Ordinal regression falls between discrete-valued classification and continuous-valued regression. Ordinal target variables can be associated with ranked random variables. These random variables are known as order statistics and they are closely related to ordinal regression. However, the challenge of using order statistics for ordinal regression prediction is finding a suitable parent distribution. In this work, we provide a case study of a real-world orienteering relay race by viewing it as a random process. For this process, we show that accurate order statistical ordinal regression predictions of final team rankings, or places, can be obtained by assuming a lognormal distribution of individual leg times. Moreover, we apply Fenton-Wilkinson approximations to intermediate changeover times alongside an estimator for the total number of teams as in the notorious German tank problem. The purpose of this work is, in part, to spark interest in studying the applicability of order statistics in ordinal regression problems.

Double descent refers to the phase transition that is exhibited by the generalization error of unregularized learning models when varying the ratio between the number of parameters and the number of training samples. The recent success of highly over-parameterized machine learning models such as deep neural networks has motivated a theoretical analysis of the double descent phenomenon in classical models such as linear regression which can also generalize well in the over-parameterized regime. We build on recent advances in Randomized Numerical Linear Algebra (RandNLA) to provide the first exact non-asymptotic expressions for double descent of the minimum norm linear estimator. Our approach involves constructing what we call a surrogate random design to replace the standard i.i.d. design of the training sample. This surrogate design admits exact expressions for the mean squared error of the estimator while preserving the key properties of the standard design. We also establish an exact implicit regularization result for over-parameterized training samples. In particular, we show that, for the surrogate design, the implicit bias of the unregularized minimum norm estimator precisely corresponds to solving a ridge-regularized least squares problem on the population distribution.

Demonstrated experience in PySpark is one of the most desirable competencies that employers are looking for when building data science teams, because it enables these teams to own live data products. While I've previously blogged about PySpark, Parallelization, and UDFs, I wanted to provide a proper overview of this topic as a book chapter. I'm sharing this complete chapter, because I want to encourage the adoption of PySpark as a tool for data scientists. All code examples from this post are available here, and all prerequisites are covered in the sample chapters here. You might want to grab some snacks before diving in! Spark is a general-purpose computing framework that can scale to massive data volumes. It builds upon prior big data tools such as Hadoop and MapReduce, while providing significant improvements in the expressivity of the languages it supports. One of the core components of Spark is resilient distributed datasets (RDD), which enable clusters of machines to perform workloads in a coordinated, and fault-tolerant process. In more recent versions of Spark, the Dataframe API provides an abstraction on top of RDDs that resembles the same data structure in R and Pandas. PySpark is the Python interface to Spark, and it provides an API for working with large-scale datasets in a distributed computing environment. PySpark is an extremely valuable tool for data scientists, because it can streamline the process for translating prototype models into production-grade model workflows. At Zynga, our data science team owns a number of production-grade systems that provide useful signals to our game and marketing teams. By using PySpark, we've been able to reduce the amount of support we need from engineering teams to scale up models from concept to production.

Interpretability is the degree to which a human can understand the cause of a decision – Miller, Tim[1] Explainable AI (XAI) is a sub-field of AI which has been gaining ground in the recent past. And as I machine learning practitioner dealing with customers day in and day out, I can see why. I've been an analytics practitioner for more than 5 years and I swear, the hardest part of a machine learning project is not creating the perfect model which beats all the benchmarks. It's the part where you convince the customer why and how it works. Humans always had a dichotomy when faced with the unknown.

Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and extend both of them using weighted lattices and a max-$\star$ algebra with an arbitrary binary operation $\star$ that distributes over max, and outline applications to geometry, machine learning, and optimization. Further, we generalize tropical geometrical objects using weighted lattices. Finally, we provide the optimal solution of max-$\star$ equations using morphological adjunctions that are projections on weighted lattices, and apply it to optimal piecewise-linear regression for fitting max-$\star$ tropical curves and surfaces to arbitrary data that constitute polygonal or polyhedral shape approximations. This also includes an efficient algorithm for solving the convex regression problem of data fitting with max-affine functions.

Differential privacy allows quantifying privacy loss from computations on sensitive personal data. This loss grows with the number of accesses to the data, making it hard to open the use of such data while respecting privacy. To avoid this limitation, we propose privacy-preserving release of a synthetic version of a data set, which can be used for an unlimited number of analyses with any methods, without affecting the privacy guarantees. The synthetic data generation is based on differentially private learning of a generative probabilistic model which can capture the probability distribution of the original data. We demonstrate empirically that we can reliably reproduce statistical discoveries from the synthetic data. We expect the method to have broad use in sharing anonymized versions of key data sets for research.

High-dimensional prediction is a challenging problem setting for traditional statistical models. Although regularization improves model performance in high dimensions, it does not sufficiently leverage knowledge on feature importances held by domain experts. As an alternative to standard regularization techniques, we propose Distance Metric Learning Regularization (DMLreg), an approach for eliciting prior knowledge from domain experts and integrating that knowledge into a regularized linear model. First, we learn a Mahalanobis distance metric between observations from pairwise similarity comparisons provided by an expert. Then, we use the learned distance metric to place prior distributions on coefficients in a linear model. Through experimental results on a simulated high-dimensional prediction problem, we show that DMLreg leads to improvements in model performance when the domain expert is knowledgeable.

We propose a novel algorithm for large-scale regression problems named histogram transform ensembles (HTE), composed of random rotations, stretchings, and translations. First of all, we investigate the theoretical properties of HTE when the regression function lies in the H\"{o}lder space $C^{k,\alpha}$, $k \in \mathbb{N}_0$, $\alpha \in (0,1]$. In the case that $k=0, 1$, we adopt the constant regressors and develop the na\"{i}ve histogram transforms (NHT). Within the space $C^{0,\alpha}$, although almost optimal convergence rates can be derived for both single and ensemble NHT, we fail to show the benefits of ensembles over single estimators theoretically. In contrast, in the subspace $C^{1,\alpha}$, we prove that if $d \geq 2(1+\alpha)/\alpha$, the lower bound of the convergence rates for single NHT turns out to be worse than the upper bound of the convergence rates for ensemble NHT. In the other case when $k \geq 2$, the NHT may no longer be appropriate in predicting smoother regression functions. Instead, we apply kernel histogram transforms (KHT) equipped with smoother regressors such as support vector machines (SVMs), and it turns out that both single and ensemble KHT enjoy almost optimal convergence rates. Then we validate the above theoretical results by numerical experiments. On the one hand, simulations are conducted to elucidate that ensemble NHT outperform single NHT. On the other hand, the effects of bin sizes on accuracy of both NHT and KHT also accord with theoretical analysis. Last but not least, in the real-data experiments, comparisons between the ensemble KHT, equipped with adaptive histogram transforms, and other state-of-the-art large-scale regression estimators verify the effectiveness and accuracy of our algorithm.

Logistic regression models are a popular and effective method to predict the probability of categorical response data. However inference for these models can become computationally prohibitive for large datasets. Here we adapt ideas from symbolic data analysis to summarise the collection of predictor variables into histogram form, and perform inference on this summary dataset. We develop ideas based on composite likelihoods to derive an efficient one-versus-rest approximate composite likelihood model for histogram-based random variables, constructed from low-dimensional marginal histograms obtained from the full histogram. We demonstrate that this procedure can achieve comparable classification rates compared to the standard full data multinomial analysis and against state-of-the-art subsampling algorithms for logistic regression, but at a substantially lower computational cost. Performance is explored through simulated examples, and analyses of large supersymmetry and satellite crop classification datasets.

The boons of machine learning have been leveraged in the industry in the past many years. With its increasing implementation, the ML tools have also evolved with time. Today, people can easily work with machine learning owing to its easy-to-use, user-friendly tools. As the gathering of data and turning it into actionable insights has been automated enough, people with some knowledge of technology and motivation can work with ML. These tools possess the strength to handle the mundane work of collecting data, adding structure and consistency where possible, and then starting the calculation.