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<Paper uid="W06-1666">
  <Title>Loss Minimization in Parse Reranking</Title>
  <Section position="3" start_page="0" end_page="560" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> The reranking approach is widely used in parsing (Collins and Koo, 2005; Koo and Collins, 2005; Henderson and Titov, 2005; Shen and Joshi, 2003) as well as in other structured classification problems. For structured classification tasks, where labels are complex and have an internal structure of interdependency, the 0-1 loss considered in classical formulation of classification algorithms is not a natural choice and different loss functions are normally employed. To tackle this problem, several approaches have been proposed to accommodate loss functions in learning algorithms (Tsochantaridis et al., 2004; Taskar et al., 2004; Henderson and Titov, 2005). A very different use of loss functions was considered in the areas of signal processing and machine translation, where direct minimization of expected loss (Minimum Bayes Risk decoding) on word sequences was considered (Kumar and Byrne, 2004; Stolcke et al., 1997). The only attempt to use Minimum Bayes Risk (MBR) decoding in parsing was made in (Goodman, 1996), where a parsing algorithm for constituent recall minimization was constructed. However, their approach is limited to binarized PCFG models and, consequently, is not applicable to state-of-the-art parsing methods (Charniak and Johnson, 2005; Henderson, 2004; Collins, 2000). In this paper we consider several approaches to loss approximation on the basis of a candidate list provided by a baseline probabilistic model.</Paragraph>
    <Paragraph position="1"> The intuitive motivation for expected loss minimization can be seen from the following example.</Paragraph>
    <Paragraph position="2"> Consider the situation where there are a group of several very similar candidates and one very different candidate whose probability is just slightly larger than the probability of any individual candidate in the group, but much smaller than their total probability. A method which chooses the maximum probability candidate will choose this outlier candidate, which is correct if you are only interested in getting the label exactly correct (i.e. 0-1 loss), and you think the estimates are accurate. But if you are interested in a loss function where the loss is small when you choose a candidate which is similar to the correct candidate, then it is better to choose one of the candidates in the group. With this choice the loss will only be large if the outlier turns out to be correct, while if the outlier is chosen then the loss will be large if any of the group are correct. In other words, the expected loss of  choosing a member of the group will be smaller than that for the outlier.</Paragraph>
    <Paragraph position="3"> More formally, the Bayes risk of a model y =</Paragraph>
    <Paragraph position="5"> where the expectation is taken over all the possible inputs x and labels y and [?](y, yprime) denotes a loss incurred by assigning x to yprime when the correct label is y. We assume that the loss function possesses values within the range from 0 to 1, which is equivalent to the requirement that the loss function is bounded in (Tsochantaridis et al., 2004). It follows that an optimal reranker hstar is one which chooses the label y that minimizes the expected loss:</Paragraph>
    <Paragraph position="7"> where G(x) denotes a candidate list provided by a baseline probabilistic model for the input x.</Paragraph>
    <Paragraph position="8"> In this paper we propose different approaches to loss approximation. We apply them to the parse reranking problem where the baseline probabilistic model is a neural network parser (Henderson, 2003), and to parse reranking of candidates provided by the (Collins, 1999) model. The resulting reranking method achieves very significant improvement in the considered loss function and improvement in most other standard measures of accuracy. null In the following three sections we will discuss three approaches to learning such a classifier. The first two derive a classification criteria for use with a predefined probability model (the first generative, the second discriminative). The third defines a kernel for use with a classification method for minimizing loss. All use previously proposed learning algorithms and optimization criteria.</Paragraph>
  </Section>
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