  Hashing, or learning binary embeddings of data, is frequently used in nearest
neighbor retrieval. In this paper, we develop learning to rank formulations for
hashing, aimed at directly optimizing ranking-based evaluation metrics such as
Average Precision (AP) and Normalized Discounted Cumulative Gain (NDCG). We
first observe that the integer-valued Hamming distance often leads to tied
rankings, and propose to use tie-aware versions of AP and NDCG to evaluate
hashing for retrieval. Then, to optimize tie-aware ranking metrics, we derive
their continuous relaxations, and perform gradient-based optimization with deep
neural networks. Our results establish the new state-of-the-art for image
retrieval by Hamming ranking in common benchmarks.
