Query optimizer is at the heart of the database systems. Cost-based optimizer studied in this paper is adopted in almost all current database systems. A cost-based optimizer introduces a plan enumeration algorithm to find a (sub)plan, and then uses a cost model to obtain the cost of that plan, and selects the plan with the lowest cost. In the cost model, cardinality, the number of tuples through an operator, plays a crucial role. Due to the inaccuracy in cardinality estimation, errors in cost model, and the huge plan space, the optimizer cannot find the optimal execution plan for a complex query in a reasonable time. In this paper, we first deeply study the causes behind the limitations above. Next, we review the techniques used to improve the quality of the three key components in the cost-based optimizer, cardinality estimation, cost model, and plan enumeration. We also provide our insights on the future directions for each of the above aspects.

Bio: Pier Paolo Ippolito is a SAS Data Scientist and MSc in Artificial Intelligence graduate from the University of Southampton. He has a strong interest in AI advancements and machine learning applications (such as finance and medicine).

This study introduces the Territory Design for Dynamic Multi-Period Vehicle Routing Problem with Time Windows (TD-DMPVRPTW), motivated by a real-world application at a food company's distribution center. This problem deals with the design of contiguous and compact territories for delivery of orders from a depot to a set of customers, with time windows, over a multi-period planning horizon. Customers and their demands vary dynamically over time. The problem is modeled as a mixed-integer linear program (MILP) and solved by a proposed heuristic. The heuristic solutions are compared with the proposed MILP solutions on a set of small artificial instances and the food company's solutions on a set of real-world instances. Computational results show that the proposed algorithm can yield high-quality solutions within moderate running times.

Mathematics: If you want to thrive in the field of data science then you need to have a certain familiarity with calculus, probability, linear algebra, and mathematics. Various standard models are essential to construct ML algorithms. In general, a data scientist should know something about probability and statistics theory as the rest depends on the job you apply for. Computer science: It is a study dealing with software systems and includes their theory, development, design, and application. It takes a scientific approach to do computation and carry out its applications. Computer science is considered as a foundation that makes achievements and obtaining more knowledge in the field easier.

Spectral methods have emerged as a simple yet surprisingly effective approach for extracting information from massive, noisy and incomplete data. In a nutshell, spectral methods refer to a collection of algorithms built upon the eigenvalues (resp. singular values) and eigenvectors (resp. singular vectors) of some properly designed matrices constructed from data. A diverse array of applications have been found in machine learning, data science, and signal processing. Due to their simplicity and effectiveness, spectral methods are not only used as a stand-alone estimator, but also frequently employed to initialize other more sophisticated algorithms to improve performance. While the studies of spectral methods can be traced back to classical matrix perturbation theory and methods of moments, the past decade has witnessed tremendous theoretical advances in demystifying their efficacy through the lens of statistical modeling, with the aid of non-asymptotic random matrix theory. This monograph aims to present a systematic, comprehensive, yet accessible introduction to spectral methods from a modern statistical perspective, highlighting their algorithmic implications in diverse large-scale applications. In particular, our exposition gravitates around several central questions that span various applications: how to characterize the sample efficiency of spectral methods in reaching a target level of statistical accuracy, and how to assess their stability in the face of random noise, missing data, and adversarial corruptions? In addition to conventional $\ell_2$ perturbation analysis, we present a systematic $\ell_{\infty}$ and $\ell_{2,\infty}$ perturbation theory for eigenspace and singular subspaces, which has only recently become available owing to a powerful "leave-one-out" analysis framework.

Generative moment matching networks (GMMNs) are suggested for modeling the cross-sectional dependence between stochastic processes. The stochastic processes considered are geometric Brownian motions and ARMA-GARCH models. Geometric Brownian motions lead to an application of pricing American basket call options under dependence and ARMA-GARCH models lead to an application of simulating predictive distributions. In both types of applications the benefit of using GMMNs in comparison to parametric dependence models is highlighted and the fact that GMMNs can produce dependent quasi-random samples with no additional effort is exploited to obtain variance reduction.

Linear algebra, via the use of matrices and vectors, along with linear algebra libraries (such as NumPy in Python), allows us to perform a large number of calculations in a more computationally efficient way while using simpler code. Knowing at least the numeric operations of linear algebra is crucial to further understanding what happens in our machine learning models. Although having the geometric intuition behind linear algebra can be incredibly useful in visualizing the operations we will discuss below, it is not required to understand most machine learning algorithms. In this tutorial, we will discuss scalars, vectors, matrices, matrix-matrix addition and subtraction, scalar multiplication and division, matrix-vector multiplication, matrix-matrix multiplication, identity matrices, matrix inverses, and matrix transposes. In addition, we will very briefly discuss some of the geometric intuition behind some of these numeric operations.

Boolean matrix factorization (BMF) is a variant of the standard matrix factorization problem in the Boolean semiring: given a binary matrix, the task is to find two smaller binary matrices so that their product, taken over the Boolean semiring, is as close to the original matrix as possible. Because the matrix product is not done over a field but over a semiring, many standard matrix factorization techniques fail to work. Indeed, finding the best Boolean factorization is computationally hard. The computational hardness of the problem has not prevented people from studying it. In psychometrics, some of the first algorithms appeared in the 1980's (see Bělohlávek and Trnecka (2018)). Even before that, mathematicians studying combinatorics had studied the "Boolean linear algebra" (Kim, 1982; Monson et al., 1995).

The large communication cost for exchanging gradients between different nodes significantly limits the scalability of distributed training for large-scale learning models. Motivated by this observation, there has been significant recent interest in techniques that reduce the communication cost of distributed Stochastic Gradient Descent (SGD), with gradient sparsification techniques such as top-k and random-k shown to be particularly effective. The same observation has also motivated a separate line of work in distributed statistical estimation theory focusing on the impact of communication constraints on the estimation efficiency of different statistical models. The primary goal of this paper is to connect these two research lines and demonstrate how statistical estimation models and their analysis can lead to new insights in the design of communication-efficient training techniques. We propose a simple statistical estimation model for the stochastic gradients which captures the sparsity and skewness of their distribution. The statistically optimal communication scheme arising from the analysis of this model leads to a new sparsification technique for SGD, which concatenates random-k and top-k, considered separately in the prior literature. We show through extensive experiments on both image and language domains with CIFAR-10, ImageNet, and Penn Treebank datasets that the concatenated application of these two sparsification methods consistently and significantly outperforms either method applied alone.

We present new algorithms and fast implementations to find efficient approximations for modelling stochastic processes. For many numerical computations it is essential to develop finite approximations for stochastic processes. While the goal is always to find a finite model, which represents a given knowledge about the real data process as accurate as possible, the ways of estimating the discrete approximating model may be quite different: (i) if the stochastic model is known as a solution of a stochastic differential equation, e.g., one may generate the scenario tree directly from the specified model; (ii) if a simulation algorithm is available, which allows simulating trajectories from all conditional distributions, a scenario tree can be generated by stochastic approximation; (iii) if only some observed trajectories of the scenario process are available, the construction of the approximating process can be based on non-parametric conditional density estimates.