, which is much larger than the sets learned by conventional rule learning algorithms. A new example is classified by selecting the best among the rules that cover this example. In our experiments on small to very large datasets, the approach's average classification accuracy is higher than that of state-of-the-art rule learning algorithms. Moreover, the algorithm is highly efficient and can inherently be processed in parallel without affecting the learned rule set and so the classification accuracy. We thus believe that it closes an important gap for large-scale classification rule induction.

In this paper, we apply the median-of-means principle to derive robust versions of local averaging rules in non-parametric regression. For various estimates, including nearest neighbors and kernel procedures, we obtain non-asymptotic exponential inequalities, with only a second moment assumption on the noise. We then show that these bounds cannot be significantly improved by establishing a corresponding lower bound on tail probabilities.

Discrete data are abundant and often arise as counts or rounded data. These data commonly exhibit complex distributional features such as zero-inflation, over- or under-dispersion, boundedness, and heaping, which render many parametric models inadequate. Yet even for parametric regression models, approximations such as MCMC typically are needed for posterior inference. This paper introduces a Bayesian modeling and algorithmic framework that enables semiparametric regression analysis for discrete data with Monte Carlo (not MCMC) sampling. The proposed approach pairs a nonparametric marginal model with a latent linear regression model to encourage both flexibility and interpretability, and delivers posterior consistency even under model misspecification. For a parametric or large-sample approximation of this model, we identify a class of conjugate priors with (pseudo) closed-form posteriors. All posterior and predictive distributions are available analytically or via Monte Carlo sampling. These tools are broadly useful for linear regression, nonlinear models via basis expansions, and variable selection with discrete data. Simulation studies demonstrate significant advantages in computing, prediction, estimation, and selection relative to existing alternatives. This novel approach is applied to self-reported mental health data that exhibit zero-inflation, overdispersion, boundedness, and heaping.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.