Second-order learning algorithms based on quasi-Newton methods have two problems. First, standard quasi-Newton methods are impractical for large-scale problems because they require N2 storage space to maintain an approximation to an inverse Hessian matrix (N is the number of weights). Second, a line search to calculate areasonably accurate step length is indispensable for these algorithms. In order to provide desirable performance, an efficient and reasonably accurate line search is needed. To overcome these problems, we propose a new second-order learning algorithm. Descent direction is calculated on the basis of a partial Broydon-Fletcher-Goldfarb-Shanno (BFGS) update with 2Ns memory space (s « N), and a reasonably accurate step length is efficiently calculated as the minimal point of a second-order approximation to the objective function with respect to the step length. Our experiments, which use a parity problem and a speech synthesis problem, have shown that the proposed algorithm outperformed major learning algorithms. Moreover, it turned out that an efficient and accurate step-length calculation plays an important role for the convergence of quasi-Newton algorithms, and a partial BFGS update greatly saves storage space without losing the convergence performance.
A Class of Diagonal Quasi-Newton Methods for Large-Scale Convex Minimization