Regularization schemes for regression have been widely studied in learning theory and inverse problems. In this paper, we study distribution regression (DR) which involves two stages of sampling, and aims at regressing from probability measures to real-valued responses over a reproducing kernel Hilbert space (RKHS). Recently, theoretical analysis on DR has been carried out via kernel ridge regression and several learning behaviors have been observed. However, the topic has not been explored and understood beyond the least square based DR. By introducing a robust loss function $l_{\sigma}$ for two-stage sampling problems, we present a novel robust distribution regression (RDR) scheme. With a windowing function $V$ and a scaling parameter $\sigma$ which can be appropriately chosen, $l_{\sigma}$ can include a wide range of popular used loss functions that enrich the theme of DR. Moreover, the loss $l_{\sigma}$ is not necessarily convex, hence largely improving the former regression class (least square) in the literature of DR. The learning rates under different regularity ranges of the regression function $f_{\rho}$ are comprehensively studied and derived via integral operator techniques. The scaling parameter $\sigma$ is shown to be crucial in providing robustness and satisfactory learning rates of RDR.

Why can the Expectation-Maximization (EM) algorithm for mixture models converge? Why can different initial parameters cause various convergence difficulties? The Q-L synchronization theory explains that the observed data log-likelihood L and the complete data log-likelihood Q are positively correlated; we can achieve maximum L by maximizing Q. According to this theory, the Deterministic Annealing EM (DAEM) algorithm's authors make great efforts to eliminate locally maximal Q for avoiding L's local convergence. However, this paper proves that in some cases, Q may and should decrease for L to increase; slow or local convergence exists only because of small samples and unfair competition. This paper uses marriage competition to explain different convergence difficulties and proposes the Fair Competition Principle (FCP) with an initialization map for improving initializations. It uses the rate-verisimilitude function, extended from the rate-distortion function, to explain the convergence of the EM and improved EM algorithms. This convergence proof adopts variational and iterative methods that Shannon et al. used for analyzing rate-distortion functions. The initialization map can vastly save both algorithms' running times for binary Gaussian mixtures. The FCP and the initialization map are useful for complicated mixtures but not sufficient; we need further studies for specific methods.

In practical data analysis under noisy environment, it is common to first use robust methods to identify outliers, and then to conduct further analysis after removing the outliers. In this paper, we consider statistical inference of the model estimated after outliers are removed, which can be interpreted as a selective inference (SI) problem. To use conditional SI framework, it is necessary to characterize the events of how the robust method identifies outliers. Unfortunately, the existing methods cannot be directly used here because they are applicable to the case where the selection events can be represented by linear/quadratic constraints. In this paper, we propose a conditional SI method for popular robust regressions by using homotopy method. We show that the proposed conditional SI method is applicable to a wide class of robust regression and outlier detection methods and has good empirical performance on both synthetic data and real data experiments.

We introduce a density-based clustering method called skeleton clustering that can detect clusters in multivariate and even high-dimensional data with irregular shapes. To bypass the curse of dimensionality, we propose surrogate density measures that are less dependent on the dimension but have intuitive geometric interpretations. The clustering framework constructs a concise representation of the given data as an intermediate step and can be thought of as a combination of prototype methods, density-based clustering, and hierarchical clustering. We show by theoretical analysis and empirical studies that the skeleton clustering leads to reliable clusters in multivariate and high-dimensional scenarios.

The emergence of Machine Learning (ML) methods as an effective and easy to use tool for modeling and prediction has opened up new horizons for users in all aspects of business, finance, health, and research [18, 7]. In stark contrast to more traditional statistical methods that require relevant expertise, ML methods constitute largely black-boxes and are based on minimal assumptions. This resulted in adoption of ML from an audience of users with possible expertise in the domain of application but little or no expertise in statistics or computer science. Despite these features, effective use of ML does require good knowledge of the method used. The choice of the method (algorithm), the tuning of its hyperparameters and the architecture requires experience and good understanding of those methods and the data. Naturally, trial and error might provide possible solutions; however it can also result in sub-optimal use of ML and wasted computational resources. The goal of this research is to find optimal values of hyperparameters for a given dataset and a modelling objective given that the relationship between the hyperparameters, model quality (model score) and training cost is not known in advance.

With increasingly more data and computation involved in their training, machine learning models constitute valuable intellectual property. This has spurred interest in model stealing, which is made more practical by advances in learning with partial, little, or no supervision. Existing defenses focus on inserting unique watermarks in a model's decision surface, but this is insufficient: the watermarks are not sampled from the training distribution and thus are not always preserved during model stealing. In this paper, we make the key observation that knowledge contained in the stolen model's training set is what is common to all stolen copies. The adversary's goal, irrespective of the attack employed, is always to extract this knowledge or its by-products. This gives the original model's owner a strong advantage over the adversary: model owners have access to the original training data. We thus introduce $dataset$ $inference$, the process of identifying whether a suspected model copy has private knowledge from the original model's dataset, as a defense against model stealing. We develop an approach for dataset inference that combines statistical testing with the ability to estimate the distance of multiple data points to the decision boundary. Our experiments on CIFAR10, SVHN, CIFAR100 and ImageNet show that model owners can claim with confidence greater than 99% that their model (or dataset as a matter of fact) was stolen, despite only exposing 50 of the stolen model's training points. Dataset inference defends against state-of-the-art attacks even when the adversary is adaptive. Unlike prior work, it does not require retraining or overfitting the defended model.

Personalized optimal decision making, finding the optimal decision rule (ODR) based on individual characteristics, has attracted increasing attention recently in many fields, such as education, economics, and medicine. Current ODR methods usually require the primary outcome of interest in samples for assessing treatment effects, namely the experimental sample. However, in many studies, treatments may have a long-term effect, and as such the primary outcome of interest cannot be observed in the experimental sample due to the limited duration of experiments, which makes the estimation of ODR impossible. This paper is inspired to address this challenge by making use of an auxiliary sample to facilitate the estimation of ODR in the experimental sample. We propose an auGmented inverse propensity weighted Experimental and Auxiliary sample-based decision Rule (GEAR) by maximizing the augmented inverse propensity weighted value estimator over a class of decision rules using the experimental sample, with the primary outcome being imputed based on the auxiliary sample. The asymptotic properties of the proposed GEAR estimators and their associated value estimators are established. Simulation studies are conducted to demonstrate its empirical validity with a real AIDS application.

We consider the optimal decision-making problem in a primary sample of interest with multiple auxiliary sources available. The outcome of interest is limited in the sense that it is only observed in the primary sample. In reality, such multiple data sources may belong to different populations and thus cannot be combined directly. This paper proposes a novel calibrated optimal decision rule (CODR) to address the limited outcome, by leveraging the shared pattern in multiple data sources. Under a mild and testable assumption that the conditional means of intermediate outcomes in different samples are equal given baseline covariates and the treatment information, we can show that the calibrated mean outcome of interest under the CODR is unbiased and more efficient than using the primary sample solely. Extensive experiments on simulated datasets demonstrate empirical validity and improvement of the proposed CODR, followed by a real application on the MIMIC-III as the primary sample with auxiliary data from eICU.

Advances in object-centric generative models (OCGMs) have culminated in the development of a broad range of methods for unsupervised object segmentation and interpretable object-centric scene generation. These methods, however, are limited to simulated and real-world datasets with limited visual complexity. Moreover, object representations are often inferred using RNNs which do not scale well to large images or iterative refinement which avoids imposing an unnatural ordering on objects in an image but requires the a priori initialisation of a fixed number of object representations. In contrast to established paradigms, this work proposes an embedding-based approach in which embeddings of pixels are clustered in a differentiable fashion using a stochastic, non-parametric stick-breaking process. Similar to iterative refinement, this clustering procedure also leads to randomly ordered object representations, but without the need of initialising a fixed number of clusters a priori. This is used to develop a new model, GENESIS-V2, which can infer a variable number of object representations without using RNNs or iterative refinement. We show that GENESIS-V2 outperforms previous methods for unsupervised image segmentation and object-centric scene generation on established synthetic datasets as well as more complex real-world datasets.

Reliability of machine learning (ML) systems is crucial in safety-critical applications such as healthcare, and uncertainty estimation is a widely researched method to highlight the confidence of ML systems in deployment. Sequential and parallel ensemble techniques have shown improved performance of ML systems in multi-modal settings by leveraging the feature sets together. We propose an uncertainty-aware boosting technique for multi-modal ensembling in order to focus on the data points with higher associated uncertainty estimates, rather than the ones with higher loss values. We evaluate this method on healthcare tasks related to Dementia and Parkinson's disease which involve real-world multi-modal speech and text data, wherein our method shows an improved performance. Additional analysis suggests that introducing uncertainty-awareness into the boosted ensembles decreases the overall entropy of the system, making it more robust to heteroscedasticity in the data, as well as better calibrating each of the modalities along with high quality prediction intervals. We open-source our entire codebase at https://github.com/usarawgi911/Uncertainty-aware-boosting