Scalable Learning of Bayesian Networks Using Feedback Arc Set-Based Heuristics

Bayesianske nettverk er en viktig klasse av probabilistiske grafiske modeller. De består av en struktur (en rettet asyklisk graf) som beskriver betingede uavhengighet mellom stokastiske variabler og deres parametere (lokale sannsynlighetsfordelinger). Med andre ord er Bayesianske nettverk generative modeller som beskriver simultanfordelingene på en kompakt form. Den største utfordringen med å lære et Bayesiansk nettverk skyldes selve strukturen, og på grunn av den kombinatoriske karakteren til asyklisitetsegenskapen er det ingen overraskelse at strukturlæringsproblemet generelt er NP-hardt. Det eksisterer algoritmer som løser dette problemet eksakt: dynamisk programmering og heltalls lineær programmering er de viktigste kandidatene når man ønsker å finne strukturen til små til mellomstore Bayesianske nettverk fra data. På den annen side er heuristikk som bakkeklatringsvarianter ofte brukt når man forsøker å lære strukturen til større nettverk med tusenvis av variabler, selv om disse heuristikkene vanligvis ikke har teoretiske garantier og ytelsen i praksis kan bli uforutsigbar når man arbeider med storskala læring. Denne oppgaven tar for seg utvikling av skalerbare metoder som takler det strukturlæringsproblemet av Bayesianske nettverk, samtidig som det forsøkes å opprettholde et nivå av teoretisk kontroll. Dette ble oppnådd ved bruk av relaterte kombinatoriske problemer, nemlig det maksimale asykliske subgrafproblemet (maximum acyclic subgraph) og det duale problemet (feedback arc set). Selv om disse problemene er NP-harde i seg selv, er de betydelig mer håndterbare i praksis. Denne oppgaven utforsker måter å kartlegge Bayesiansk nettverksstrukturlæring til maksimale asykliske subgrafforekomster og trekke ut omtrentlige løsninger for det første problemet, basert på løsninger oppnådd for det andre. Vår forskning tyder på at selv om økt skalerbarhet kan oppnås på denne måten, er det adskillig mer utfordrende å opprettholde den teoretisk forståelsen med denne tilnærmingen. Videre fant vi ut at å lære strukturen til Bayesianske nettverk basert på maksimal asyklisk subgraf kanskje ikke er den beste metoden generelt, men vi identifiserte en kontekst - lineære strukturelle ligningsmodeller - der vi eksperimentelt kunne validere fordelene med denne tilnærmingen, som fører til rask og skalerbar identifisering av strukturen og med mulighet til å lære komplekse strukturer på en måte som er konkurransedyktig med moderne metoder.Bayesian networks form an important class of probabilistic graphical models. They consist of a structure (a directed acyclic graph) expressing conditional independencies among random variables, as well as parameters (local probability distributions). As such, Bayesian networks are generative models encoding joint probability distributions in a compact form. The main difficulty in learning a Bayesian network comes from the structure itself, owing to the combinatorial nature of the acyclicity property; it is well known and does not come as a surprise that the structure learning problem is NP-hard in general. Exact algorithms solving this problem exist: dynamic programming and integer linear programming are prime contenders when one seeks to recover the structure of small-to-medium sized Bayesian networks from data. On the other hand, heuristics such as hill climbing variants are commonly used when attempting to approximately learn the structure of larger networks with thousands of variables, although these heuristics typically lack theoretical guarantees and their performance in practice may become unreliable when dealing with large scale learning. This thesis is concerned with the development of scalable methods tackling the Bayesian network structure learning problem, while attempting to maintain a level of theoretical control. This was achieved via the use of related combinatorial problems, namely the maximum acyclic subgraph problem and its dual problem the minimum feedback arc set problem. Although these problems are NP-hard themselves, they exhibit significantly better tractability in practice. This thesis explores ways to map Bayesian network structure learning into maximum acyclic subgraph instances and extract approximate solutions for the first problem, based on the solutions obtained for the second. Our research suggests that although increased scalability can be achieved this way, maintaining theoretical understanding based on this approach is much more challenging. Furthermore, we found that learning the structure of Bayesian networks based on maximum acyclic subgraph/minimum feedback arc set may not be the go-to method in general, but we identified a setting - linear structural equation models - in which we could experimentally validate the benefits of this approach, leading to fast and scalable structure recovery with the ability to learn complex structures in a competitive way compared to state-of-the-art baselines.Doktorgradsavhandlin

Tropicalizing the simplex algorithm

We develop a tropical analog of the simplex algorithm for linear programming. In particular, we obtain a combinatorial algorithm to perform one tropical pivoting step, including the computation of reduced costs, in O(n(m+n)) time, where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39 pages, 9 figures, 4 algorithm

Classes of Intersection Digraphs with Good Algorithmic Properties

While intersection graphs play a central role in the algorithmic analysis of hard problems on undirected graphs, the role of intersection digraphs in algorithms is much less understood. We present several contributions towards a better understanding of the algorithmic treatment of intersection digraphs. First, we introduce natural classes of intersection digraphs that generalize several classes studied in the literature. Second, we define the directed locally checkable vertex (DLCV) problems, which capture many well-studied problems on digraphs such as (Independent) Dominating Set, Kernel, and H-Homomorphism. Third, we give a new width measure of digraphs, bi-mim-width, and show that the DLCV problems are polynomial-time solvable when we are provided a decomposition of small bi-mim-width. Fourth, we show that several classes of intersection digraphs have bounded bi-mim-width, implying that we can solve all DLCV problems on these classes in polynomial time given an intersection representation of the input digraph. We identify reflexivity as a useful condition to obtain intersection digraph classes of bounded bi-mim-width, and therefore to obtain positive algorithmic results

Nonpermutation flow line scheduling by ant colony optimization

A flow line is a conventional manufacturing system where all jobs must be processed on all machines with the same operation sequence. Line buffers allow nonpermutation flowshop scheduling and job sequences to be changed on different machines. A mixed-integer linear programming model for nonpermutation flowshop scheduling and the buffer requirement along with manufacturing implication is proposed. Ant colony optimization based heuristic is evaluated against Taillard's (1993) well-known flowshop benchmark instances, with 20 to 500 jobs to be processed on 5 to 20 machines (stages). Computation experiments show that the proposed algorithm is incumbent to the state-of-the-art ant colony optimization for flowshop with higher job to machine ratios, using the makespan as the optimization criterion

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Economic Networks: Theory and Computation

This textbook is an introduction to economic networks, intended for students and researchers in the fields of economics and applied mathematics. The textbook emphasizes quantitative modeling, with the main underlying tools being graph theory, linear algebra, fixed point theory and programming. The text is suitable for a one-semester course, taught either to advanced undergraduate students who are comfortable with linear algebra or to beginning graduate students.Comment: Textbook homepage is https://quantecon.github.io/book-networks/intro.htm