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. This kernel enables us to perform clustering to identify the homogeneous subpopulations. Indeed, we prove 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. We can then use existing methods to learn a causal structure for each of these subpopulations. We demonstrate using our kernel for causal clustering with an application in genetics, allowing us to reason about the latent transcription factor networks regulating measured gene expression levels.

Previously, statistical textbook wisdom has held that interpolating noisy data will generalize poorly, but recent work has shown that data interpolation schemes can generalize well. This could explain why overparameterized deep nets do not necessarily overfit. Optimal data interpolation schemes have been exhibited that achieve theoretical lower bounds for excess risk in any dimension for large data (Statistically Consistent Interpolation). These are non-parametric Nadaraya-Watson estimators with singular kernels. The recently proposed weighted interpolating nearest neighbors method (wiNN) is in this class, as is the previously studied Hilbert kernel interpolation scheme, in which the estimator has the form $\hat{f}(x)=\sum_i y_i w_i(x)$, where $w_i(x)= \|x-x_i\|^{-d}/\sum_j \|x-x_j\|^{-d}$. This estimator is unique in being completely parameter-free. While statistical consistency was previously proven, convergence rates were not established. Here, we comprehensively study the finite sample properties of Hilbert kernel regression. We prove that the excess risk is asymptotically equivalent pointwise to $\sigma^2(x)/\ln(n)$ where $\sigma^2(x)$ is the noise variance. We show that the excess risk of the plugin classifier is less than $2|f(x)-1/2|^{1-\alpha}\,(1+\varepsilon)^\alpha \sigma^\alpha(x)(\ln(n))^{-\frac{\alpha}{2}}$, for any $0<\alpha<1$, where $f$ is the regression function $x\mapsto\mathbb{E}[y|x]$. We derive asymptotic equivalents of the moments of the weight functions $w_i(x)$ for large $n$, for instance for $\beta>1$, $\mathbb{E}[w_i^{\beta}(x)]\sim_{n\rightarrow \infty}((\beta-1)n\ln(n))^{-1}$. We derive an asymptotic equivalent for the Lagrange function and exhibit the nontrivial extrapolation properties of this estimator. We present heuristic arguments for a universal $w^{-2}$ power-law behavior of the probability density of the weights in the large $n$ limit.

Blockwise missing data occurs frequently when we integrate multisource or multimodality data where different sources or modalities contain complementary information. In this paper, we consider a high-dimensional linear regression model with blockwise missing covariates and a partially observed response variable. Under this semi-supervised framework, we propose a computationally efficient estimator for the regression coefficient vector based on carefully constructed unbiased estimating equations and a multiple blockwise imputation procedure, and obtain its rates of convergence. Furthermore, building upon an innovative semi-supervised projected estimating equation technique that intrinsically achieves bias-correction of the initial estimator, we propose nearly unbiased estimators for the individual regression coefficients that are asymptotically normally distributed under mild conditions. By carefully analyzing these debiased estimators, asymptotically valid confidence intervals and statistical tests about each regression coefficient are constructed. Numerical studies and application analysis of the Alzheimer's Disease Neuroimaging Initiative data show that the proposed method performs better and benefits more from unsupervised samples than existing methods.

Target networks are at the core of recent success in Reinforcement Learning. They stabilize the training by using old parameters to estimate the $Q$-values, but this also limits the propagation of newly-encountered rewards which could ultimately slow down the training. In this work, we propose an alternative training method based on functional regularization which does not have this deficiency. Unlike target networks, our method uses up-to-date parameters to estimate the target $Q$-values, thereby speeding up training while maintaining stability. Surprisingly, in some cases, we can show that target networks are a special, restricted type of functional regularizers. Using this approach, we show empirical improvements in sample efficiency and performance across a range of Atari and simulated robotics environments.

In the past decade, contextual bandit and reinforcement learning algorithms have been successfully used in various interactive learning systems such as online advertising, recommender systems, and dynamic pricing. However, they have yet to be widely adopted in high-stakes application domains, such as healthcare. One reason may be that existing approaches assume that the underlying mechanisms are static in the sense that they do not change over time or over different environments. In many real world systems, however, the mechanisms are subject to shifts across environments which may invalidate the static environment assumption. In this paper, we tackle the problem of environmental shifts under the framework of offline contextual bandits. We view the environmental shift problem through the lens of causality and propose multi-environment contextual bandits that allow for changes in the underlying mechanisms. We adopt the concept of invariance from the causality literature and introduce the notion of policy invariance. We argue that policy invariance is only relevant if unobserved confounders are present and show that, in that case, an optimal invariant policy is guaranteed, under certain assumptions, to generalize across environments. Our results do not only provide a solution to the environmental shift problem but also establish concrete connections among causality, invariance and contextual bandits.

Nonuniform subsampling methods are effective to reduce computational burden and maintain estimation efficiency for massive data. Existing methods mostly focus on subsampling with replacement due to its high computational efficiency. If the data volume is so large that nonuniform subsampling probabilities cannot be calculated all at once, then subsampling with replacement is infeasible to implement. This paper solves this problem using Poisson subsampling. We first derive optimal Poisson subsampling probabilities in the context of quasi-likelihood estimation under the A- and L-optimality criteria. For a practically implementable algorithm with approximated optimal subsampling probabilities, we establish the consistency and asymptotic normality of the resultant estimators. To deal with the situation that the full data are stored in different blocks or at multiple locations, we develop a distributed subsampling framework, in which statistics are computed simultaneously on smaller partitions of the full data. Asymptotic properties of the resultant aggregated estimator are investigated. We illustrate and evaluate the proposed strategies through numerical experiments on simulated and real data sets.

Bayesian statistics is a statistical theory based on the Bayesian interpretation of probability. To understand Bayesian Statistics, we need to first understand conditional probability and Bayes' theorem. Conditional probability measures the probability of an event occurring based on the fact that another event has already occurred. Just as the formula below shows, event A occurs given that event B occurred, is the division of the joint probability of event A and B and the probability of event B occurring. There are two notes regarding this formula.

An ensemble learning method involves combining the predictions from multiple contributing models. Nevertheless, not all techniques that make use of multiple machine learning models are ensemble learning algorithms. It is common to divide a prediction problem into subproblems. For example, some problems naturally subdivide into independent but related subproblems and a machine learning model can be prepared for each. It is less clear whether these represent examples of ensemble learning, although we might distinguish these methods from ensembles given the inability for a contributing ensemble member to produce a solution (however weakly) to the overall prediction problem.

Radar-based recognition and localization of people and things in the home environment has certain advantages over computer vision, including increased user privacy, low power consumption, zero-light operation and more sensor flexible placement. Shallow machine learning techniques such as Support Vector Machines and Logistic Regression can be used to classify images from radar, and in my previous work, Teaching Radar to Understand the Home and Using Stochastic Gradient Descent to Train Linear Classifiers I shared how to apply some of these methods. In this article, you will learn how to develop Deep Neural Networks (DNN)and train them to classify objects in radar images. In addition, you will learn how to use a Semi-Supervised Generative Adversarial Network (SGAN) [1] that only needs a small number of labeled data to train a DNN classifier. This is important in dealing with radar data sets because of the dearth of large training sets, in contrast to those available for camera-based images (e.g., ImageNet) which has helped to make computer vision ubiquitous.

Machine learning involves using an algorithm to learn and generalize from historical data in order to make predictions on new data. This problem can be described as approximating a function that maps examples of inputs to examples of outputs. Approximating a function can be solved by framing the problem as function optimization. This is where a machine learning algorithm defines a parameterized mapping function (e.g. a weighted sum of inputs) and an optimization algorithm is used to fund the values of the parameters (e.g. This means that each time we fit a machine learning algorithm on a training dataset, we are solving an optimization problem.