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lossgrad : Automatic Learning Rate in Gradient Descent

Wójcik Bartosz, Maziarka Łukasz, Tabor Jacek

Schedae Informaticae

2018

Vol. 27

47--57

In this paper, we propose a simple, fast and easy to implement algorithm lossgrad (locally optimal step-size in gradient descent), which au- tomatically modifies the step-size in gradient descent during neural networks training. Given a function f, a point x, and the gradient rxf of f, we aim to nd the step-size h which is (locally) optimal, i.e. satisfies: h = arg min t0 f(x 􀀀 trxf): Making use of quadratic approximation, we show that the algorithm satisfies the above assumption. We experimentally show that our method is insensitive to the choice of initial learning rate while achieving results comparable to other methods.

A local truncation error estimation for a SubIval solver

Sowa M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2018

Vol. 66, nr 4

475--484

The paper concerns an analysis for SubIval (the subinterval-based method for fractional derivative computations in initial value problems). A time step size adaptive solver is discussed, for which the formula of a local truncation error is derived. A general form for a system of linear equations is given for the considered class of problems (for which the analysis is performed in the paper). Two circuit examples are introduced to display the usefulness of the SubIval solver. For the examples that have been chosen it is possible to obtain referential solutions through completely different methods. The results obtained through the numerical solver are compared with evaluations of the referential solutions. The error estimation results obtained for the time steps of the SubIval solver are compared with the actual errors, being the differences between the numerical solutions and the referential solutions. The paper also contains a comparison of the accuracy of results obtained through the SubIval solver with the accuracies of other solvers.