Linear Algebra is a branch of mathematics that is extremely useful in data science and machine learning. Linear algebra is the most important math skill in machine learning. Most machine learning models can be expressed in matrix form. A dataset itself is often represented as a matrix. Linear algebra is used in data preprocessing, data transformation, and model evaluation.

Game theory has by now found numerous applications in various fields, including economics, industry, jurisprudence, and artificial intelligence, where each player only cares about its own interest in a noncooperative or cooperative manner, but without obvious malice to other players. However, in many practical applications, such as poker, chess, evader pursuing, drug interdiction, coast guard, cyber-security, and national defense, players often have apparently adversarial stances, that is, selfish actions of each player inevitably or intentionally inflict loss or wreak havoc on other players. Along this line, this paper provides a systematic survey on three main game models widely employed in adversarial games, i.e., zero-sum normal-form and extensive-form games, Stackelberg (security) games, zero-sum differential games, from an array of perspectives, including basic knowledge of game models, (approximate) equilibrium concepts, problem classifications, research frontiers, (approximate) optimal strategy seeking techniques, prevailing algorithms, and practical applications. Finally, promising future research directions are also discussed for relevant adversarial games.

Machine learning consists of several algorithms suited for different real-life problems. Anyone with a solid programming foundation can become a good machine learning engineer using ready-made tools, libraries, and models. But if you want to become a real specialist in the field, you cannot escape learning some of the concepts of Linear Algebra. All the magic that happens under the hood of any machine learning algorithm, especially Deep Learning, is mostly Linear Algebra math. So, before I started learning ML, many were dead set that Linear Algebra is a big prerequisite.

Security-constrained unit commitment (SCUC) is solved for power system day-ahead generation scheduling, which is a large-scale mixed-integer linear programming problem and is very computationally intensive. Model reduction of SCUC may bring significant time savings. In this work, a novel approach is proposed to effectively utilize machine learning (ML) to reduce the problem size of SCUC. An ML model using logistic regression (LR) algorithm is proposed and trained with historical nodal demand profiles and the respective commitment schedules. The ML outputs are processed and analyzed to reduce variables and constraints in SCUC. The proposed approach is validated on several standard test systems including IEEE 24-bus system, IEEE 73-bus system, IEEE 118-bus system, synthetic South Carolina 500-bus system and Polish 2383-bus system. Simulation results demonstrate that the use of the prediction from the proposed LR model in SCUC model reduction can substantially reduce the computing time while maintaining solution quality.

We introduce the Poisson Binomial mechanism (PBM), a discrete differential privacy mechanism for distributed mean estimation (DME) with applications to federated learning and analytics. We provide a tight analysis of its privacy guarantees, showing that it achieves the same privacy-accuracy trade-offs as the continuous Gaussian mechanism. Our analysis is based on a novel bound on the R\'enyi divergence of two Poisson binomial distributions that may be of independent interest. Unlike previous discrete DP schemes based on additive noise, our mechanism encodes local information into a parameter of the binomial distribution, and hence the output distribution is discrete with bounded support. Moreover, the support does not increase as the privacy budget $\varepsilon \rightarrow 0$ as in the case of additive schemes which require the addition of more noise to achieve higher privacy; on the contrary, the support becomes smaller as $\varepsilon \rightarrow 0$. The bounded support enables us to combine our mechanism with secure aggregation (SecAgg), a multi-party cryptographic protocol, without the need of performing modular clipping which results in an unbiased estimator of the sum of the local vectors. This in turn allows us to apply it in the private FL setting and provide an upper bound on the convergence rate of the SGD algorithm. Moreover, since the support of the output distribution becomes smaller as $\varepsilon \rightarrow 0$, the communication cost of our scheme decreases with the privacy constraint $\varepsilon$, outperforming all previous distributed DP schemes based on additive noise in the high privacy or low communication regimes.

This article was published as a part of the Data Science Blogathon. Linear Algebra, a branch of mathematics, is very much useful in Data Science. We can mathematically operate on large amounts of data by using Linear Algebra. Most algorithms used in ML use Linear Algebra, especially matrices. As most of the data is represented in matrix form.

Computer algebra can answer various questions about partial differential equations using symbolic algorithms. However, the inclusion of data into equations is rare in computer algebra. Therefore, recently, computer algebra models have been combined with Gaussian processes, a regression model in machine learning, to describe the behavior of certain differential equations under data. While it was possible to describe polynomial boundary conditions in this context, we extend these models to analytic boundary conditions. Additionally, we describe the necessary algorithms for Gröbner and Janet bases of Weyl algebras with certain analytic coefficients. Using these algorithms, we provide examples of divergence-free flow in domains bounded by analytic functions and adapted to observations. Keywords: Gaussian processes boundary conditions Gröbner bases partial differential equations.

Causality is a fundamental part of the scientific endeavour to understand the world. Unfortunately, causality is still taboo in much of psychology and social science. Motivated by a growing number of recommendations for the importance of adopting causal approaches to research, we reformulate the typical approach to research in psychology to harmonize inevitably causal theories with the rest of the research pipeline. We present a new process which begins with the incorporation of techniques from the confluence of causal discovery and machine learning for the development, validation, and transparent formal specification of theories. We then present methods for reducing the complexity of the fully specified theoretical model into the fundamental submodel relevant to a given target hypothesis. From here, we establish whether or not the quantity of interest is estimable from the data, and if so, propose the use of semi-parametric machine learning methods for the estimation of causal effects. The overall goal is the presentation of a new research pipeline which can (a) facilitate scientific inquiry compatible with the desire to test causal theories (b) encourage transparent representation of our theories as unambiguous mathematical objects, (c) to tie our statistical models to specific attributes of the theory, thus reducing under-specification problems frequently resulting from the theory-to-model gap, and (d) to yield results and estimates which are causally meaningful and reproducible. The process is demonstrated through didactic examples with real-world data, and we conclude with a summary and discussion of limitations.

Plotkin, M. Binary codes with specified minimum distance.

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