Machine Learning research, as most of Statistics, heavily relies on the concept of a data-generating probability distribution. The standard presumption is that since data points are `sampled from' such a distribution, one can learn from observed data about this distribution and, thus, predict future data points which, it is presumed, are also drawn from it. Drawing on scholarship across disciplines, we here argue that this framework is not always a good model. Not only do such true probability distributions not exist; the framework can also be misleading and obscure both the choices made and the goals pursued in machine learning practice. We suggest an alternative framework that focuses on finite populations rather than abstract distributions; while classical learning theory can be left almost unchanged, it opens new opportunities, especially to model sampling. We compile these considerations into five reasons for modelling machine learning -- in some settings -- with finite populations rather than generative distributions, both to be more faithful to practice and to provide novel theoretical insights.

The Poisson process, especially the nonhomogeneous Poisson process (NHPP), is an essentially important counting process with numerous real-world applications. Up to date, almost all works in the literature have been on the estimation of NHPPs with infinite data using non-data driven binning methods. In this paper, we formulate the problem of estimation of NHPPs from finite and limited data as a learning generalization problem. We mathematically show that while binning methods are essential for the estimation of NHPPs, they pose a threat of overfitting when the amount of data is limited. We propose a framework for regularized learning of NHPPs with two new adaptive and data-driven binning methods that help to remove the ad-hoc tuning of binning parameters. Our methods are experimentally tested on synthetic and real-world datasets and the results show their effectiveness.

In this paper we present a Twin Parametric-Margin Support Vector Machine (TPMSVM) model to tackle the problem of multiclass classification. In the spirit of one-versus-all paradigm, for each class we construct a classifier by solving a TPMSVM-type model. Once all classifiers have been determined, they are combined into an aggregate decision function. We consider the cases of both linear and nonlinear kernel-induced classifiers. In addition, we robustify the proposed approach through robust optimization techniques. Indeed, in real-world applications observations are subject to measurement errors and noise, affecting the quality of the solutions. Consequently, data uncertainties need to be included within the model in order to prevent low accuracies in the classification process. Preliminary computational experiments on real-world datasets show the good performance of the proposed approach.

A new regression technique based on Vapnik's concept of support vectors is introduced. We compare support vector regression (SVR) with a committee regression technique (bagging) based on regression trees and ridge regression done in feature space. On the basis of these experiments, it is expected that SVR will have advantages in high dimensionality space because SVR optimization does not depend on the dimensionality of the input space.

Vapnik's result that the expectation of the generalisation error ofthe opti(cid:173) mal hyperplane is bounded by the expectation of the ratio of the number of support vectors to the number of training examples is extended to a broad class of kernel machines. The class includes Support Vector Ma(cid:173) chines for soft margin classification and regression, and Regularization Networks with a variety of kernels and cost functions. We show that key inequalities in Vapnik's result become equalities once "the classification error" is replaced by "the margin error", with the latter defined as an in(cid:173) stance with positive cost. In particular we show that expectations of the true margin error and the empirical margin error are equal, and that the sparse solutions for kernel machines are possible only if the cost function is "partially" insensitive.

In contrast to inductive learning, in the transductive setting the learner is given both the training and test sets prior to learning. The goal of the learner is to infer (or "transduce") the labels of the test points. The transduction setting was introduced by Vapnik [1, 2] who proposed basic bounds and an algorithm for this setting. Clearly, inferring the labels of points in the test set can be done using an inductive scheme. However, as pointed out in [2], it makes little sense to solve an easier problem by'reducing' it to a much more difficult one. In particular, the prior knowledge carried by the (unlabeled) test points can be incorporated into an algorithm, potentially leading to superior performance. Indeed, a number of papers have demonstrated empirically that transduction can offer substantial advantage over induction whenever the training set is small or moderate (see e.g.

Here, I'll be giving an overview and theoretical concepts of the statistical learning. Supervised learning can play a key role in learning from examples. From this algorithm, useful information can be easily extracted from large datasets, the problem of learning from examples consecutively involves approximating functions from a sparse and noisy data. In supervised learning, network is trained on a dataset of the form, T {xk, dk} from k 1 to Q. It is observed that using MLP multilayer perceptron with sufficient number of hidden neurons, it is possible to approximate a given function to any arbitrary degree of accuracy.

Domain generalization (DG) aims at generalizing a classifier trained on multiple source domains to an unseen target domain with domain shift. A common pervasive theme in existing DG literature is domain-invariant representation learning with various invariance assumptions. However, prior works restrict themselves to a radical assumption for realworld challenges: If a mapping induced by a deep neural network (DNN) could align the source domains well, then such a mapping aligns a target domain as well. In this paper, we simply take DNNs as feature extractors to relax the requirement of distribution alignment. Specifically, we put forward a novel angular invariance and the accompanied norm shift assumption. Based on the proposed term of invariance, we propose a novel deep DG method called Angular Invariance Domain Generalization Network (AIDGN). The optimization objective of AIDGN is developed with a von-Mises Fisher (vMF) mixture model. Extensive experiments on multiple DG benchmark datasets validate the effectiveness of the proposed AIDGN method.

There has been growing interest in generalization performance of large multilayer neural networks that can be trained to achieve zero training error, while generalizing well on test data. This regime is known as'second descent' and it appears to contradict the conventional view that optimal model complexity should reflect an optimal balance between underfitting and overfitting, i.e., the bias-variance trade-off. This paper presents a VC-theoretical analysis of double descent and shows that it can be fully explained by classical VC-generalization bounds. We illustrate an application of analytic VC-bounds for modeling double descent for classification, using empirical results for several learning methods, such as SVM, Least Squares, and Multilayer Perceptron classifiers. In addition, we discuss several reasons for the misinterpretation of VC-theoretical results in Deep Learning community. There have been many recent successful applications of Deep Learning (DL). However, at present, various DL methods are driven mainly by heuristic improvements, while theoretical and conceptual understanding of this technology remains limited. For example, large neural networks can be trained to fit available data (achieving zero training error) and still achieve good generalization for test data. This contradicts the conventional statistical wisdom that overfitting leads to poor generalization. This phenomenon has been systematically described by Belkin et al. (2019) who introduced the term'double descent' and pointed out the difference between the classical regime (first descent) and the modern one (second descent).

We present a series of new and more favorable margin-based learning guarantees that depend on the empirical margin loss of a predictor. We give two types of learning bounds, both distribution-dependent and valid for general families, in terms of the Rademacher complexity or the empirical $\ell_\infty$ covering number of the hypothesis set used. Furthermore, using our relative deviation margin bounds, we derive distribution-dependent generalization bounds for unbounded loss functions under the assumption of a finite moment. We also briefly highlight several applications of these bounds and discuss their connection with existing results.