مرکز منطقه ای اطلاع رساني علوم و فناوري - Distribution-free inequalities for the deleted and holdout error estimates

Abstract :

In the discrimination problem the random variable $\\theta$ , known to take values in ${1 ,\\ldots ,M}$ , is estimated from the random vector $X$ taking values in ${\\bf R}^{d}$ . Ali that is known about the joint distribution of $(X,O)$ is that which can be inferred from a sample $(X_{1} , \\theta_{1}, \\ldots , (X_{n}, \\theta_{n})$ of size $n$ drawn from that distribution. A discrimination rule is any procedure which determines a decision $\\hat{\\theta}$ for $\\theta$ from $X$ and $(X_{1},\\theta_{1}) , \\ldots , (X_{n}, \\theta_{n})$ . The rule is called $k$ -local if the decision $\\hat{\\theta}$ depends only on $X$ and the pairs $(X_{i}, \\theta_{i})$ ,for which $X_{i}$ is one of the $k$ closest to $X$ from $X_{1} , \\ldots ,X_{n}$ . If $L_{n}$ denotes the probability of error for a $k$ -local rule given the sample, then estimates $\\hat{L}_{n}$ of $L_{n}$ , are determined for which $P {| \\hat{L}_{n} - L_{n} \\geq \\epsilon} \\exp (- Bn)$ , where $A$ and $B$ are positive constants depending only on $d$ , $M$ , and $\\epsilon$ .