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Memory-limited non-U-shaped learning with solved open problems

Memory-limited non-U-shaped learning with solved open proble...

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作者： Case, John Kötzing, Timo Department of Computer and Information Science University of Delaware Newark DE 19716 United States Department 1: Algorithms and Complexity Max-Planck-Institut für Informatik 66123 Saarbrücken Germany

In empirical cognitive science, for human learning, a semantic or behavioral U-shape occurs when a learner first learns, then unlearns, and, finally, relearns, some target concept. Within the formal framework of Inductive Inference, for learning from positive data, previous results have shown, for example, that such U-shapes are unnecessary for explanatory learning, but are necessary for behaviorally correct and non-trivial vacillatory learning. Herein we also distinguish between semantic and syntactic U-shapes. We answer a number of open questions in the prior literature as well as provide new results regarding syntactic U-shapes. Importantly for cognitive science, we see more of a previously noticed pattern that, for parameterized learning criteria, beyond very few initial parameter values, U-shapes are necessary for full learning power. We analyze the necessity of U-shapes in two memory-limited settings. The first setting is Bounded Memory State (BMS) learning, where a learner has an explicitly-bounded state memory, and otherwise only knows its current datum. We show that there are classes learnable with three (or more) memory states that are not learnable non-U-shapedly with any finite number of memory states. This result is surprising, since, for learning with one or two memory states, U-shapes are known to be unnecessary. This solves an open question from the literature. The second setting is that of Memoryless feedback (MLf) learning, where a learner may ask a bounded number of questions about what data has been seen so far, and otherwise only knows its current datum. We show that there is a class learnable memorylessly with a single feedback query such that this class is not learnable non-U-shapedly memorylessly with any finite number of feedback queries. We employ self-learning classes together with the Operator recursion Theorem for many of our results, but we also introduce two new techniques for obtaining results. The first is for transferring inclusion

关键词： Syntactics

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Two Proofs for Shallow Packings 31

Two Proofs for Shallow Packings

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31st International Symposium on Computational Geometry, SoCG 2015

作者： dutta, Kunal Ezra, Esther Ghosh, Arijit D1: Algorithms and Complexity Max-Planck-Institut für Informatik Saarbrücken66123 Germany Department of Computer Science and Engineering Polytechnic Institute of NYU BrooklynNY11201-3840 United States School of Mathematics Georgia Institute of Technology AtlantaGA30332 United States

ISBN: (纸本)9783939897835

We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X;we view V as a set of indicator vectors over the n-dimensional unit cube. A δ-separated set of V is a subcollection W, s.t. The Hamming distance between each pair u, v 2 W is greater than δ, where δ > 0 is an integer parameter. The δ-packing number is then defined as the cardinality of the largest δ-separated subcollection of V. Haussler showed an asymptotically tight bound of φ((n/δ)d) on the δ-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X′ ⊆ X of size m ≤ n and for any parameter 1 ≤ κ ≤ m, the number of vectors of length at most κ in the restriction of V to X′ is only O(md1kd-d1 ), for a fixed integer d > 0 and a real parameter 1 ≤ d1 ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d1 = d). In this case when V is "κ-shallow" (all vector lengths are at most κ), we show that its δ-packing number is O(nd1kd-d1/δd), matching Haussler s bound for the special cases where d1 = d or κ = n. We present two proofs, the first is an extension of Haussler s approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler s proof.

关键词： Set theory

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Plateaus can be harder in multi-objective optimization

Plateaus can be harder in multi-objective optimization

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Congress on Evolutionary Computation

作者： Tobias friedrich Nils Hebbinghaus frank Neumann Department 1: Algorithms and Complexity Max Planck Institut für Informatik Saarbruecken Germany

In recent years a lot of progress has been made in understanding the behavior of evolutionary computation methods for single- and multi-objective problems. Our aim is to analyze the diversity mechanisms that are implicitly used in evolutionary algorithms for multi-objective problems by rigorous runtime analyses. We show that, even if the population size is small, the runtime can be exponential where corresponding single-objective problems are optimized within polynomial time. To illustrate this behavior we analyze a simple plateau function in a first step and extend our result to a class of instances of the well-known SETCOVEr problem.

关键词： Evolutionary computation Polynomials Optimization methods Algorithm design and analysis runtime Pareto optimization Time measurement

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Ant colony optimization and the minimum cut problem 10

Ant colony optimization and the minimum cut problem

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12th Annual Genetic and Evolutionary Computation Conference, GECCO-2010

作者： Kötzing, Timo Lehre, Per Kristian Neumann, frank Oliveto, Pietro S. Department 1 : Algorithms and Complexity Max-Planck-Institut für Informatik 66123 Saarbrücken Germany School of Computer Science University of Birmingham B15 2TT Birmingham United Kingdom

ISBN: (纸本)9781450300728

Ant Colony Optimization (ACO) is a powerful metaheuristic for solving combinatorial optimization problems. With this paper we contribute to the theoretical understanding of this kind of algorithm by investigating the classical minimum cut problem. An ACO algorithm similar to the one that was proved successful for the minimum spanning tree problem is studied. Using rigorous runtime analyses we show how the ACO algorithm behaves similarly to Karger and Stein's algorithm for the minimum cut problem as long as the use of pheromone values is limited. Hence optimal solutions are obtained in expected polynomial time. On the other hand, we show that high use of pheromones has a negative effect, and the ACO algorithm may get trapped in local optima resulting in an exponential runtime to obtain an optimal solution. This result indicates that ACO algorithms may be inappropriate for finding minimum cuts. Copyright 2010 ACM.

关键词： Ant colony optimization

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A proof of the Boyd-Carr conjecture

A proof of the Boyd-Carr conjecture

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23rd Annual ACM-SIAM Symposium on discrete algorithms, SOdA 2012

作者： Schalekamp, frans Williamson, david P. Van Zuylen, Anke School of Operations Research and Information Engineering Cornell University Ithaca NY 14853 United States Max-Planck-Institut für Informatik Department 1: Algorithms and Complexity Campus El 4 66123 Saarbrücken Germany

ISBN: (纸本)9781611972108

determining the precise integrality gap for the subtour LP relaxation of the traveling salesman problem is a significant open question, with little progress made in thirty years in the general case of symmetric costs that obey triangle inequality. Boyd and Carr [3] observe that we do not even know the worst-case upper bound on the ratio of the optimal 2-matching to the subtour LP;they conjecture the ratio is at most 10/9. In this paper, we prove the Boyd-Carr conjecture. In the case that a fractional 2-matching has no cut edge, we can further prove that an optimal 2-matching is at most 10/9 times the cost of the fractional 2-matching. Copyright © SIAM.

关键词： Traveling salesman problem

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A central limit theorem for the number of degree-k vertices in random maps 9

A central limit theorem for the number of degree-k vertices ...

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9th Meeting on Analytic Algorithmics and Combinatorics, ANALCO 2012

作者： drmota, Michael Panagiotou, Konstantinos Iinstitute of Discrete Mathematics and Geometry TU Wien Wiedner Hauptstr. 8-10 WienA-1040 Austria Max-Planck-Institut für Informatik Department 1: Algorithms and Complexity Campus E1.4 Saarbrücken66123 Germany

ISBN: (纸本)9781618396235

We prove that the number of vertices of given degree in random planar maps satisfies a central limit theorem with mean and variance that are asymptotically linear in the number of edges. The proof relies on an analytic version of the quadratic method and singularity analysis of multivariate generating functions. © Copyright (2012) by SIAM: Society for Industrial and Applied Mathematics. All rights reserved.

关键词： Multivariant analysis

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