# Coordinate-wise Armijo's condition: General case

Let z=(x,y) be coordinates for the product space R^m_1×R^m_2. Let f:R^m_1×R^m_2→R be a C^1 function, and ∇ f=(∂ _xf,∂ _yf) its gradient. Fix 0<α <1. For a point (x,y) ∈R^m_1×R^m_2, a number δ >0 satisfies Armijo's condition at (x,y) if the following inequality holds: f(x-δ∂ _xf,y-δ∂ _yf)-f(x,y)≤ -αδ (||∂ _xf||^2+||∂ _yf||^2). In one previous paper, we proposed the following 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: [f(x-δ _1∂ _xf(x,y), y-δ _2∂ _y f(x,y))]-[f(x,y)]≤ -α (δ _1||∂ _xf(x,y)||^2+δ _2||∂ _yf(x,y)||^2). Previously we applied this condition for functions of the form f(x,y)=f(x)+g(y), and proved various convergent results for them. For a general function, it is crucial - for being able to do real computations - to have a systematic algorithm for obtaining δ _1 and δ _2 satisfying the coordinate-wise version of Armijo's condition, much like Backtracking for the usual Armijo's condition. In this paper we propose such an algorithm, and prove according convergent results. We then analyse and present experimental results for some functions such as f(x,y)=a|x|+y (given by Asl and Overton in connection to Wolfe's method), f(x,y)=x^3 sin (1/x) + y^3 sin(1/y) and Rosenbrock's function.

## Authors

• 9 publications
• ### Coordinate-wise Armijo's condition

Let z=(x,y) be coordinates for the product space R^m_1×R^m_2. Let f:R^m_...
11/18/2019 ∙ by Tuyen Trung Truong, et al. ∙ 0

• ### Some convergent results for Backtracking Gradient Descent method on Banach spaces

Our main result concerns the following condition: Condition C. Let X ...
01/16/2020 ∙ by Tuyen Trung Truong, et al. ∙ 0

• ### Finding best possible constant for a polynomial inequality

Given a multi-variant polynomial inequality with a parameter, how to fin...
03/04/2016 ∙ by Lu Yang, et al. ∙ 0

• ### Biased halfspaces, noise sensitivity, and relative Chernoff inequalities (extended version)

In analysis of Boolean functions, a halfspace is a function f:{-1,1}^n →...
10/20/2017 ∙ by Nathan Keller, et al. ∙ 0

• ### Efficient coordinate-wise leading eigenvector computation

We develop and analyze efficient "coordinate-wise" methods for finding t...
02/25/2017 ∙ by Jialei Wang, et al. ∙ 0

• ### An improved bound on ℓ_q norms of noisy functions

Let T_ϵ, 0 ≤ϵ≤ 1/2, be the noise operator acting on functions on the boo...
10/06/2020 ∙ by Alex Samorodnitsky, et al. ∙ 0

• ### Asymptotic behaviour of learning rates in Armijo's condition

Fix a constant 0<α <1. For a C^1 function f:ℝ^k→ℝ, a point x and a posit...
07/07/2020 ∙ by Tuyen Trung Truong, et al. ∙ 0

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