We consider a high-dimensional mixture of two Gaussians in the noisy regime where even an oracle knowing the centers of the clusters misclassifies a small but finite fraction of the points. We provide a rigorous analysis of the generalization error of regularized convex classifiers, including ridge, hinge and logistic regression, in the high-dimensional limit where the number $n$ of samples and their dimension $d$ go to infinity while their ratio is fixed to $\alpha= n/d$. We discuss surprising effects of the regularization that in some cases allows to reach the Bayes-optimal performances. We also illustrate the interpolation peak at low regularization, and analyze the role of the respective sizes of the two clusters.

Shape-constrained convex regression problem deals with fitting a convex function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. This paper provides a comprehensive mechanism for computing the least squares estimator of a multivariate shape-constrained convex regression function in $\mathbb{R}^d$. We prove that the least squares estimator is computable via solving a constrained convex quadratic programming (QP) problem with $(n+1)d$ variables and at least $n(n-1)$ linear inequality constraints, where $n$ is the number of data points. For solving the generally very large-scale convex QP, we design two efficient algorithms, one is the symmetric Gauss-Seidel based alternating direction method of multipliers ({\tt sGS-ADMM}), and the other is the proximal augmented Lagrangian method ({\tt pALM}) with the subproblems solved by the semismooth Newton method ({\tt SSN}). Comprehensive numerical experiments, including those in the pricing of basket options and estimation of production functions in economics, demonstrate that both of our proposed algorithms outperform the state-of-the-art algorithm. The {\tt pALM} is more efficient than the {\tt sGS-ADMM} but the latter has the advantage of being simpler to implement.

Meta-learning, or learning-to-learn, seeks to design algorithms that can utilize previous experience to rapidly learn new skills or adapt to new environments. Representation learning---a key tool for performing meta-learning---learns a data representation that can transfer knowledge across multiple tasks, which is essential in regimes where data is scarce. Despite a recent surge of interest in the practice of meta-learning, the theoretical underpinnings of meta-learning algorithms are lacking, especially in the context of learning transferable representations. In this paper, we focus on the problem of multi-task linear regression---in which multiple linear regression models share a common, low-dimensional linear representation. Here, we provide provably fast, sample-efficient algorithms to address the dual challenges of (1) learning a common set of features from multiple, related tasks, and (2) transferring this knowledge to new, unseen tasks. Both are central to the general problem of meta-learning. Finally, we complement these results by providing information-theoretic lower bounds on the sample complexity of learning these linear features, showing that our algorithms are optimal up to logarithmic factors.

Building machine learning models that are fair with respect to an unprivileged group is a topical problem. Modern fairness-aware algorithms often ignore causal effects and enforce fairness through modifications applicable to only a subset of machine learning models. In this work, we propose a new definition of fairness that incorporates causality through the Controlled Direct Effect (CDE). We develop regularizations to tackle classical fairness measures and present a causal regularization that satisfies our new fairness definition by removing the impact of unprivileged group variables on the model outcomes as measured by the CDE. These regularizations are applicable to any model trained using by iteratively minimizing a loss through differentiation. We demonstrate our approaches using both gradient boosting and logistic regression on: a synthetic dataset, the UCI Adult (Census) Dataset, and a real-world credit-risk dataset. Our results were found to mitigate unfairness from the predictions with small reductions in model performance.

In the previous article, we studied Artificial Intelligence, its functions, and its python implementations. In this article, we will be studying Machine Learning. One thing that I believe is that if we are able to correlate anything with us or our life, there are greater chances of understanding the concept. So I will try to explain everything by relating it to humans.

The statistical analysis of Randomized Numerical Linear Algebra (RandNLA) algorithms within the past few years has mostly focused on their performance as point estimators. However, this is insufficient for conducting statistical inference, e.g., constructing confidence intervals and hypothesis testing, since the distribution of the estimator is lacking. In this article, we develop an asymptotic analysis to derive the distribution of RandNLA sampling estimators for the least-squares problem. In particular, we derive the asymptotic distribution of a general sampling estimator with arbitrary sampling probabilities. The analysis is conducted in two complementary settings, i.e., when the objective of interest is to approximate the full sample estimator or is to infer the underlying ground truth model parameters. For each setting, we show that the sampling estimator is asymptotically normally distributed under mild regularity conditions. Moreover, the sampling estimator is asymptotically unbiased in both settings. Based on our asymptotic analysis, we use two criteria, the Asymptotic Mean Squared Error (AMSE) and the Expected Asymptotic Mean Squared Error (EAMSE), to identify optimal sampling probabilities. Several of these optimal sampling probability distributions are new to the literature, e.g., the root leverage sampling estimator and the predictor length sampling estimator. Our theoretical results clarify the role of leverage in the sampling process, and our empirical results demonstrate improvements over existing methods.

Despite breakthrough performance, modern learning models are known to be highly vulnerable to small adversarial perturbations in their inputs. While a wide variety of recent \emph{adversarial training} methods have been effective at improving robustness to perturbed inputs (robust accuracy), often this benefit is accompanied by a decrease in accuracy on benign inputs (standard accuracy), leading to a tradeoff between often competing objectives. Complicating matters further, recent empirical evidence suggest that a variety of other factors (size and quality of training data, model size, etc.) affect this tradeoff in somewhat surprising ways. In this paper we provide a precise and comprehensive understanding of the role of adversarial training in the context of linear regression with Gaussian features. In particular, we characterize the fundamental tradeoff between the accuracies achievable by any algorithm regardless of computational power or size of the training data. Furthermore, we precisely characterize the standard/robust accuracy and the corresponding tradeoff achieved by a contemporary mini-max adversarial training approach in a high-dimensional regime where the number of data points and the parameters of the model grow in proportion to each other. Our theory for adversarial training algorithms also facilitates the rigorous study of how a variety of factors (size and quality of training data, model overparametrization etc.) affect the tradeoff between these two competing accuracies.

Deleting data from a trained machine learning (ML) model is a critical task in many applications. For example, we may want to remove the influence of training points that might be out of date or outliers. Regulations such as EU's General Data Protection Regulation also stipulate that individuals can request to have their data deleted. The naive approach to data deletion is to retrain the ML model on the remaining data, but this is too time consuming. Moreover there is no known efficient algorithm that exactly deletes data from most ML models. In this work, we evaluate several approaches for approximate data deletion from trained models. For the case of linear regression, we propose a new method with linear dependence on the feature dimension $d$, a significant gain over all existing methods which all have superlinear time dependence on the dimension. We also provide a new test for evaluating data deletion from linear models.

In many predictive decision-making scenarios, such as credit scoring and academic testing, a decision-maker must construct a model (predicting some outcome) that accounts for agents' incentives to "game" their features in order to receive better decisions. Whereas the strategic classification literature generally assumes that agents' outcomes are not causally dependent on their features (and thus strategic behavior is a form of lying), we join concurrent work in modeling agents' outcomes as a function of their changeable attributes. Our formulation is the first to incorporate a crucial phenomenon: when agents act to change observable features, they may as a side effect perturb hidden features that causally affect their true outcomes. We consider three distinct desiderata for a decision-maker's model: accurately predicting agents' post-gaming outcomes (accuracy), incentivizing agents to improve these outcomes (improvement), and, in the linear setting, estimating the visible coefficients of the true causal model (causal precision). As our main contribution, we provide the first algorithms for learning accuracy-optimizing, improvement-optimizing, and causal-precision-optimizing linear regression models directly from data, without prior knowledge of agents' possible actions. These algorithms circumvent the hardness result of Miller et al. (2019) by allowing the decision maker to observe agents' responses to a sequence of decision rules, in effect inducing agents to perform causal interventions for free.

Many imbalanced classification tasks require a skillful model that predicts a crisp class label, where both classes are equally important. An example of an imbalanced classification problem where a class label is required and both classes are equally important is the detection of oil spills or slicks in satellite images. The detection of a spill requires mobilizing an expensive response, and missing an event is equally expensive, causing damage to the environment. One way to evaluate imbalanced classification models that predict crisp labels is to calculate the separate accuracy on the positive class and the negative class, referred to as sensitivity and specificity. These two measures can then be averaged using the geometric mean, referred to as the G-mean, that is insensitive to the skewed class distribution and correctly reports on the skill of the model on both classes. In this tutorial, you will discover how to develop a model to predict the presence of an oil spill in satellite images and evaluate it using the G-mean metric. Develop an Imbalanced Classification Model to Detect Oil Spills Photo by Lenny K Photography, some rights reserved. In this project, we will use a standard imbalanced machine learning dataset referred to as the "oil spill" dataset, "oil slicks" dataset or simply "oil."