We study the problem of on-line classification in which the prediction algorithm, for each “significance level ” δ, is required to output as its prediction a range of labels (intuitively, those labels deemed compatible with the available data at the level δ) rather than just one label; as usual, the examples are assumed to be generated independently from the same probability distribution P. The prediction algorithm is said to be “well-calibrated ” for P and δ if the long-run relative frequency of errors does not exceed δ almost surely w.r. to P. For well-calibrated algorithms we take the number of “uncertain ” predictions (i.e., those containing more than one label) as the principal measure of predictive performance. The main result of this paper is the construction of a prediction algorithm which, for any (unknown) P and any δ: (a) makes errors independently and with probability δ at every trial (in particular, is well-calibrated for P and δ); (b) makes in the long run no more uncertain predictions than any other prediction algorithm that is well-calibrated for P and δ; (c) processes example n in time O(logn).

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This thesis is devoted to relaxing certain theoretical assumptions in pattern recognition models. In pattern recognition a predictor is trying to guess a discrete label of some object (usually a real vector), based on given examples of object-label pairs. Pattern recognition was

Conformal predictors, introduced by Vovk et al. [16], serve to build prediction intervals by exploiting a notion of conformity of the new data point with previously observed data. In the present paper, we propose a novel method for constructing prediction intervals for the response variable in multivariate linear models. The main emphasis is on sparse linear models, where only few of the covariates have significant influence on the response variable even if their number is very large. Our approach is based on combining the principle of conformal prediction with the ℓ1 penalized least squares estimator (LASSO). The resulting confidence set depends on a parameter ε> 0 and has a coverage probability larger than or equal to 1 − ε. The numerical experiments reported in the paper show that the length of the confidence set is small. Furthermore, as a by-product of the proposed approach, we provide a data-driven procedure for choosing the LASSO penalty. The selection power of the method is illustrated on simulated data.