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Statistical learning based on Markovian data maximal deviation inequalities and learning rates
Annals of Mathematics and Artificial Intelligence ( IF 1.2 ) Pub Date : 2019-08-29 , DOI: 10.1007/s10472-019-09670-6
Stephan Clémençon , Patrice Bertail , Gabriela Ciołek

In statistical learning theory, numerous works established non-asymptotic bounds assessing the generalization capacity of empirical risk minimizers under a large variety of complexity assumptions for the class of decision rules over which optimization is performed, by means of sharp control of uniform deviation of i.i.d. averages from their expectation, while fully ignoring the possible dependence across training data in general. It is the purpose of this paper to show that similar results can be obtained when statistical learning is based on a data sequence drawn from a (Harris positive) Markov chain X , through the running example of estimation of minimum volume sets (MV-sets) related to X ’s stationary distribution, an unsupervised statistical learning approach to anomaly/novelty detection. Based on novel maximal deviation inequalities we establish, using the regenerative method , learning rate bounds that depend not only on the complexity of the class of candidate sets but also on the ergodicity rate of the chain X , expressed in terms of tail conditions for the length of the regenerative cycles. In particular, this approach fully tailored to Markovian data permits to interpret the rate bound results obtained in frequentist terms, in contrast to alternative coupling techniques based on mixing conditions: the larger the expected number of cycles over a trajectory of finite length, the more accurate the MV-set estimates. Beyond the theoretical analysis, this phenomenon is supported by illustrative numerical experiments.

中文翻译:

基于马尔可夫数据最大偏差不等式和学习率的统计学习

在统计学习理论中,许多工作建立了非渐近界限,通过对 iid 平均值的均匀偏差的严格控制,在各种复杂性假设下评估经验风险最小化器的泛化能力,用于执行优化的决策规则类别与他们的期望不同,同时完全忽略一般训练数据之间可能的依赖性。本文的目的是通过最小体积集(MV-sets)估计的运行示例,表明当统计学习基于从(哈里斯正)马尔可夫链 X 中提取的数据序列时,可以获得类似的结果。与 X 的平稳分布相关,这是一种用于异常/新颖性检测的无监督统计学习方法。基于新的最大偏差不等式,我们使用再生方法建立了学习率界限,该界限不仅取决于候选集类别的复杂性,还取决于链 X 的遍历率,以长度的尾部条件表示再生循环。特别是,与基于混合条件的替代耦合技术相比,这种完全针对马尔可夫数据量身定制的方法允许解释以频率论术语获得的速率界限结果:有限长度轨迹上的预期循环数越大,精度越高MV 集估计。除了理论分析之外,这种现象还得到了说明性数值实验的支持。学习率界限不仅取决于候选集类别的复杂性,还取决于链 X 的遍历率,以再生循环长度的尾部条件表示。特别是,与基于混合条件的替代耦合技术相比,这种完全针对马尔可夫数据量身定制的方法允许解释以频率论术语获得的速率界限结果:有限长度轨迹上的预期循环数越大,精度越高MV 集估计。除了理论分析之外,这种现象还得到了说明性数值实验的支持。学习率界限不仅取决于候选集类别的复杂性,还取决于链 X 的遍历率,以再生循环长度的尾部条件表示。特别是,与基于混合条件的替代耦合技术相比,这种完全针对马尔可夫数据量身定制的方法允许解释以频率论术语获得的速率界限结果:有限长度轨迹上的预期循环数越大,精度越高MV 集估计。除了理论分析之外,这种现象还得到了说明性数值实验的支持。与基于混合条件的替代耦合技术相比,这种完全针对马尔可夫数据量身定制的方法允许解释以频率论术语获得的速率界限结果:有限长度轨迹上的预期循环数越大,MV 越准确设定估计。除了理论分析之外,这种现象还得到了说明性数值实验的支持。与基于混合条件的替代耦合技术相比,这种完全针对马尔可夫数据量身定制的方法允许解释以频率论术语获得的速率界限结果:有限长度轨迹上的预期循环数越大,MV 越准确设定估计。除了理论分析之外,这种现象还得到了说明性数值实验的支持。
更新日期:2019-08-29
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