TY - GEN
T1 - Mixability in statistical learning
AU - Van Erven, Tim
AU - Grünwald, Peter D.
AU - Reid, Mark D.
AU - Williamson, Robert C.
PY - 2012
Y1 - 2012
N2 - Statistical learning and sequential prediction are two different but related formalisms to study the quality of predictions. Mapping out their relations and transferring ideas is an active area of investigation. We provide another piece of the puzzle by showing that an important concept in sequential prediction, the mixability of a loss, has a natural counterpart in the statistical setting, which we call stochastic mixability. Just as ordinary mixability characterizes fast rates for the worst-case regret in sequential prediction, stochastic mixability characterizes fast rates in statistical learning. We show that, in the special case of log-loss, stochastic mixability reduces to a well-known (but usually unnamed) martingale condition, which is used in existing convergence theorems for minimum description length and Bayesian inference. In the case of 0/1-loss, it reduces to the margin condition of Mammen and Tsybakov, and in the case that the model under consideration contains all possible predictors, it is equivalent to ordinary mixability.
AB - Statistical learning and sequential prediction are two different but related formalisms to study the quality of predictions. Mapping out their relations and transferring ideas is an active area of investigation. We provide another piece of the puzzle by showing that an important concept in sequential prediction, the mixability of a loss, has a natural counterpart in the statistical setting, which we call stochastic mixability. Just as ordinary mixability characterizes fast rates for the worst-case regret in sequential prediction, stochastic mixability characterizes fast rates in statistical learning. We show that, in the special case of log-loss, stochastic mixability reduces to a well-known (but usually unnamed) martingale condition, which is used in existing convergence theorems for minimum description length and Bayesian inference. In the case of 0/1-loss, it reduces to the margin condition of Mammen and Tsybakov, and in the case that the model under consideration contains all possible predictors, it is equivalent to ordinary mixability.
UR - http://www.scopus.com/inward/record.url?scp=84877768099&partnerID=8YFLogxK
M3 - Conference contribution
SN - 9781627480031
T3 - Advances in Neural Information Processing Systems
SP - 1691
EP - 1699
BT - Advances in Neural Information Processing Systems 25
T2 - 26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012
Y2 - 3 December 2012 through 6 December 2012
ER -