Analysis of Hannan Consistent Selection for Monte Carlo Tree Search in Simultaneous Move Games

Hannan consistency, or no external regret, is a~key concept for learning in games. An action selection algorithm is Hannan consistent (HC) if its performance is eventually as good as selecting the~best fixed action in hindsight. If both players in a~zero-sum normal form game use a~Hannan consistent algorithm, their average behavior converges to a~Nash equilibrium (NE) of the~game. A similar result is known about extensive form games, but the~played strategies need to be Hannan consistent with respect to the~counterfactual values, which are often difficult to obtain. We study zero-sum extensive form games with simultaneous moves, but otherwise perfect information. These games generalize normal form games and they are a special case of extensive form games. We study whether applying HC algorithms in each decision point of these games directly to the~observed payoffs leads to convergence to a~Nash equilibrium. This learning process corresponds to a~class of Monte Carlo Tree Search algorithms, which are popular for playing simultaneous-move games but do not have any known performance guarantees. We show that using HC algorithms directly on the~observed payoffs is not sufficient to guarantee the~convergence. With an~additional averaging over joint actions, the~convergence is guaranteed, but empirically slower. We further define an~additional property of HC algorithms, which is sufficient to guarantee the~convergence without the~averaging and we empirically show that commonly used HC algorithms have this property.Comment: arXiv admin note: substantial text overlap with arXiv:1509.0014

Absolutně a neabsolutně F-borelovské prostory

Zabýváme se F-borelovskou složitostí topologických prostorů a tím, jak se tato složitost liší v závislosti na tom, kam je daný topologický prostor vnořen. Obzvláště nás pak zajímá, kdy je tato složitost absolutní, tj. stejná ve všech kompaktifikacích. Ukazujeme, že složitost metrizovatelných prostorů je abso- lutní. Dále odvozujeme postačující podmínku pro to, aby byl prostor absolutně Fσδ. Studujeme vztah lokální a globální složitosti a závádíme různé reprezentace F-borelovských množin. Tyto nástroje používáme k důkazu několika různých výsledků, zejména pak k získání hierarchie prostorů, jejichž složitost je neabso- lutní. 1We investigate F-Borel topological spaces. We focus on finding out how a complexity of a space depends on where the space is embedded. Of a particular interest is the problem of determining whether a complexity of given space X is absolute (that is, the same in every compactification of X). We show that the complexity of metrizable spaces is absolute and provide a sufficient condition for a topological space to be absolutely Fσδ. We then investigate the relation between local and global complexity. To improve our understanding of F-Borel spaces, we introduce different ways of representing an F-Borel set. We use these tools to construct a hierarchy of F-Borel spaces with non-absolute complexity, and to prove several other results. 1Katedra matematické analýzyDepartment of Mathematical AnalysisFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Strategy 3axis Milling of Cast Spindle in CAM System with Support of Multimedia Application

Import 26/06/2013Tato bakalářská práce se zabývá strategií 3osého frézování odlitku vřetene. V teoretické části práce je zpracovaná problematika strategie frézování, technologie obrábění a inovací v obrábění. Hlavní částí je multimediální podpora vytvořena programem Microsoft FrontPage. V závěru jsou shrnuty všechny zjištěné poznatky.This thesis discusses strategies 3osého milling spindle casting. The theoretical part is treated the issue of strategy milling, machining technology and innovation in machining. The main part of the multimedia support generated by Microsoft FrontPage. The conclusion summarizes all the lessons learned.346 - Katedra obrábění a montáževelmi dobř