具有次指数包络类的一般非精确Oracle不等式
General nonexact oracle inequalities for classes with a subexponential envelope
摘要：
我们表明，经验风险最小化程序和正规化的经验风险最小化程序满足非精确的预言不等式在一个无界的框架下，假设类有一个次指数包络函数。主要的新颖之处，除了有界性假设自由设置，是这些不平等可以产生快速的速度，即使在确切的预言不平等的情况下，只持有较慢的速度。我们应用这些结果表明，程序的基础上$\ell_1$和核范数正则化函数满足甲骨文不等式的剩余项，减少像$1/n$为每个$L_q$损失函数（$q\geq2$），而只假设的尾部行为的输入和输出变量表现良好。特别是，不需要RIP类型的假设或“不相干条件”来获得这些设置中的快速残差项。我们也将这些结果应用到凸聚集和模型选择的问题。
英文摘要：
We show that empirical risk minimization procedures and regularized empirical risk minimization procedures satisfy nonexact oracle inequalities in an unbounded framework, under the assumption that the class has a subexponential envelope function. The main novelty, in addition to the boundedness assumption free setup, is that those inequalities can yield fast rates even in situations in which exact oracle inequalities only hold with slower rates. We apply these results to show that procedures based on $\ell_1$ and nuclear norms regularization functions satisfy oracle inequalities with a residual term that decreases like $1/n$ for every $L_q$-loss functions ($q\geq2$), while only assuming that the tail behavior of the input and output variables are well behaved. In particular, no RIP type of assumption or "incoherence condition" are needed to obtain fast residual terms in those setups. We also apply these results to the problems of convex aggregation and model selection.
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