COS 511 : Theoretical Machine Learning
نویسنده
چکیده
1 Review of the Bayes Algorithm Last time we talked about the Bayes algorithm, in which we give priors π i to each expert. The algorithm maintains weights w t,i for each expert. The π i values serve as the initial weights for the experts. Experts predict the distributions p t,i over the same set X, and the algorithm predicts the q t distribution as a mixture of those distributions. To restate it mathematically: N experts π i = prior, π i ≥ 0, i π i = 1 w 1,i = π i for t = 1,. .. T expert i predicts p t,i (distribution over X) master predicts q t q t (x) = N i=1 w t,i p t,i (x) observe x t normalization We showed: − t ln q t (x t) ≤ min i − t ln p t,i − ln π i. We derived the algorithm and its analysis by pretending the data was generated by a random process in which (1) Pr[i * = i] = π i (2) Pr[x t |x t−1 1 , i * = i] = p i (x t |x t−1 1) = p t,i (x t). We then defined: q t (x t) = q(x t |x t−1 1) = P r[x t |x t−1 1 ]. In the last part, we have shown the upper bound for the loss of the algorithm by pretending that everything is random. Also we note that the algorithm's loss is bounded with respect to the best expert. Now, we will look at the case where the error is not with respect to a single best expert, but a switching sequence of a subset of the experts.
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