Randomized Forward Mode of Automatic Differentiation For Optimization Algorithms

TL;DR

This paper introduces a randomized forward mode gradient (RFG) as an alternative to backpropagation, providing a new way to compute gradients efficiently for optimization algorithms with theoretical and experimental validation.

Contribution

It proposes RFG as a novel gradient estimator based on directional derivatives, analyzing its statistical properties and integrating it into optimization algorithms like GD and PHB.

Findings

Distribution with smallest kurtosis minimizes expected error

RFG-based algorithms converge on quadratic functions

Experimental results confirm theoretical predictions

Abstract

We present a randomized forward mode gradient (RFG) as an alternative to backpropagation. RFG is a random estimator for the gradient that is constructed based on the directional derivative along a random vector. The forward mode automatic differentiation (AD) provides an efficient computation of RFG. The probability distribution of the random vector determines the statistical properties of RFG. Through the second moment analysis, we found that the distribution with the smallest kurtosis yields the smallest expected relative squared error. By replacing gradient with RFG, a class of RFG-based optimization algorithms is obtained. By focusing on gradient descent (GD) and Polyak's heavy ball (PHB) methods, we present a convergence analysis of RFG-based optimization algorithms for quadratic functions. Computational experiments are presented to demonstrate the performance of the proposed algorithms and verify the theoretical findings.