No fast exponential deviation inequalities for the progressive mixture rule

Details No fast exponential deviation inequalities for the progressive mixture rule No fast exponential deviation inequalities for the progressive mixture rule

2007-03-28

arxiv.org/abs/math/0703848v1

Mathematics

Statistics Theory

Scientific paper

We consider the learning task consisting in predicting as well as the best function in a finite reference set G up to the smallest possible additive term. If R(g) denotes the generalization error of a prediction function g, under reasonable assumptions on the loss function (typically satisfied by the least square loss when the output is bounded), it is known that the progressive mixture rule g_n satisfies E R(g_n) < min_{g in G} R(g) + C (log|G|)/n where n denotes the size of the training set, E denotes the expectation w.r.t. the training set distribution and C denotes a positive constant. This work mainly shows that for any training set size n, there exist a>0, a reference set G and a probability distribution generating the data such that with probability at least a R(g_n) > min_{g in G} R(g) + c sqrt{[log(|G|/a)]/n}, where c is a positive constant. In other words, surprisingly, for appropriate reference set G, the deviation convergence rate of the progressive mixture rule is only of order 1/sqrt{n} while its expectation convergence rate is of order 1/n. The same conclusion holds for the progressive indirect mixture rule. This work also emphasizes on the suboptimality of algorithms based on penalized empirical risk minimization on G.

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