Online Learning in Variable Feature Spaces under Incomplete Supervision

This paper explores a new online learning problem where the input sequence lives in an over-time varying feature space and the ground-truth label of any input point is given only occasionally, making online learners less restrictive and more applicable. The crux in this setting lies in how to exploit the very limited labels to efficiently update the online learners. Plausible ideas such as propagating labels from labeled points to their neighbors through uncovering the point-wise geometric relations face two challenges: (1) distance measurement fails to work as different points may be described by disparate sets of features and (2) storing the geometric shape, which is formed by all arrived points, is unrealistic in an online setting. To address these challenges, we first construct a universal feature space that accumulates all observed features, making distance measurement feasible. Then, we use manifolds to represent the geometric shapes and approximate them in a sparse means, making manifolds computational and memory tractable in online learning. We frame these two building blocks into a regularized risk minimization algorithm. Theoretical analysis and empirical evidence substantiate the viability and effectiveness of our proposal.