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The Broyden-Fletcher-Goldfarb-Shanno One-StepMemoryless Quasi-Newton Method

Steepest descent suffers from a bad reputation with researchers in optimization. From the literature (e.g., [Gill:81a]), we found a wide selection of more appropriate optimization  techniques. Following the ``decision tree'' and considering the characteristics of large supervised learning problems (large memory requirements and time-consuming calculations of the energy and the gradient), the Broyden-Fletcher-Goldfarb-Shanno one-step memoryless quasi-Newton method (all adjectives are necessary to define it) is a good candidate in the competition and performed very efficiently on different problems.

Let's define the following vectors: , and . The one-dimensional search direction for the nth iteration is a modification of the gradient , as follows:

Every N steps (N being the number of weights in the network), the search is restarted in the direction of the negative gradient.

The coefficients and are combinations of scalar products:

The one-dimensional minimization  used in this work is based on quadratic interpolation and tuned to back-propagation where, in a single step, both the energy value and the negative gradient can be efficiently obtained. Details on this step are contained in [Williams:87b].

The computation during each step requires operations (the same behavior as standard batch back-propagation), while the CPU time for each step increases by an average factor of three for the problems considered. Because the total number of steps for convergence  is much smaller, we measured a large net benefit in computing time.

Last but not least, this method can be efficiently implemented on MIMD parallel computers.

Guy Robinson
Wed Mar 1 10:19:35 EST 1995