# Shortcut Defensive Forecasting

The Defensive Forecasting tecnique cannot provide regret term better than $\sqrt{N}$. 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 $\eta > 0, (\Omega=\{0,1\},\Gamma=[0,1],\lambda)$ - is the standard $\eta$-mixable game of prediction and $k \in [0,\eta]$. Then the process

$S_N := \exp{ (k\sum_{n=1}^N (\lambda(\omega_n,\gamma_n)-\lambda(\omega_n,\gamma_n^k)))}$

is a supermartingale (in the sense that expected (by $\gamma$) value of $S_N$ is less than $S_{N-1}$).

Then he uses the Levin Lemma:

Lemma (Levin, Takemura). For any forecast-continuous supermartingale $S: (\Gamma \times \Gamma \times \Omega)^* \to \mathbb{R}$ there exists a strategy for Forecaster ensuring that $S_0 \ge S_1 \ge \dots$ 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 $K$ experts, that guarantees $L_N \le L_N^k + \frac{\ln K}{\eta}$ for all $N=1,2,\dots$ and all $k=1,\dots,K$.

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