Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)
Cong Han Lim
We study a generalization of the classic isotonic regression problem where we allow separable nonconvex objective functions, focusing on the case of estimators used in robust regression. A simple dynamic programming approach allows us to solve this problem to within ε-accuracy (of the global minimum) in time linear in 1/ε and the dimension. We can combine techniques from the convex case with branch-and-bound ideas to form a new algorithm for this problem that naturally exploits the shape of the objective function. Our algorithm achieves the best bounds for both the general nonconvex and convex case (linear in log (1/ε)), while performing much faster in practice than a straightforward dynamic programming approach, especially as the desired accuracy increases.