Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plucker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.

https://www.springer.com/productFlyer_978-0-387-22356-8.pdf?SGWID=0-0-1297-34952457-0

Combinatorial Commutative Algebra

Mobile users typically have high demand on localized and location-based information services. To always retrieve the localized data from the remote cloud, however, tends to be inefficient, which motivates fog computing. The fog computing, also known as edge computing, extends cloud computing by deploying localized computing facilities at the premise of users, which prestores cloud data and distributes to mobile users with fast-rate local connections. As such, fog computing introduces an intermediate fog layer between mobile users and cloud, and complements cloud computing toward low-latency high-rate services to mobile users. In this fundamental framework, it is important to study the interplay and cooperation between the edge (fog) and the core (cloud). In this paper, the tradeoff between power consumption and transmission delay in the fog-cloud computing system is investigated. We formulate a workload allocation problem which suggests the optimal workload allocations between fog and cloud toward the minimal power consumption with the constrained service delay. The problem is then tackled using an approximate approach by decomposing the primal problem into three subproblems of corresponding subsystems, which can be, respectively, solved. Finally, based on simulations and numerical results, we show that by sacrificing modest computation resources to save communication bandwidth and reduce transmission latency, fog computing can significantly improve the performance of cloud computing.

Optimal Workload Allocation in Fog-Cloud Computing Toward Balanced Delay and Power Consumption

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

/pdf/mixed-integer-nonlinear-optimization-3yf57zxyv3.pdf

Mixed-integer nonlinear optimization

Associated to any toric ideal are two special generating sets: the universal Grobner basis and the Graver basis, which encode polyhedral and combinatorial properties of the ideal, or equivalently, its defining matrix. If the two sets coincide, then the complexity of the Graver bases of the higher Lawrence liftings of the toric matrices is bounded. While a general classification of all matrices for which both sets agree is far from known, we identify all such matrices within two families of nonunimodular matrices, namely, those defining rational normal scrolls and those encoding homogeneous primitive colored partition identities. This also allows us to show that higher Lawrence liftings of matrices with fixed Grobner and Graver complexities do not preserve equality of the two bases. The proof of our classification combines computations with the theoretical tool of Graver complexity of a pair of matrices.

/pdf/universal-grobner-bases-of-colored-partition-identities-3gzq984tem.pdf

Universal Gröbner Bases of Colored Partition Identities

Both the combinatorial and the circuit diameter of polyhedra are of interest to the theory of linear programming for their intimate connection to a best-case performance of linear programming algorithms. We study the diameters of dual network flow polyhedra associated to b-flows on directed graphs $$G=(V,E)$$G=(V,E) and prove quadratic upper bounds for both of them: the minimum of $$(|V|-1)\cdot |E|$$(|V|-1)·|E| and $$\frac{1}{6}|V|^3$$16|V|3 for the combinatorial diameter, and $$\frac{|V|\cdot (|V|-1)}{2}$$|V|·(|V|-1)2 for the circuit diameter. Previously, bounds on these diameters have only been known for bipartite graphs. The situation is much more involved for general graphs. In particular, we construct a family of dual network flow polyhedra with members that violate the circuit diameter bound for bipartite graphs by an arbitrary additive constant.

Quadratic diameter bounds for dual network flow polyhedra

This paper presents three kinds of algebraic-analytic algorithms for solving integer and mixed integer programming problems. We report both theoretical and experimental results. We use the generating function techniques introduced by A. Barvinok.

Three kinds of integer programming algorithms based on Barvinok's rational functions

The multi-transshipment problem is NP-hard already for two commodities over bipartite networks. Nonetheless, using our recent theory of n-fold integer programming and extensions developed herein, we are able to establish the polynomial time solvability of the problem in two broad situations. First, for any fixed number of commodities and number of suppliers, we solve the problem over bipartite networks with variable number of consumers in polynomial time. This is very natural in operations research applications where few facilities serve many customers. Second, for every fixed network, we solve the problem with variable number of commodities in polynomial time.

N-fold integer programming and nonlinear multi-transshipment

In this paper, we present a construction that turns certain relations on Graver basis elements of an M-fold matrix $${{A^{(M)}}}$$
 into relations on Graver basis elements of an $${(M+1)}$$
 -fold matrix $${{A^{(M+1)}}}$$
 . In doing so, we strengthen the bound on the Graver complexity of the M-fold matrix $${{A_{3\times{M}}}}$$
 from $${{g(A_{3\times{M}}) \geq 17\cdot2^{M-3}-7}}$$
 (Berstein and Onn) to $${{g(A_{3\times{M}}) \geq 24\cdot2^{M-3}-21}}$$
 , for $${M \geq 4}$$
 . Moreover, we give a lower bound on the Graver complexity $${{g(A^{(M)})}}$$
 of general $${M}$$
 -fold matrices $${{A^{(M)}}}$$
 and we prove that the bound for $${g(A_{3\times{M}})}$$
 is not tight.

Raymond Hemmecke

Papers

Universal Gröbner Bases of Colored Partition Identities

Quadratic diameter bounds for dual network flow polyhedra

Three kinds of integer programming algorithms based on Barvinok's rational functions

N-fold integer programming and nonlinear multi-transshipment

Lower Bounds on the Graver Complexity of M -Fold Matrices