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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)
For the prediction with expert advice setting, we consider methods to construct algorithms that have low adaptive regret. The adaptive regret of an algorithm on a time interval [t 1 , t 2 ] is the loss of the algorithm minus the loss of the best expert over that interval. Adaptive regret measures how well the algorithm approximates the best expert locally,(More)
Classical probability theory considers probability distributions that assign probabilities to all events (at least in the finite case). However, there are natural situations where only part of the process is controlled by some probability distribution while for the other part we know only the set of possibilities without any probabilities assigned. We adapt(More)
We bound the future loss when predicting any (computably) stochastic sequence online. Solomonoff finitely bounded the total deviation of his universal predictor M from the true distribution µ by the algorithmic complexity of µ. Here we assume we are at a time t > 1 and already observed x = x 1 ...x t. We bound the future prediction performance on x t+1 x(More)
We bound the future loss when predicting any (computably) stochastic sequence online. Solomonoff finitely bounded the total deviation of his universal predictor M from the true distribution µ by the algorithmic complexity of µ. Here we assume that we are at a time t > 1 and have already observed x = x 1 ...x t. We bound the future prediction performance on(More)
Randomness in the sense of Martin-Löf can be defined in terms of lower semi-computable supermartingales. We show that such a supermartingale cannot be replaced by a pair of supermartingales that bet only on the even bits (the first one) and on the odd bits (the second one) knowing all preceding bits. 1 Randomness and lower semicomputable super-martingales(More)