 
Summary: No fast exponential deviation inequalities for the
progressive mixture rule
JeanYves Audibert
CERTIS  Ecole des Ponts
19, rue Alfred Nobel  Cit´e Descartes
77455 MarnelaVall´ee  France
audibert@certis.enpc.fr
Abstract. 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 satisfies
ER(^g) mingG R(g) + C log G
n
, (1)
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 > 0,
a reference set G and a probability distribution generating the data such
