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Many large-scale applications amount to finding relevant results from anenormous output space of potential candidates. For example, finding the bestmatching product from a large catalog or suggesting related search phrases on asearch engine. The size of the output space for these problems can range frommillions to billions, and can even be infinite in some applications. Moreover,training data is often limited for the long-tail items in the output space.Fortunately, items in the output space are often correlated thereby presentingan opportunity to alleviate the data sparsity issue. In this paper, we proposethe Prediction for Enormous and Correlated Output Spaces (PECOS) framework, aversatile and modular machine learning framework for solving predictionproblems for very large output spaces, and apply it to the eXtreme MultilabelRanking (XMR) problem: given an input instance, find and rank the most relevantitems from an enormous but fixed and finite output space. We propose a threephase framework for PECOS: (i) in the first phase, PECOS organizes the outputspace using a semantic indexing scheme, (ii) in the second phase, PECOS usesthe indexing to narrow down the output space by orders of magnitude using amachine learned matching scheme, and (iii) in the third phase, PECOS ranks thematched items using a final ranking scheme. The versatility and modularity ofPECOS allows for easy plug-and-play of various choices for the indexing,matching, and ranking phases. We also develop very fast inference procedureswhich allow us to perform XMR predictions in real time; for example, inferencetakes less than 1 millisecond per input on the dataset with 2.8 million labels.The PECOS software is available at https://libpecos.org.

PECOS: Prediction for Enormous and Correlated Output Spaces