Fast Algorithms for Segmented Regression

We study the fixed design segmented regression problem – a classical statistical inference task that encompasses several natural problems as a special case. Given noisy samples from a piecewise linear function f , we want to recover f up to a desired accuracy in mean-squared error. Previous rigorous approaches for this problem rely on dynamic programming (DP) and, while sample efficient, have running time quadratic in the sample size. As our main contribution, we provide new nearly-linear time algorithms for the problem that – while not being minimax optimal – achieve a significantly better sample-time tradeoff on large data sets compared to the DP approach. Our experimental evaluation shows that our algorithms provide a convergence rate that is only off by a factor of 2 to 4 compared to the DP approach, while achieving a speedup of three orders of magnitude.