An efficient, provably exact algorithm for the 0-1 loss linear classification problem

Algorithms for solving the linear classification problem have a long history, dating back at least to 1936 with linear discriminant analysis. For linearly separable data, many algorithms can obtain the exact solution to the corresponding 0-1 loss classification problem efficiently, but for data which is not linearly separable, it has been shown that this problem, in full generality, is NP-hard. Alternative approaches all involve approximations of some kind, including the use of surrogates for the 0-1 loss (for example, the hinge or logistic loss) or approximate combinatorial search, none of which can be guaranteed to solve the problem exactly. Finding efficient algorithms to obtain an exact i.e. globally optimal solution for the 0-1 loss linear classification problem with fixed dimension, remains an open problem. In research we report here, we detail the construction of a new algorithm, incremental cell enumeration (ICE), that can solve the 0-1 loss classification problem exactly in polynomial time. To our knowledge, this is the first, rigorously-proven polynomial time algorithm for this long-standing problem.