Trading Off Worst and Expected Cost in Decision Tree Problems
Trading Off Worst and Expected Cost in Decision Tree Problems
复制标题
权衡决策树问题中的最坏成本和预期成本
作者：
A. Saettler;E. Laber;F. Cicalese
我们描述了在优化决策树的构造时，相对于最坏成本和预期成本，可以实现的最佳可能的权衡。众所周知，达到最小可能最坏情况代价的决策树的预期表现非常差(甚至指数级地比最优差)，反之亦然。在确定哪种优化标准可能并不容易的应用程序的带领下，几位作者最近专注于决策树的双标准优化。在这里，我们尖锐地定义了预期成本和最坏情况成本之间可能的最佳权衡的界限。更准确地说，我们证明了对于每个ρ&>0DocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}，例如{Document}$
Ho>；0$$end{Document}存在测试成本最差至多(1+ρ)的决策树D OPTWDocumentclass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$(1+
Ho)Extit{opt}_W$$End{Document}和预期测试成本至多11-e-ρ选项，DocumentClass[12pt]{Minimal}UsPackage{amsath}UsPack{wa ysym}Usepackage{amsFonts}UsPack{amssymb}UsPack{amsbsy}Usepackage{mathsfs}UsPackage{upgreek}setLength{oddsidemargin}{-69pt}例如{Document}$$frac{1}{1-e^
Ho}}退出{opt}_E，$$end{Document}其中OPTWDocumentclass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$$extit{opt}_W$$end{Documentclass[12pt]{}usepackage{amsath}usepackack{waysym}usepackage{amssymb}usepackage{masssy}usepackage{greupek{greupek{setododargemin}in$$extipt}{}usepackage{amepackage{waysym}usepackage{amssy}usepackage{masssy}usepackage{greupek{setododemin}in{$extit{egt_end}E表示给定实例的决策树的最小最差测试成本和最小预期测试成本。我们还证明了这是最好的折衷方案，因为有无限多个实例无法获得最差测试成本小于(1+ρ)OPTW DocumentClass[12pt]{Minimum}usepackage{amsath}usepackage{amsFonts}usepackage{amssymb}usepackage{amsbsy}usepackage{matrsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$(1+
Ho)现有{OPT}_W$$END{DOCUMENT}且预期测试成本小于11-e-ρOPTE文档类[12pt]{Minimal}UsPack{amsath}UsPack{wa ysym}UsPack{amsFonts}UsPack{amssymb}UsPack{amsbsy}UsPack{mathsfs}UsPackage{upgreek}setLength{oddsidemargin}{-69pt}例如{文档}$$FRAC{1}{1-e^-
Ho}}退出{opt}_E$$end{文档}。
We characterize the best possible trade-off achievable when optimizing the construction of a decision tree with respect to both the worst and the expected cost. It is known that a decision tree achieving the minimum possible worst case cost can behave very poorly in expectation (even exponentially worse than the optimal), and the vice versa is also true. Led by applications where deciding which optimization criterion might not be easy, several authors recently have focussed on the bicriteria optimization of decision trees. Here we sharply define the limits of the best possible trade-offs between expected and worst case cost. More precisely, we show that for every ρ>0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$
ho >0$$end{document} there is a decision tree D with worst testing cost at most (1+ρ)OPTWdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(1 +
ho ) extit{OPT}_W$$end{document} and expected testing cost at most 11-e-ρOPTE,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$frac{1}{1 - e^{-
ho }} extit{OPT}_E,$$end{document} where OPTWdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ extit{OPT}_W$$end{document} and OPTEdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$ extit{OPT}_E$$end{document} denote the minimum worst testing cost and the minimum expected testing cost of a decision tree for the given instance. We also show that this is the best possible trade-off in the sense that there are infinitely many instances for which we cannot obtain a decision tree with both worst testing cost smaller than (1+ρ)OPTWdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(1 +
ho ) extit{OPT}_W$$end{document} and expected testing cost smaller than 11-e-ρOPTEdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$frac{1}{1 - e^{-
ho }} extit{OPT}_E$$end{document}.