Over the last few decades, many distinct lines of research aimed at automating mathematics have been developed, including computer algebra systems (CASs) for mathematical modelling, automated theorem provers for first-order logic, SAT/SMT solvers aimed at program verification, and higher-order proof assistants for checking mathematical proofs. More recently, some of these lines of research have started to converge in complementary ways. One success story is the combination of SAT solvers and CASs (SAT+CAS) aimed at resolving mathematical conjectures. Many conjectures in pure and applied mathematics are not amenable to traditional proof methods. Instead, they are best addressed via computational methods that involve very large combinatorial search spaces. SAT solvers are powerful methods to search through such large combinatorial spaces---consequently, many problems from a variety of mathematical domains have been reduced to SAT in an attempt to resolve them. However, solvers traditionally lack deep repositories of mathematical domain knowledge that can be crucial to pruning such large search spaces. By contrast, CASs are deep repositories of mathematical knowledge but lack efficient general search capabilities. By combining the search power of SAT with the deep mathematical knowledge in CASs we can solve many problems in mathematics that no other known methods seem capable of solving. We demonstrate the success of the SAT+CAS paradigm by highlighting many conjectures that have been disproven, verified, or partially verified using our tool MathCheck. These successes indicate that the paradigm is positioned to become a standard method for solving problems requiring both a significant amount of search and deep mathematical reasoning. For example, the SAT+CAS paradigm has recently been used by Heule, Kauers, and Seidl to find many new algorithms for $3\times3$ matrix multiplication.

Reviewed by Douglas Farenick, University of Regina Undergraduate mathematics textbooks are not what they used to be, and Gilbert Strang's superb new edition of Introduction to Linear Algebra is an example of everything that a modern textbook could possibly be, and more. First, let us consider the book itself. As with his classic Linear Algebra and its Applications (Academic Press) from forty years ago, Strang's new edition of Introduction to Linear Algebra keeps one eye on the theory, the other on applications, and has thestated goal of "opening linear algebra to the world" (Preface, page x).Aimed at the serious undergraduate student - though not just thoseundergraduates who fill the lecture halls of MIT, Strang's homeinstitution - the writing is engaging and personal, and the presentation is exceptionally clear and informative (even seasoned instructors maybenefit from Strang's insights). The first six chapters offer atraditional first course that covers vector algebra and geometry,systems of linear equations, vector spaces and subspaces, orthogonality, determinants, and eigenvalues and eigenvectors. The next three chapters are devoted to the singular value decomposition, lineartransformations, and complex numbers and complex matrices, followed bychapters that address a wide range of contemporary applications andcomputational issues. The book concludes with a brief but cogenttreatment of linear statistical analysis. I would like to stress that there is arichness to the material that goes beyond most texts at this level.Included are guides to websites and to OpenCourseWare, which I shallcomment upon later in this review.

Linear algebra is an important subject not only to mathematicians but also to an ever widening range of disciplines.

From machine learning and data science to engineering and finance, linear algebra is an important prerequisite for the careers of today and of the future. There aren't many resources out there that give simple detailed examples and that walk you through the topics step by step. Many resources out there are either too dry or too difficult. This book aims to teach linear algebra step-by-step with examples that are simple but concrete.

A neural network is built around simple linear equations like Y WX B, which contain something called as weights W. These weights multiply with the input X and play a crucial in how the model predicts. The prediction scores can go downhill if a wrong weight gets updated and as the network gets deeper i.e addition of more layers (columns of connected nodes), the error magnifies and the results miss the target. There is no denying the fact that building ML algorithms from scratch is a thing of the past. Modern-day programming platforms offer plenty of options where a single line of code would invoke a monstrous algorithm in the background. This works for those who want to get an idea of how ML plays out.

Finite-horizon sequential decision problems arise naturally in many machine learning contexts; examples include Bayesian optimization and Bayesian quadrature. Computing the optimal policy for such problems requires solving Bellman equations, which are generally intractable. Most existing work resorts to myopic approximations by limiting the horizon to only a single time-step, which can perform poorly in balancing exploration and exploitation. We propose a general framework for efficient, nonmyopic approximation of the optimal policy by drawing a connection between the optimal adaptive policy and its non-adaptive counterpart. Our proposal is to compute an optimal batch of points, then select a single point from within this batch to evaluate. We realize this idea for both Bayesian optimization and Bayesian quadrature and demonstrate that our proposed method significantly outperforms common myopic alternatives on a variety of tasks.

It depends on what you are trying to do with this model. The question, as currently posed, might be better suited for /r/statistics. If you're just trying to maximize predictive accuracy, why use polynomial regression at all? Try a boosting tree. If you really care about explaining this model, you're doing statistical inference and this question is not well posed. Would it be "wrong" do use lasso to select your polynomial degree?

It was written as an advanced text to serve as a basis for a graduate course, and/or as a reference to the researcher diving into this fascinating world at the intersection of optimization and machine learning. Such a course was given at the Technion in the years 2010-2014 with slight variations from year to year, and later at Princeton University in the years 2015-2016. The core material in these courses is fully covered in this book, along with exercises that allow the students to complete parts of proofs, or that were found illuminating and thought-provoking. Most of the material is given with examples of applications, which are interlaced throughout different topics. These include prediction from expert advice, portfolio selection, matrix completion and recommendation systems, SVM training and more.

Network complexity has been studied for over half a century and has found a wide range of applications. Many methods have been developed to characterize and estimate the complexity of networks. However, there has been little research with statistical guarantees. In this paper, we develop a statistical theory of graph complexity in a general model of random graphs, the so-called graphon model. Given a graphon, we endow the latent space of the nodes with the so-called neighborhood distance that measures the propensity of two nodes to be connected with similar nodes. Our complexity index is then based on the covering number and the Minkowski dimension of (a purified version of) this metric space. Although the latent space is not identifiable, these indices turn out to be identifiable. This notion of complexity has simple interpretations on popular examples of random graphs: it matches the number of communities in stochastic block models; the dimension of the Euclidean space in random geometric graphs; the regularity of the link function in H\"older graphon models. From a single observation of the graph, we construct an estimator of the neighborhood-distance and show universal non-asymptotic bounds for its risk, matching minimax lower bounds. Based on this estimated distance, we compute the corresponding covering number and Minkowski dimension and we provide optimal non-asymptotic error bounds for these two plug-in estimators.

We prove that the norm version of the adaptive stochastic gradient method (AdaGrad-Norm) achieves a linear convergence rate for a subset of either strongly convex functions or non-convex functions that satisfy the Polyak-Lojasiewicz (PL) inequality. The paper introduces the notion of Restricted Uniform Inequality of Gradients (RUIG), which describes the uniform lower bound for the norm of the stochastic gradients with respect to the distance to the optimal solution. RUIG plays the key role in proving the robustness of AdaGrad-Norm to its hyper-parameter tuning. On top of RUIG, we develop a novel two-stage framework to prove linear convergence of AdaGrad-Norm without knowing the parameters of the objective functions: Stage I: the step-size decrease fast such that it reaches to Stage II; Stage II: the step-size decreases slowly and converges. This framework can likely be extended to other adaptive stepsize algorithms. The numerical experiments show desirable agreement with our theories.