The training phase of a neural network consists of two parts. Feedforward is the process where data flows through the neural network from the input node towards the output node. In other words, this is where the network is trying to make predictions. These predictions are then evaluated by comparing them to the the actual values. The error can be found as the difference between the two values.

In traditional logistic regression models, the link function is often assumed to be linear and continuous in predictors. Here, we consider a threshold model that all continuous features are discretized into ordinal levels, which further determine the binary responses. Both the threshold points and regression coefficients are unknown and to be estimated. For high dimensional data, we propose a fusion penalized logistic threshold regression (FILTER) model, where a fused lasso penalty is employed to control the total variation and shrink the coefficients to zero as a method of variable selection. Under mild conditions on the estimate of unknown threshold points, we establish the non-asymptotic error bound for coefficient estimation and the model selection consistency. With a careful characterization of the error propagation, we have also shown that the tree-based method, such as CART, fulfill the threshold estimation conditions. We find the FILTER model is well suited in the problem of early detection and prediction for chronic disease like diabetes, using physical examination data. The finite sample behavior of our proposed method are also explored and compared with extensive Monte Carlo studies, which supports our theoretical discoveries.

A supervised machine learning algorithm determines a model from a learning sample that will be used to predict new observations. To this end, it aggregates individual characteristics of the observations of the learning sample. But this information aggregation does not consider any potential selection on unobservables and any status-quo biases which may be contained in the training sample. The latter bias has raised concerns around the so-called \textit{fairness} of machine learning algorithms, especially towards disadvantaged groups. In this chapter, we review the issue of fairness in machine learning through the lenses of structural econometrics models in which the unknown index is the solution of a functional equation and issues of endogeneity are explicitly accounted for. We model fairness as a linear operator whose null space contains the set of strictly {\it fair} indexes. A {\it fair} solution is obtained by projecting the unconstrained index into the null space of this operator or by directly finding the closest solution of the functional equation into this null space. We also acknowledge that policymakers may incur a cost when moving away from the status quo. Achieving \textit{approximate fairness} is obtained by introducing a fairness penalty in the learning procedure and balancing more or less heavily the influence between the status quo and a full fair solution.

It has now been well established that empirical risk minimization (ERM) can empirically achieve high test performance on a variety of tasks, particularly with modern overparameterized models where the number of parameters is much larger than the number of training samples. This strong performance of ERM however has been shown to degrade under distributional shift, where the training and test distributions are different [HS15, BGO16, Tat17]. There are two broad categories of distribution shift studied in recent years. The first is domain generalization, where the training distribution is a mixture of environments, while the test distribution contains new environments that do not appear in the training distribution. The hope in such cases is to learn "invariant features" that do not change across environments, in contrast to spurious features, such as the background in image classification instead of the object, and negation words such as "not" and "never" in language sentiment analysis instead of the sentence meaning itself. However, it has been empirically shown that overparameterized models trained via ERM tend to learn spurious features. The second is subpopulation shift, where the training distribution consists of a number of groups, and the test distribution is the groupconditional distribution of any group (or more generally, an arbitrary mixture of the training groups). Such subpopulation shift occurs in the context of fair machine learning, where the dataset is divided into demographic groups, and it is of interest to perform well on all such groups; as well as in learning with imbalanced classes, where each class is a group, and the model needs to perform well on all classes. While overparameterized models trained via ERM can achieve high average performance over the entire data domain, they have been shown to have low performance on underrepresented data subpopulations.

We consider the distributed stochastic gradient descent problem, where a main node distributes gradient calculations among $n$ workers from which at most $b \leq n$ can be utilized in parallel. By assigning tasks to all the workers and waiting only for the $k$ fastest ones, the main node can trade-off the error of the algorithm with its runtime by gradually increasing $k$ as the algorithm evolves. However, this strategy, referred to as adaptive k sync, can incur additional costs since it ignores the computational efforts of slow workers. We propose a cost-efficient scheme that assigns tasks only to $k$ workers and gradually increases $k$. As the response times of the available workers are unknown to the main node a priori, we utilize a combinatorial multi-armed bandit model to learn which workers are the fastest while assigning gradient calculations, and to minimize the effect of slow workers. Assuming that the mean response times of the workers are independent and exponentially distributed with different means, we give empirical and theoretical guarantees on the regret of our strategy, i.e., the extra time spent to learn the mean response times of the workers. Compared to adaptive k sync, our scheme achieves significantly lower errors with the same computational efforts while being inferior in terms of speed.

For high dimensional data, where P features for N objects (P >> N) are represented in an NxP matrix X, we describe a clustering algorithm based on the normalized left Gram matrix, G = XX'/P. Under certain regularity conditions, the rows in G that correspond to objects in the same cluster converge to the same mean vector. By clustering on the row means, the algorithm does not require preprocessing by dimension reduction or feature selection techniques and does not require specification of tuning or hyperparameter values. Because it is based on the NxN matrix G, it has a lower computational cost than many methods based on clustering the feature matrix X. When compared to 14 other clustering algorithms applied to 32 benchmarked microarray datasets, the proposed algorithm provided the most accurate estimate of the underlying cluster configuration more than twice as often as its closest competitors.

Bayesian optimization (BO) is a sample-efficient approach for tuning design parameters to optimize expensive-to-evaluate, black-box performance metrics. In many manufacturing processes, the design parameters are subject to random input noise, resulting in a product that is often less performant than expected. Although BO methods have been proposed for optimizing a single objective under input noise, no existing method addresses the practical scenario where there are multiple objectives that are sensitive to input perturbations. In this work, we propose the first multi-objective BO method that is robust to input noise. We formalize our goal as optimizing the multivariate value-at-risk (MVaR), a risk measure of the uncertain objectives. Since directly optimizing MVaR is computationally infeasible in many settings, we propose a scalable, theoretically-grounded approach for optimizing MVaR using random scalarizations. Empirically, we find that our approach significantly outperforms alternative methods and efficiently identifies optimal robust designs that will satisfy specifications across multiple metrics with high probability.

Rapid progress in representation learning has led to a proliferation of embedding models, and to associated challenges of model selection and practical application. It is non-trivial to assess a model's generalizability to new, candidate datasets and failure to generalize may lead to poor performance on downstream tasks. Distribution shifts are one cause of reduced generalizability, and are often difficult to detect in practice. In this paper, we use the embedding space geometry to propose a non-parametric framework for detecting distribution shifts, and specify two tests. The first test detects shifts by establishing a robustness boundary, determined by an intelligible performance criterion, for comparing reference and candidate datasets. The second test detects shifts by featurizing and classifying multiple subsamples of two datasets as in-distribution and out-of-distribution. In evaluation, both tests detect model-impacting distribution shifts, in various shift scenarios, for both synthetic and real-world datasets.

For the last part of the Statistics series, we will cover the ANOVA, Post-hoc Pairwise Comparison, Two-way ANOVA, and R-squared. Previously, our study focused on one or two groups of subjects. How can we handle the concept of multiple groups with multiple factors? For example, the dose level and gender may impact the effectiveness of a vaccine. How can we determine whether it is statistically significant for particular combinations?

There are so many machine learning algorithms out there, and we can find different kinds of overviews and cheat sheets. When trying to connect the dots of the multitude of machine learning algorithms, I discovered is that there are often several approaches to build/understand an algorithm. There is no better way, and each approach allows us to understand one aspect. One approach can be easier than another depending on what you already know. My aim in this article is to find a meaningful way to connect them.