Online Isotonic Regression
This work addresses the challenge of efficient online learning with monotonicity constraints, which is incremental as it adapts existing methods to a specific problem with new regret bounds.
The paper tackles the online isotonic regression problem, where a learner predicts labels sequentially against adversarial positions and is evaluated by squared loss compared to the best isotonic function in hindsight; it proves that an Exponential Weights algorithm over a covering net achieves a regret bound of O(T^{1/3} log^{2/3}(T)) with a matching lower bound of Ω(T^{1/3}), and improves this to O(log T) or O(1) in noise-free cases.
We consider the online version of the isotonic regression problem. Given a set of linearly ordered points (e.g., on the real line), the learner must predict labels sequentially at adversarially chosen positions and is evaluated by her total squared loss compared against the best isotonic (non-decreasing) function in hindsight. We survey several standard online learning algorithms and show that none of them achieve the optimal regret exponent; in fact, most of them (including Online Gradient Descent, Follow the Leader and Exponential Weights) incur linear regret. We then prove that the Exponential Weights algorithm played over a covering net of isotonic functions has a regret bounded by $O\big(T^{1/3} \log^{2/3}(T)\big)$ and present a matching $Ω(T^{1/3})$ lower bound on regret. We provide a computationally efficient version of this algorithm. We also analyze the noise-free case, in which the revealed labels are isotonic, and show that the bound can be improved to $O(\log T)$ or even to $O(1)$ (when the labels are revealed in isotonic order). Finally, we extend the analysis beyond squared loss and give bounds for entropic loss and absolute loss.