Abstract: MAP Structured Output Prediction by Sampling

MAP Structured Output Prediction by Sampling Shankar Vembu, Thomas Gärtner, and Mario Boley [email protected] Fraunhofer IAIS, Schloß Birlinghoven, 53754 Sankt Augustin, Germany We consider maximum a posteriori parameter estimation for structured output prediction with exponential family models. In this setting the main difficulty lies in the computation of the partition function and the first-order moment of the sufficient statistics. We consider the case that efficient algorithms for exact uniform sampling from the output space exist. This assumption is orthogonal to the typical assumptions made in structured output learning. It holds, in particular, for the highly relevant problem of sampling potent drugs. Under our uniform sampling assumption we show that exactly computing the partition function is intractable (Section 2) but it can be approximated efficiently (Section 3). Furthermore, we show that also the first-order moment of the sufficient statistics can be approximated (Section 4) and that we can sample according to the estimated distribution (Section 5). After a few simple application settings (Section 6), we discuss related and future work (Section 7). 1. Preliminaries and Problems We use [[n]] to denote {1, . . . , n}. Let X and Y be the input and the output space, respectively, where Y is parameterised by some finite alphabet Σ. For instance, Y can consist of strings, trees, or graphs over of Σ. Let {xi, yi}i∈[[m]] ⊆ X × Y be a set of observations. Our goal is to find θ as the maximum a posteriori parameters of the conditional exponential family model: p(y | x, θ) = exp(〈φ(x, y), θ〉)/Z(θ|x) , where φ(x, y) are the joint sufficient statistics of x and y, and Z(θ|x) = ∑ y∈Y exp(〈φ(x, y), θ〉) is the partition function. Imposing a normal prior on θ, this leads to minimising the following function: