Arithmetic Simplicity in Stochastic Gradient Methods

A gradient descent method is arithmetically simple if the operations are limited to +,-, ×, and division x/2t with integer t. An arthmetically simple gradient method is easy to implement in chip design. We show how to transform AdaGrad, Adam, and AdamW into arithmetically simple. AdamW is based on the recursion xt+1=(1-λη)xt-fracη smt and Adam is the special case of AdamW with λ=0. We transform them into a static case with s=S(T), where T is the number of iterations, and S(T) is a fixed function. The convergence analysis is given for a static Adam, which is also arithmetically simple.