Smart Gradient - An adaptive technique for improving gradient estimation

Esmail Abdul Fattah; Janet Van Niekerk; Håvard Rue

doi:10.3934/fods.2021037

Article Contents

2022, Volume 4, Issue 1: 123-136. Doi: 10.3934/fods.2021037

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Smart Gradient - An adaptive technique for improving gradient estimation

CEMSE Division, King Abdullah University of Science and Technology, Kingdom of Saudi Arabia

^* Corresponding author: Esmail Abdul Fattah
^* Corresponding author: Esmail Abdul Fattah

Received: June 2021

Revised: December 2021

Early access: January 11, 2022

Published online: January 11, 2022

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Abstract

Computing the gradient of a function provides fundamental information about its behavior. This information is essential for several applications and algorithms across various fields. One common application that requires gradients are optimization techniques such as stochastic gradient descent, Newton's method and trust region methods. However, these methods usually require a numerical computation of the gradient at every iteration of the method which is prone to numerical errors. We propose a simple limited-memory technique for improving the accuracy of a numerically computed gradient in this gradient-based optimization framework by exploiting (1) a coordinate transformation of the gradient and (2) the history of previously taken descent directions. The method is verified empirically by extensive experimentation on both test functions and on real data applications. The proposed method is implemented in the $\texttt{R} $ package $ \texttt{smartGrad}$ and in C$ \texttt{++} $.

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Mathematics Subject Classification: Primary: 03-08; Secondary: 62-08.

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Figure 1. 2-dimensional Rosenbrock Function and a contour plot around the point p (-0.29, 0.4). At the same point p, the MSE and magnitude of gradient are plotted using different directions.

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Figure 2. Directions used to compute Vanilla Gradient (black) - Smart Gradient (iteration 1 - red) and Smart Gradient (iteration 2 - blue).

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Figure 3. MSE for Smart and Vanilla Gradients at each iteration using Different Dimensional Rosenbrock and Roth Functions.

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Figure 4. MSE and computational time (to reach optimum) as the function of step-size for different dimensional Rosenbrock Function.

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Figure 5. Triangulated mesh with 1000 locations on the unit square.

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Table 1. The average MSE when using Vanilla and Smart Gradient approaches compared to the exact gradient for different dimensional Rosenbrock and Roth functions.

$ \pmb x $ dimension	Average MSE Vanilla Gradient	Average MSE Smart Gradient	Improvement
Extended Rosenbrock Function
5	2.60e-04	1.04e-04	2.5
10	2.71e-04	0.78e-04	3.47
25	2.80e-04	0.49e-04	5.71
Extended Freudenstein Roth Function
5	2.26e-04	1.39e-04	1.63
10	2.78e-04	1.42e-04	1.96
25	2.84e-04	1.25e-04	2.27

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