Nonconvex One-bit Single-label Multi-label Learning

Abstract

We study an extreme scenario in multi-label learning where each training instance is endowed with a single one-bit label out of multiple labels. We formulate this problem as a non-trivial special case of one-bit rank-one matrix sensing and develop an efficient non-convex algorithm based on alternating power iteration. The proposed algorithm is able to recover the underlying low-rank matrix model with linear convergence. For a rank-$k$ model with $d_1$ features and $d_2$ classes, the proposed algorithm achieves $O(\hat{I}\mu )$ recovery error after retrieving $O(k^{1.5}{d}_1{d}_2/\hat{I}\mu )$ one-bit labels within $O(kd)$ memory. Our bound is nearly optimal in the order of $O(1/\hat{I}\mu )$. This significantly improves the state-of-the-art sampling complexity of one-bit multi-label learning. We perform experiments to verify our theory and evaluate the performance of the proposed algorithm.