A Co-Evolutionary Computing for Statistical Learning Theory

Nelson Goodman’s new riddle of induction forcefully illustrates a challenge that must be confronted by any adequate theory of inductive inference: provide some basis for choosing among alternative hypotheses that fit past data but make divergent predictions. One response to this challenge is to distinguish among alternatives by means of some epistemically significant characteristic beyond fit with the data. Statistical learning theory takes this approach by showing how a concept similar to Popper’s notion of degrees of testability is linked to minimizing expected predictive error. In contrast, formal learning theory appeals to Ockham’s razor, which it justifies by reference to the goal of enhancing efficient convergence to the truth. In this essay, I show that, despite their differences, statistical and formal learning theory yield precisely the same result for a class of inductive problems that I call strongly VC ordered, of which Goodman’s riddle is just one example.

The best algorithm for a computational problem generally depends on the “relevant inputs,” a concept that depends on the application domain and often defies formal articulation. While there is a large body of literature on empirical approaches to selecting the best algorithm for a given application domain, there has been surprisingly little theoretical analysis of the problem. This paper adapts concepts from statistical and online learning theory to reason about application-specific algorithm selection. Our models capture several state-of-the-art empirical and theoretical approaches to the problem, ranging from self-improving algorithms to empirical performance models, and our results identify conditions under which these approaches are guaranteed to perform well. We present one framework that models algorithm selection as a statistical learning problem, and our work here shows that dimension notions from statistical learning theory, historically used to measure the complexity of classes of binary- and re...

This paper uses statistical learning theory and big data analysis to study the energy consumption structure of China from qualitative and quantitative aspects during COVID-19. According to the domestic and foreign scholars’ research on the optimization of energy consumption structure, the carbon emission factor is considered in the optimization of energy consumption structure. Taking the minimum energy consumption cost and carbon dioxide emission as the objective function, the carbon dioxide emission is taken as the objective function, and the total energy consumption and various energy consumption proportions as the constraint conditions, the multi-objective planning method is used to evaluate the energy consumption structure of China. The optimization model of source consumption structure is analyzed, and the medium and long-term energy transformation path and optimization model under low-carbon constraints are studied. Combined with the experimental algorithms related to big data, it is concluded that China’s economic development mainly depends on a large amount of energy consumption during the COVID-19 period. On this basis, some suggestions are put forward to realize the sustainable development of China’s economy and energy.

Although statistical learning theory provides a robust framework to understand supervised learning, many theoretical aspects of deep learning remain unclear;in particular, how different architectures may lead to inductive bias when trained using gradient-based methods. The goal of these lectures is to provide an overview of some of the main questions that arise when attempting to understand deep learning from a learning theory perspective. After a brief reminder on statistical learning theory and stochastic optimization, we discuss implicit bias in the context of benign overfitting. We then move to a general description of the mirror descent algorithm, showing how we may go back and forth between a parameter space and the corresponding function space for a given learning problem, as well as how the geometry of the learning problem may be represented by a metric tensor. Building on this framework, we provide a detailed study of the implicit bias of gradient descent on linear diagonal networks for various regression tasks, showing how the loss function, scale of parameters at initialization and depth of the network may lead to various forms of implicit bias; in particular, transitioning between kernel and feature learning regimes.

With the increasing demand for high-resolution remote sensing images for mapping and monitoring the Earth’s environment, geometric positioning accuracy improvement plays a significant role in the image preprocessing step. Based on the statistical learning theory, we propose a new method to improve the geometric positioning accuracy without ground control points (GCPs). Multi-temporal images from the ZY-3 satellite are tested and the bias-compensated rational function model (RFM) is applied as the block adjustment model in our experiment. An easy and stable weight strategy and the fast iterative shrinkage-thresholding (FIST) algorithm which is widely used in the field of compressive sensing are improved and utilized to define the normal equation matrix and solve it. Then, the residual errors after traditional block adjustment are acquired and tested with the newly proposed inherent error compensation model based on statistical learning theory. The final results indicate that the geometric positioning accuracy of ZY-3 satellite imagery can be improved greatly with our proposed method.

Statistical learning theory is regarded as an appropriate theory to deal with learning problems on small samples, and it has now become a novel research interest of the machine learning field. However, the theory is based on real-valued random samples and established on probability measure space; it rarely deals with learning problems based on fuzzy random samples and established on Sugeno measure space. It is well known that fuzzy random samples and Sugeno measure space are interesting and important extensions of real-valued random samples and probability measure space, respectively. Therefore, the statistical learning theory based on fuzzy random samples in Sugeno measure space is further discussed in this paper. Firstly, based on definitions of the distribution function and the expected value of fuzzy random variables in Sugeno measure space, the Hoeffding inequality of fuzzy random variables is proved. Secondly, for the sake of completeness of the paper, the key theorem of learning theory based on fuzzy random samples in Sugeno measure space is introduced. Finally, the bounds on the rate of uniform convergence of a learning process based on fuzzy random samples in Sugeno measure space are constructed.

<title>Abstract</title> Brain morphology emerges from the interplay of genetic programming and mechanical forces. However, its fractal-like folding patterns, regional anisotropy, and uneven curvature distributions pose longstanding challenges for quantitative analysis in medical and developmental studies. These difficulties are exacerbated by the limited availability of labeled data for existing statistical learning approaches. Here, we introduce a theory-grounded physics-transfer (PT) learning framework that enables generalization analysis and prediction in complex physical systems by leveraging the continuity of underlying physical laws across different levels of complexity. Specifically, mechanistic insights into nonlinear elasticity of simple, analytically tractable geometries are embedded directly into neural networks and then transferred to brain models through a sequence of physics-anchored domains, providing, for the first time, a formulation that integrates explicit physics constraints into learning theory. In this sense, PT advances classical statistical learning and domain adaptation theories by enabling generalization analysis in regimes where purely statistical assumptions are insufficient. Building on PT, we derive a generalization bound that explains the model’s strong performance in feature characterization and morphogenesis prediction for brain. The results reveal that localized deformation modes, rather than global geometry, dominantly shape cortical folding dynamics. Beyond predictive accuracy, the framework also yields reduced-dimensional evolutionary representations that distill the essential physics of the highly folded cerebral cortex. Validation against medical imaging data and expert clinical assessment demonstrates the promise of physics-aware digital-twin technologies for understanding, diagnosing, and ultimately intervening in the morphological complexity of the developing and diseased brain.

The traditional approach of statistical physics to supervised learning routinely assumes unrealistic generative models for the data: Usually inputs are independent random variables, uncorrelated with their labels. Only recently, statistical physicists started to explore more complex forms of data, such as equally labeled points lying on (possibly low-dimensional) object manifolds. Here we provide a bridge between this recently established research area and the framework of statistical learning theory, a branch of mathematics devoted to inference in machine learning. The overarching motivation is the inadequacy of the classic rigorous results in explaining the remarkable generalization properties of deep learning. We propose a way to integrate physical models of data into statistical learning theory and address, with both combinatorial and statistical mechanics methods, the computation of the Vapnik-Chervonenkis entropy, which counts the number of different binary classifications compatible with the loss class. As a proof of concept, we focus on kernel machines and on two simple realizations of data structure introduced in recent physics literature: k-dimensional simplexes with prescribed geometric relations and spherical manifolds (equivalent to margin classification). Entropy, contrary to what happens for unstructured data, is nonmonotonic in the sample size, in contrast with the rigorous bounds. Moreover, data structure induces a transition beyond the storage capacity, which we advocate as a proxy of the nonmonotonicity, and ultimately a cue of low generalization error. The identification of a synaptic volume vanishing at the transition allows a quantification of the impact of data structure within replica theory, applicable in cases where combinatorial methods are not available, as we demonstrate for margin learning.

In statistical learning theory (SLT), the key theorem and the bounds on the rate of uniform convergence of learning processes provide theoretical basis for the applied research of support vector machine etc., so they play important roles in SLT. In the study of two aspects, samples which we deal with are supposed to be noise-free. But it is not always the case because of the influence of human or environmental factors. With a view of this, we propose and prove the key theorem and discuss the bounds on the rate of uniform convergence of learning processes when samples are corrupted by noise

It has been more than 30 years that statistical learning theory (SLT) has been introduced in the field of machine learning. Its objective is to provide a framework for studying the problem of inference that is of gaining knowledge, making predictions, making decisions or constructing models from a set of data. Support Vector Machine, a method based on SLT, then emerged and becoming a widely accepted method for solving real-world problems. This paper overviews the pattern recognition techniques and describes the state of art in SVM in the field of pattern recognition.

Using information technology such as search engine for collecting and monitoring of network public opinion is a practical and effective method. This paper puts forward an improved algorithm of SVM classifier based on incremental learning online, and then implements a topic crawler system for network public opinion and to grab the public opinion.

Model complexity of Genetic Programming (GP) as a learning machine is currently attracting considerable interest from the research community. Here we provide an up-to-date overview of the research concerning complexity measure techniques in GP learning. The scope of this review includes methods based on information theory techniques, such as the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC); plus those based on statistical machine learning theory on generalization error bound, namely, Vapnik-Chervonenkis (VC) theory; and some based on structural complexity. The research contributions from each of these are systematically summarized and compared, allowing us to clearly define existing research challenges, and to highlight promising new research directions. The findings of this review provides valuable insights into the current GP literature and is a good source for anyone who is interested in the research on model complexity and applying statistical learning theory to GP.

The article applies statistical learning theory to the supervised learning problem. Pattern recognition is covered, including Vapnik-Chervonenkis (VC) theory and the implications for support vector machines (SVMs), neural networks and decision trees. Real predictions are given for scale-sensitive dimensions. The article concludes by analysing large margin classification.

By now it is known that several problems in control and matrix theory are NP-hard. These include matrix problems that arise in control theory, as well as other problems in the robustness analysis and synthesis of control systems. These negative results force us to modify our notion of “solving” a given problem. If we cannot solve a problem exactly because it is NP-hard, then we must settle for solving it approximately. If we cannot solve all instances of a problem, we must settle for solving “almost all” instances of a problem. An approach that is recently gaining popularity is that of using randomized algorithms. The notion of a randomized algorithm as defined here is somewhat different from that in the computer science literature, and enlarges the class of problems that can be efficiently solved. We begin with the premise that many problems in robustness analysis and synthesis can be formulated as the minimization of an objective function with respect to the controller parameters. It is argued that, in order to assess the performance of a controller as the plant varies over a prespecified family, it is better to use the average performance of the controller as the objective function to be minimized, rather than its worstcase performance, as the worst-case objective function usually leads to rather conservative designs. Then it is shown that a property from statistical learning theory known as uniform convergence of empirical means (UCEM) plays an important role in allowing us to construct efficient randomized algorithms for a wide variety of controller synthesis problems. In particular, whenever the UCEM property holds, there exists an efficient (i.e., polynomial-time) randomized algorithm. Using very recent results in VC-dimension theory, it is shown that the UCEM property holds in several problems such as robust stabilization and weighted H∞-norm minimization. Hence it is possible to solve such problems efficiently using randomized algorithms. The paper is concluded by showing that the statistical learning methodology is also applicable to some NP-hard matrix problems.

Abstract: Scaling laws have emerged as a fundamental principle in characterizing the performance of transformer architectures across domains, revealing power-law relationships between model size, data availability, and predictive accuracy. While these laws are well-established in high-dimensional, data-rich environments, their validity in low-dimensional settings remains underexplored. This study examines transformer scalability on low-dimensional data through a dual lens of statistical learning theory and approximation theory, thereby establishing a principled framework that extends beyond empirical heuristics. The research problem addressed is the lack of theoretical grounding for scaling laws in constrained feature spaces, where over-parameterization and limited sample regimes alter approximation behaviors. Methodologically, the study integrates theoretical derivations with controlled empirical evaluations on representative low-dimensional datasets, enabling comparison between predicted and observed scaling dynamics. Results indicate that conventional power-law behaviors deviate under low-dimensional constraints, with approximation-theoretic limits exerting dominant influence on generalization. The findings propose reformulated scaling laws that capture these dynamics, demonstrating superior predictive alignment with empirical performance. The impact of this study lies in its contribution to a unified theoretical framework of transformer scalability, informing efficient model design in resource-constrained domains such as econometrics, biomedical analysis, and scientific computing. By bridging statistical theory and approximation analysis, this research advances both theoretical understanding and practical applicability of scaling laws across diverse data regimes. Keywords transformer architectures, scaling laws, low-dimensional data, statistical learning theory, approximation theory, generalization, model complexity, power-law behavior, constrained datasets, empirical validation