Using the chain rule, we get equations (4)-(8), where the summation in Equation 6 is over all source-target phrase pairs in the derivation of hypothesis h sm , ξ is the decoding weight assigned to the log phrase translation score, and the summation in Equation 7 is over all training sentences 2 .
in order to compute the derivative of the objective function we first need to calculate for every phrase pair (s k , t k ) in the translation N-best based on Equations 7 and 8, which requires time proportional to the number of occurrences of these phrases in the parallel training data.