Suspiciousness of loading problems

We introduce the notion of suspect families of loading problems in the attempt of formalizing situations in which classical learning algorithms based on local optimization are likely to fail (because of local minima or numerical precision problems). We show that any loading problem belonging to a nonsuspect family can be solved with optimal complexity by a canonical form of gradient descent with forced dynamics (i.e., for this class of problems no algorithm exhibits a better computational complexity than a slightly modified form of backpropagation). The analyses of this paper suggest intriguing links between the shape of the error surface attached to parametrical learning systems (like neural networks) and the computational complexity of the corresponding optimization problem.