Accelerated experimental design for pairwise comparisons

Pairwise comparison labels are more informative and less variable than class labels, but generating them poses a challenge: their number grows quadratically in the dataset size. We study a natural experimental design objective, namely, D-optimality, that can be used to identify which K pairwise comparisons to generate. This objective is known to perform well in practice, and is submodular, making the selection approximable via the greedy algorithm. A naïve greedy implementation has O(N²d²K) complexity, where N is the dataset size, d is the feature space dimension, and K is the number of generated comparisons. We show that, by exploiting the inherent geometry of the dataset–namely, that it consists of pairwise comparisons–the greedy algorithm’s complexity can be reduced to O(N²(K + d) + N(dK + d²) + d²K). We apply the same acceleration also to the so-called lazy greedy algorithm. When combined, the above improvements lead to an execution time of less than 1 hour for a dataset with 10⁸ comparisons; the naïve greedy algorithm on the same dataset would require more than 10 days to terminate.