These concepts form the foundation of many machine learning algorithms. Initially, I decided against writing an article on these topics because they are so widely covered. However, I have changed my mind because understanding these concepts is essential for understanding more advanced topics like Neural Networks (that I plan on tackling in the near future). In addition, this series will be divided into two parts to make it more manageable and organized for better understanding. So make yourself comfortable, grab a cup of coffee, and get ready to embark on a magical journey of machine learning. As with any machine learning problem, we begin with a specific question we want to answer.

Abstract: The problem of domain generalization is to learn, given data from different source distributions, a model that can be expected to generalize well on new target distributions which are only seen through unlabeled samples. In this paper, we study domain generalization as a problem of functional regression. Our concept leads to a new algorithm for learning a linear operator from marginal distributions of inputs to the corresponding conditional distributions of outputs given inputs. Our algorithm allows a source distribution-dependent construction of reproducing kernel Hilbert spaces for prediction, and, satisfies finite sample error bounds for the idealized risk. Abstract: eed-forward neural networks (NN) are a staple machine learning method widely used in many areas of science and technology.

The beauty of learning complex things is by breaking them down into smaller simple things. Nobody was born an expert, just like the writer did not become a data geek until after campus -- without even a Data Science background. Nevertheless, you should be in love with mathematics and coding to even appreciate the most difficult concepts in Data Science. To be a Data Science pro, you should be skilled in Statistics, Machine Learning,Deep Leaning; capable of knowing the right tools in those fields. Apparently, there is more to Data than collecting, preparing and cleaning using tools like MS Excel, R, SQL and Tableau that you will find in any Data Analytics course. Data Analytics answers questions pertaining descriptive, diagnostic and prescriptive analytics while Data Science involves an additional field known as predictive analytics.

That said, let's see what the code for writing a logistic regression model looks like. Our class inherits from nn.Module. This class provides the methods behind the scenes that make the model work. The __init__ method of a class contains the logic that runs when instantiating a class in Python. Here we pass two arguments: the number of features and the number of classes to predict.

The goal of a learning algorithm is to receive a training data set as input and provide a hypothesis that can generalize to all possible data points from a domain set. The hypothesis is chosen from hypothesis classes with potentially different complexities. Linear regression modeling is an important category of learning algorithms. The practical uncertainty of the target samples affects the generalization performance of the learned model. Failing to choose a proper model or hypothesis class can lead to serious issues such as underfitting or overfitting. These issues have been addressed by alternating cost functions or by utilizing cross-validation methods. These approaches can introduce new hyperparameters with their own new challenges and uncertainties or increase the computational complexity of the learning algorithm. On the other hand, the theory of probably approximately correct (PAC) aims at defining learnability based on probabilistic settings. Despite its theoretical value, PAC does not address practical learning issues on many occasions. This work is inspired by the foundation of PAC and is motivated by the existing regression learning issues. The proposed approach, denoted by epsilon-Confidence Approximately Correct (epsilon CoAC), utilizes Kullback Leibler divergence (relative entropy) and proposes a new related typical set in the set of hyperparameters to tackle the learnability issue. Moreover, it enables the learner to compare hypothesis classes of different complexity orders and choose among them the optimum with the minimum epsilon in the epsilon CoAC framework. Not only the epsilon CoAC learnability overcomes the issues of overfitting and underfitting, but it also shows advantages and superiority over the well known cross-validation method in the sense of time consumption as well as in the sense of accuracy.

This paper considers binary classification of high-dimensional features under a postulated model with a low-dimensional latent Gaussian mixture structure and non-vanishing noise. A generalized least squares estimator is used to estimate the direction of the optimal separating hyperplane. The estimated hyperplane is shown to interpolate on the training data. While the direction vector can be consistently estimated as could be expected from recent results in linear regression, a naive plug-in estimate fails to consistently estimate the intercept. A simple correction, that requires an independent hold-out sample, renders the procedure minimax optimal in many scenarios. The interpolation property of the latter procedure can be retained, but surprisingly depends on the way the labels are encoded.

Motivated by multi-center biomedical studies that cannot share individual data due to privacy and ownership concerns, we develop communication-efficient iterative distributed algorithms for estimation and inference in the high-dimensional sparse Cox proportional hazards model. We demonstrate that our estimator, even with a relatively small number of iterations, achieves the same convergence rate as the ideal full-sample estimator under very mild conditions. To construct confidence intervals for linear combinations of high-dimensional hazard regression coefficients, we introduce a novel debiased method, establish central limit theorems, and provide consistent variance estimators that yield asymptotically valid distributed confidence intervals. In addition, we provide valid and powerful distributed hypothesis tests for any coordinate element based on a decorrelated score test. We allow time-dependent covariates as well as censored survival times. Extensive numerical experiments on both simulated and real data lend further support to our theory and demonstrate that our communication-efficient distributed estimators, confidence intervals, and hypothesis tests improve upon alternative methods.

We investigate the generalization and optimization properties of shallow neural-network classifiers trained by gradient descent in the interpolating regime. Specifically, in a realizable scenario where model weights can achieve arbitrarily small training error $\epsilon$ and their distance from initialization is $g(\epsilon)$, we demonstrate that gradient descent with $n$ training data achieves training error $O(g(1/T)^2 /T)$ and generalization error $O(g(1/T)^2 /n)$ at iteration $T$, provided there are at least $m=\Omega(g(1/T)^4)$ hidden neurons. We then show that our realizable setting encompasses a special case where data are separable by the model's neural tangent kernel. For this and logistic-loss minimization, we prove the training loss decays at a rate of $\tilde O(1/ T)$ given polylogarithmic number of neurons $m=\Omega(\log^4 (T))$. Moreover, with $m=\Omega(\log^{4} (n))$ neurons and $T\approx n$ iterations, we bound the test loss by $\tilde{O}(1/n)$. Our results differ from existing generalization outcomes using the algorithmic-stability framework, which necessitate polynomial width and yield suboptimal generalization rates. Central to our analysis is the use of a new self-bounded weak-convexity property, which leads to a generalized local quasi-convexity property for sufficiently parameterized neural-network classifiers. Eventually, despite the objective's non-convexity, this leads to convergence and generalization-gap bounds that resemble those found in the convex setting of linear logistic regression.

Graph embedding maps graph nodes to low-dimensional vectors, and is widely adopted in machine learning tasks. The increasing availability of billion-edge graphs underscores the importance of learning efficient and effective embeddings on large graphs, such as link prediction on Twitter with over one billion edges. Most existing graph embedding methods fall short of reaching high data scalability. In this paper, we present a general-purpose, distributed, information-centric random walk-based graph embedding framework, DistGER, which can scale to embed billion-edge graphs. DistGER incrementally computes information-centric random walks. It further leverages a multi-proximity-aware, streaming, parallel graph partitioning strategy, simultaneously achieving high local partition quality and excellent workload balancing across machines. DistGER also improves the distributed Skip-Gram learning model to generate node embeddings by optimizing the access locality, CPU throughput, and synchronization efficiency. Experiments on real-world graphs demonstrate that compared to state-of-the-art distributed graph embedding frameworks, including KnightKing, DistDGL, and Pytorch-BigGraph, DistGER exhibits 2.33x-129x acceleration, 45% reduction in cross-machines communication, and > 10% effectiveness improvement in downstream tasks.

We address the problem of causal effect estimation in the presence of unobserved confounding, but where proxies for the latent confounder(s) are observed. We propose two kernel-based methods for nonlinear causal effect estimation in this setting: (a) a two-stage regression approach, and (b) a maximum moment restriction approach. We focus on the proximal causal learning setting, but our methods can be used to solve a wider class of inverse problems characterised by a Fredholm integral equation. In particular, we provide a unifying view of two-stage and moment restriction approaches for solving this problem in a nonlinear setting. We provide consistency guarantees for each algorithm, and we demonstrate these approaches achieve competitive results on synthetic data and data simulating a real-world task. In particular, our approach outperforms earlier methods that are not suited to leveraging proxy variables.