Before revision:  It is known that the sample complexity of any solution to the identification problem must be at least exponential in k. Previous results demonstrated either worse sample complexity and worse } O(k ^{2+o(1) .
After revision:  It is known that the sample complexity of any solution to the identification problem must be at least exponential in k. Previous results demonstrated either worse sample complexity and worse } O(k ^c) runtime for some c substantially larger than 2, or similar sample complexity and much worse k^{O(k^2) .