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  • 标题:Extractor-Based Time-Space Lower Bounds for Learning
  • 本地全文:下载
  • 作者:Sumegha Garg ; Ran Raz ; Avishay Tal
  • 期刊名称:Electronic Colloquium on Computational Complexity
  • 印刷版ISSN:1433-8092
  • 出版年度:2017
  • 卷号:2017
  • 出版社:Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
  • 摘要:

    A matrix M : A X − 1 1 corresponds to the following learning problem: An unknown element x X is chosen uniformly at random. A learner tries to learn x from a stream of samples, ( a 1 b 1 ) ( a 2 b 2 ) , where for every i , a i A is chosen uniformly at random and b i = M ( a i x ) .

    Assume that k r are such that any submatrix of M of at least 2 − k A rows and at least 2 − X columns, has a bias of at most 2 − r . We show that any learning algorithm for the learning problem corresponding to M requires either a memory of size at least k , or at least 2 ( r ) samples. The result holds even if the learner has an exponentially small success probability (of 2 − ( r ) ).

    In particular, this shows that for a large class of learning problems, any learning algorithm requires either a memory of size at least ( log X ) ( log A ) or an exponential number of samples, achieving a tight ( log X ) ( log A ) lower bound on the size of the memory, rather than a bound of min ( log X ) 2 ( log A ) 2 obtained in previous works [R17,MM17b].

    Moreover, our result implies all previous memory-samples lower bounds, as well as a number of new applications.

  • 关键词:branching program ; lower bounds ; PAC learning ; time-space tradeoff
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