# 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.

### Bibliography

• Vladimir Vovk. Defensive forecasting for optimal prediction with expert advice, arXiv:0708.1503v1 [cs.LG]. arXiv.org e-Print archive, August 2007