Table 3

Input:T training iterations, n training examples associating each input x_{i} with an output set (i.e., semantic stack or realization sequences). GEN(x_{i}) returns the n-best output sequences for input x_{i} based on a Viterbi search using the corresponding FLM, in which n depends on a pruning beam and a maximum value. Φ(x_{i},y) is a sparse feature vector of dimensionality d representing the number of occurrences of specific combinations of realization phrases and/or semantic stacks in (x_{i}, y), with an entry for each instantiation in the training data of each node of the backoff graph of the large context FLM in Figure 6. |

Output: a collection V of feature vectors in ℝ^{d} and their respective weights α in ℝ^{|V|}. Using a linear kernel, the algorithm is simplified as the weighted feature vectors can be represented as a single weight vector in ℝ^{d}. |

Linear kernel algorithm: |

For t = 1…T, i = 1…n |

For z in |

If w . Φ(x_{i}, z) ≥ 0 then w ←w − Φ(x_{i}, z) // incorrect positive prediction |

For y in |

If w . Φ(x_{i}, y) < 0 then w ←w + Φ(x_{i}, y) // incorrect negative prediction |

Kernelized algorithm with kernel function K: ℝ^{d} ×ℝ^{d} →ℝ: |

For t = 1…T, i = 1…n |

For z in |

If then // incorrect positive prediction |

append Φ(x_{i}, z) to V // weigh instance negatively |

append −1 to α |

For y in |

If then // incorrect negative prediction |

append Φ(x_{i}, y) to V // weigh instance positively |

append 1 to α |

Input:T training iterations, n training examples associating each input x_{i} with an output set (i.e., semantic stack or realization sequences). GEN(x_{i}) returns the n-best output sequences for input x_{i} based on a Viterbi search using the corresponding FLM, in which n depends on a pruning beam and a maximum value. Φ(x_{i},y) is a sparse feature vector of dimensionality d representing the number of occurrences of specific combinations of realization phrases and/or semantic stacks in (x_{i}, y), with an entry for each instantiation in the training data of each node of the backoff graph of the large context FLM in Figure 6. |

Output: a collection V of feature vectors in ℝ^{d} and their respective weights α in ℝ^{|V|}. Using a linear kernel, the algorithm is simplified as the weighted feature vectors can be represented as a single weight vector in ℝ^{d}. |

Linear kernel algorithm: |

For t = 1…T, i = 1…n |

For z in |

If w . Φ(x_{i}, z) ≥ 0 then w ←w − Φ(x_{i}, z) // incorrect positive prediction |

For y in |

If w . Φ(x_{i}, y) < 0 then w ←w + Φ(x_{i}, y) // incorrect negative prediction |

Kernelized algorithm with kernel function K: ℝ^{d} ×ℝ^{d} →ℝ: |

For t = 1…T, i = 1…n |

For z in |

If then // incorrect positive prediction |

append Φ(x_{i}, z) to V // weigh instance negatively |

append −1 to α |

For y in |

If then // incorrect negative prediction |

append Φ(x_{i}, y) to V // weigh instance positively |

append 1 to α |

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