This paper defines the notion of class discrepancy for families of functions. It shows that low discrepancy classes admit small offline and streaming coresets. We provide general techniques for bounding the class discrepancy of machine learning problems. As corollaries of the general technique we bound the discrepancy (and therefore coreset complexity) of logistic regression, sigmoid activation loss, matrix covariance, kernel density and any analytic function of the dot product or the squared distance. Our results prove the existence of epsilon-approximation O(sqrt{d}/epsilon) sized coresets for the above problems. This resolves the long-standing open problem regarding the coreset complexity of Gaussian kernel density estimation. We provide two more related but independent results. First, an exponential improvement of the widely used merge-and-reduce trick which gives improved streaming sketches for any low discrepancy problem. Second, an extremely simple deterministic algorithm for finding low discrepancy sequences (and therefore coresets) for any positive semi-definite kernel. This paper establishes some explicit connections between class discrepancy, coreset complexity, learnability, and streaming algorithms.

Least-mean squares (LMS) solvers such as Linear / Ridge / Lasso-Regression, SVD and Elastic-Net not only solve fundamental machine learning problems, but are also the building blocks in a variety of other methods, such as decision trees and matrix factorizations. We suggest an algorithm that gets a finite set of $n$ $d$-dimensional real vectors and returns a weighted subset of $d+1$ vectors whose sum is \emph{exactly} the same. The proof in Caratheodory's Theorem (1907) computes such a subset in $O(n^2d^2)$ time and thus not used in practice. Our algorithm computes this subset in $O(nd)$ time, using $O(\log n)$ calls to Caratheodory's construction on small but "smart" subsets. This is based on a novel paradigm of fusion between different data summarization techniques, known as sketches and coresets. As an example application, we show how it can be used to boost the performance of existing LMS solvers, such as those in scikit-learn library, up to x100. Generalization for streaming and distributed (big) data is trivial. Extensive experimental results and complete open source code are also provided.

We consider classification in the presence of class-dependent asymmetric label noise with unknown noise probabilities. In this setting, identifiability conditions are known, but additional assumptions were shown to be required for finite sample rates, and so far only the parametric rate has been obtained. Assuming these identifiability conditions, together with a measure-smoothness condition on the regression function and Tsybakov's margin condition, we show that the Robust kNN classifier of Gao et al. attains, the minimax optimal rates of the noise-free setting, up to a log factor, even when trained on data with unknown asymmetric label noise. Hence, our results provide a solid theoretical backing for this empirically successful algorithm. By contrast the standard kNN is not even consistent in the setting of asymmetric label noise. A key idea in our analysis is a simple kNN based method for estimating the maximum of a function that requires far less assumptions than existing mode estimators do, and which may be of independent interest for noise proportion estimation and randomised optimisation problems.

Depression and anxiety are critical public health issues affecting millions of people around the world. To identify individuals who are vulnerable to depression and anxiety, predictive models have been built that typically utilize data from one source. Unlike these traditional models, in this study, we leverage a rich heterogeneous data set from the University of Notre Dame's NetHealth study that collected individuals' (student participants') social interaction data via smartphones, health-related behavioral data via wearables (Fitbit), and trait data from surveys. To integrate the different types of information, we model the NetHealth data as a heterogeneous information network (HIN). Then, we redefine the problem of predicting individuals' mental health conditions (depression or anxiety) in a novel manner, as applying to our HIN a popular paradigm of a recommender system (RS), which is typically used to predict the preference that a person would give to an item (e.g., a movie or book). In our case, the items are the individuals' different mental health states. We evaluate three state-of-the-art RS approaches. Also, we model the prediction of individuals' mental health as another problem type -- that of node classification (NC) in our HIN, evaluating in the process four node features under logistic regression as a proof-of-concept classifier. We find that our RS and NC network methods produce more accurate predictions than a logistic regression model using the same NetHealth data in the traditional non-network fashion as well as a random-approach. Also, we find that RS outperforms NC. This is the first study to integrate smartphone, wearable sensor, and survey data in an HIN manner and use RS or NC on the HIN to predict individuals' mental health conditions.

We consider selection of random predictors for high-dimensional regression problem with binary response for a general loss function. Important special case is when the binary model is semiparametric and the response function is misspecified under parametric model fit. Selection for such a scenario aims at recovering the support of the minimizer of the associated risk with large probability. We propose a two-step selection procedure which consists of screening and ordering predictors by Lasso method and then selecting a subset of predictors which minimizes Generalized Information Criterion on the corresponding nested family of models. We prove consistency of the selection method under conditions which allow for much larger number of predictors than number of observations. For the semiparametric case when distribution of random predictors satisfies linear regression conditions the true and the estimated parameters are collinear and their common support can be consistently identified.

In experimental design, we are given $n$ vectors in $d$ dimensions, and our goal is to select $k\ll n$ of them to perform expensive measurements, e.g., to obtain labels/responses, for a linear regression task. Many statistical criteria have been proposed for choosing the optimal design, with popular choices including A- and D-optimality. If prior knowledge is given, typically in the form of a $d\times d$ precision matrix $\mathbf A$, then all of the criteria can be extended to incorporate that information via a Bayesian framework. In this paper, we demonstrate a new fundamental connection between Bayesian experimental design and determinantal point processes, the latter being widely used for sampling diverse subsets of data. We use this connection to develop new efficient algorithms for finding $(1+\epsilon)$-approximations of optimal designs under four optimality criteria: A, C, D and V. Our algorithms can achieve this when the desired subset size $k$ is $\Omega(\frac{d_{\mathbf A}}{\epsilon} + \frac{\log 1/\epsilon}{\epsilon^2})$, where $d_{\mathbf A}\leq d$ is the $\mathbf A$-effective dimension, which can often be much smaller than $d$. Our results offer direct improvements over a number of prior works, for both Bayesian and classical experimental design, in terms of algorithm efficiency, approximation quality, and range of applicable criteria.

Off-policy evaluation (OPE) in both contextual bandits and reinforcement learning allows one to evaluate novel decision policies without needing to conduct exploration, which is often costly or otherwise infeasible. The problem's importance has attracted many proposed solutions, including importance sampling (IS), self-normalized IS (SNIS), and doubly robust (DR) estimates. DR and its variants ensure semiparametric local efficiency if Q-functions are well-specified, but if they are not they can be worse than both IS and SNIS. It also does not enjoy SNIS's inherent stability and boundedness. We propose new estimators for OPE based on empirical likelihood that are always more efficient than IS, SNIS, and DR and satisfy the same stability and boundedness properties as SNIS. On the way, we categorize various properties and classify existing estimators by them. Besides the theoretical guarantees, empirical studies suggest the new estimators provide advantages.

Accurate estimation of the run time of computational codes has a number of significant advantages for scientific computing. It is required information for optimal resource allocation, improving turnaround times and utilization of science gateways. Furthermore, it allows users to better plan and schedule their research, streamlining workflows and improving the overall productivity of cyberinfrastructure. Predicting run time is challenging, however. The inputs to scientific codes can be complex and high dimensional. Their relationship to the run time may be highly non-linear, and, in the most general case is completely arbitrary and thus unpredictable (i.e., simply a random mapping from inputs to run time). Most codes are not so arbitrary, however, and there has been significant prior research on predicting the run time of applications and workloads. Such predictions are generally application-specific, however. In this paper, we focus on the Gaussian computational chemistry code. We characterize a data set of runs from the SEAGrid science gateway with a number of different studies. We also explore a number of different potential regression methods and present promising future directions.

Other authors suggest that specific margin instances Forests (Breiman, 2001) and rotation forests (Rodriguez hold a clue to better generalization (Shen and Li, et al., 2006), create a set of weak classifiers from 2010; Wang et al., 2011, 2012). In this article, we design a base learning algorithm B, which are typically decision algorithms to empirically test whether the state of research trees, then combine the predictions from the classifiers in in the explanation of ensemble performance translates into the form of a weighted vote, to produce an improved prediction better performing algorithms. We do not question the theoretical compared to individual classifiers (Drucker et al., soundness of the generalization error bounds, but 1994; Dietterich, 2000; Breiman, 2001; Maclin and Opitz, simply test whether evidence suggests that better performing 2011). Upper bounds based on the sample margins of the ensemble algorithms can be derived from the practical ensemble provide some explanation on why ensembles perform interpretations of the bounds. In the next section we discuss as well as they do. Schapire et al. (1998) first pointed margins, the generalization error bounds based on the to margins as a key determinant of ensemble performance.

Magnetoencephalography and electroencephalography (M/EEG) can reveal neuronal dynamics non-invasively in real-time and are therefore appreciated methods in medicine and neuroscience. Recent advances in modeling brain-behavior relationships have highlighted the effectiveness of Riemannian geometry for summarizing the spatially correlated time-series from M/EEG in terms of their covariance. However, after artefact-suppression, M/EEG data is often rank deficient which limits the application of Riemannian concepts. In this article, we focus on the task of regression with rank-reduced covariance matrices. We study two Riemannian approaches that vectorize the M/EEG covariance between-sensors through projection into a tangent space. The Wasserstein distance readily applies to rank-reduced data but lacks affine-invariance. This can be overcome by finding a common subspace in which the covariance matrices are full rank, enabling the affine-invariant geometric distance. We investigated the implications of these two approaches in synthetic generative models, which allowed us to control estimation bias of a linear model for prediction. We show that Wasserstein and geometric distances allow perfect out-of-sample prediction on the generative models. We then evaluated the methods on real data with regard to their effectiveness in predicting age from M/EEG covariance matrices. The findings suggest that the data-driven Riemannian methods outperform different sensor-space estimators and that they get close to the performance of biophysics-driven source-localization model that requires MRI acquisitions and tedious data processing. Our study suggests that the proposed Riemannian methods can serve as fundamental building-blocks for automated large-scale analysis of M/EEG.