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We develop minimax optimal risk bounds for the general learning taskconsisting in predicting as well as the best function in a reference set$\mathcal{G}$ up to the smallest possible additive term, called the convergencerate. When the reference set is finite and when $n$ denotes the size of thetraining data, we provide minimax convergence rates of the form$C(\frac{\log|\mathcal{G}|}{n})^v$ with tight evaluation of the positiveconstant $C$ and with exact $0<v\le1$, the latter value depending on theconvexity of the loss function and on the level of noise in the outputdistribution. The risk upper bounds are based on a sequential randomizedalgorithm, which at each step concentrates on functions having both low riskand low variance with respect to the previous step prediction function. Ouranalysis puts forward the links between the probabilistic and worst-caseviewpoints, and allows to obtain risk bounds unachievable with the standardstatistical learning approach. One of the key ideas of this work is to useprobabilistic inequalities with respect to appropriate (Gibbs) distributions onthe prediction function space instead of using them with respect to thedistribution generating the data. The risk lower bounds are based onrefinements of the Assouad lemma taking particularly into account theproperties of the loss function. Our key example to illustrate the upper andlower bounds is to consider the $L_q$-regression setting for which anexhaustive analysis of the convergence rates is given while $q$ ranges in$[1;+\infty[$.

Fast learning rates in statistical inference through aggregation