Shortcut Defensive Forecasting
The Defensive Forecasting tecnique cannot provide regret term better than . To avoid this, in Vovk (2007) the modified tecnique is proposed for the case when benchmark class is finite number of experts. To find a regret term, which is equal to the regret term achieved by Strong Aggregating Algorithm, author uses the following scheme. First he proves the
Lemma. Let - is the standard -mixable game of prediction and . Then the process
is a supermartingale (in the sense that expected (by ) value of is less than ).
Then he uses the Levin Lemma:
Lemma (Levin, Takemura). For any forecast-continuous supermartingale there exists a strategy for Forecaster ensuring that regardless of the other players' moves.
Finally, the author proves the theorem with the regret term.
Theorem. There exists a strategy for Learner competing with experts, that guarantees for all and all .
So the bound provided by Shortcut Defensive Forecasting tecnique is equivalent to the one provided by Strong Aggregating Algorithm.
- Vladimir Vovk. Defensive forecasting for optimal prediction with expert advice, arXiv:0708.1503v1 [cs.LG]. arXiv.org e-Print archive, August 2007