# 高级计算模型|COMP2922 Models of Computation (Adv)代写

Theorem 16.4 [LW94]. For any t, let $M_{t}$ (which is now a random variable) be the number of mistakes made so far by the randomized weighted majority algorithm, and let $m_{t}$ ( $i$ ) be (as before) the number of mistakes made so far by the ith expert. Then for every i, the expected number of mistakes of the algorithm is bounded by
$$\mathbb{E}\left[M_{t}\right] \leq(1+\epsilon) m_{t}(i)+O((\log k) / \epsilon) .$$
The performance of on-line algorithms is often expressed in terms of regret, namely, the largest gap between the performance of the on-line algorithm and the best performance in hindsight. Let us present the last result in this language. Let $i *$ be the expert with the minimum number of mistakes in some number $T$ of rounds. Thus the (expected) regret $\mathbb{E}\left[M_{T}\right]-m_{T}(i *)$ is bounded by $\epsilon m_{T}(i *)+O((\log k) / \epsilon)$. Using the trivial bound $m_{T}(i *) \leq T$, and choosing $\epsilon$ (which can be set at will) to balance the two terms in the bound, one sees that after any number $T$ of steps, the regret is bounded by
$$\mathbb{E}\left[M_{T}\right]-m_{T}(i *) \leq O(\sqrt{T \log k})$$
Thus, for very large $T$, the average regret per step is only $O(1 / \sqrt{T}$ ) (even though it could be 1 ). This dependence on $T$ is the best possible.

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