Coordinate-wise Armijo's condition

Abstract

Let $z=(x,y)$ be coordinates for the product space $\mathbb{R}^{m_1}\times \mathbb{R}^{m_2}$. Let $f:\mathbb{R}^{m_1}\times \mathbb{R}^{m_2}\rightarrow \mathbb{R}$ be a $C^1$ function, and $\nabla f=(\partial _xf,\partial _yf)$ its gradient. Fix $0<α<1$. For a point $(x,y) \in \mathbb{R}^{m_1}\times \mathbb{R}^{m_2}$, a number $δ>0$ satisfies Armijo's condition at $(x,y)$ if the following inequality holds: \begin{eqnarray*} f(x-δ\partial _xf,y-δ\partial _yf)-f(x,y)\leq -αδ(||\partial _xf||^2+||\partial _yf||^2). \end{eqnarray*} When $f(x,y)=f_1(x)+f_2(y)$ is a coordinate-wise sum map, we propose the following {\bf coordinate-wise} Armijo's condition. Fix again $0<α<1$. A pair of positive numbers $δ_1,δ_2>0$ satisfies the coordinate-wise variant of Armijo's condition at $(x,y)$ if the following inequality holds: \begin{eqnarray*} [f_1(x-δ_1\nabla f_1(x))+f_2(y-δ_2\nabla f_2(y))]-[f_1(x)+f_2(y)]\leq -α(δ_1||\nabla f_1(x)||^2+δ_2||\nabla f_2(y)||^2). \end{eqnarray*} We then extend results in our recent previous results, on Backtracking Gradient Descent and some variants, to this setting. We show by an example the advantage of using coordinate-wise Armijo's condition over the usual Armijo's condition.