We apply the method of defensive forecasting, based on the use of game-theoretic supermartingales, to prediction with expert advice. In the traditional setting of a countable number of experts and a finite number of outcomes, the Defensive Forecasting Algorithm is very close to the well-known Aggregating Algorithm. Not only the performance guarantees but… (More)
We show that the Brier game of prediction is mixable and find the optimal learning rate and substitution function for it. The resulting prediction algorithm is applied to predict results of football and tennis matches. The theoretical performance guarantee turns out to be rather tight on these data sets, especially in the case of the more extensive tennis… (More)
We study prediction with expert advice in the setting where the losses are accumulated with some discounting and the impact of old losses can gradually vanish. We generalize the Aggregating Algorithm and the Ag-gregating Algorithm for Regression, propose a new variant of exponentially weighted average algorithm, and prove bounds on the cumulative discounted… (More)
We study the problem of online regression. We do not make any assumptions about input vectors or outcomes. We prove a theoretical bound on the square loss of Ridge Regression. We also show that Bayesian Ridge Regression can be thought of as an online algorithm competing with all the Gaussian linear experts. We then consider the case of infinite-dimensional… (More)
This paper provides a probabilistic derivation of an identity connecting the square loss of ridge regression in on-line mode with the loss of a retrospectively best regressor. Some corollaries of the identity providing upper bounds for the cumulative loss of on-line ridge regression are also discussed.
Multi-class classification is one of the most important tasks in machine learning. In this paper we consider two online multi-class classification problems: classification by a linear model and by a kernelized model. The quality of predictions is measured by the Brier loss function. We suggest two computationally efficient algorithms to work with these… (More)
The paper deals with on-line regression settings with signals belonging to a Banach lattice. Our algorithms work in a semi-online setting where all the inputs are known in advance and outcomes are unknown and given step by step. We apply the Aggregating Algorithm to construct a prediction method whose cumulative loss over all the input vectors is comparable… (More)