最優化算法-梯度下降

/*****************************************
 * FileName  : grad.go
 * Author    : fredric
 * Date      : 2017.09.01
 * Note      : 梯度算法
 * History   :
*****************************************/
package grad

import(
    "fmt"
    "math"
)

//無法采用牛頓方法求得極值，主要原因在于無法確定初始值，造成導數偏差很大
func _get_argmin_newton(x1, x2, x3, grad_x1, grad_x2, grad_x3 float64) float64 {

    fmt.Printf("_get_argmin input value %f,%f,%f,%f,%f,%f\n", x1, x2, x3, grad_x1, grad_x2, grad_x3)

    //f(x - a*delta) = (x1 - a * grad_x1 - 4)^4 + (x2 - a * grad_x2 - 3)^2 + 4 * (x3 - a*grad_x3 + 5)^4
    //f'(x - a*delta) = 4 * grad_x1 * (x1 - a * grad_x1 - 4)^3
    //                + 2 * grad_x2 * (x2 - a * grad_x2 - 3)
    //                + 16* grad_x3 * (x3 - a*grad_x3 + 5)^3
    //f''(x - a*delta)= 12 * grad_x1^2 * (x1 - a * grad_x1 - 4)^2
    //                + 2  * grad_x2^2 * a
    //                + 48 * grad_x3^2 * (x3 - a*grad_x3 + 5)^2
    //采用牛頓法求取f(a)的最小值
    
    //此處的初始值還是比較疑惑，因為初始值取不對，結果差太遠
    var a0 float64 = 0.0002
    var a1 float64 = 0.0005
    delta := 0.0005

    for math.Abs(a1 - a0) > delta {

        a0 = a1

        //fmt.Printf("a0: %f\n" , a0)
        //fmt.Printf("grad_x2: %f\n" , grad_x2)
        //fmt.Printf("grad_x2 * a0: %f\n" , grad_x2 * a0)
        //fmt.Printf("grad_x2 * 0.2: %f\n" , grad_x2 * 0.2)

        f_1_v := 4 * grad_x1 * (x1 - a0 * grad_x1 - 4)* (x1 - a0 * grad_x1 - 4)* (x1 - a0 * grad_x1 - 4) + 
        2 * grad_x2 * (x2 - a0 * grad_x2 - 3) + 
        16* grad_x3 * (x3 - a0 * grad_x3 + 5)* (x3 - a0 * grad_x3 + 5) * (x3 - a0 * grad_x3 + 5)


        f_2_v := 12 * grad_x1 * grad_x1 * (x1 - a1 * grad_x1 - 4)* (x1 - a1 * grad_x1 - 4) + 2  * grad_x2* grad_x2 * a1 + 48 * grad_x3* grad_x3 * (x3 - a1 * grad_x3 + 5)* (x3 - a1 * grad_x3 + 5)

        a1 = a0 - f_1_v / f_2_v

        //fmt.Printf("----------abs = %f\n", math.Abs(a1 - a0))

    
        fmt.Printf("step value = %f f_1_v = %f, f_2_v = %f\n", (a0 + a1)/2, f_1_v, f_2_v)
    }

    return (a0 + a1)/2
}

//采用常量方式求極值
func _get_argmin_const(x1, x2, x3, grad_x1, grad_x2, grad_x3 float64) float64{


    /*
    * 不是很搞的清楚，當采用快速下降算法時如何確定固定步長，網上有一個說法實踐是正確的
    * 即滿足李普希茲條件存在L>0使得|f(x1)-f(x2)|<=L|x1-x2|，步長取1/L
    * 下面這個例子由于存在x3這個高階，所以如果步長取大的話，完全沒有辦法計算
    */

    return 0.0004
}

func DoGradAlgorithm(){

    //計算f(x1,x2,x3) = (x1 - 4)^4 + (x2 - 3)^2 + 4*(x3 + 5)^4
    //所謂梯度本質上也是導數，只是針對多維度上，取了各個維度偏導數，組成向量；
    //最速下降法就是在每次迭代時取當前負梯度方向的能獲取的函數數最小值

    //初始值x0 = [4, 2, -1]
    x1 := 4.0
    x2 := 2.0
    x3 := -1.0

    //取三次迭代
    for i := 0; i < 4; i++ {

        grad_x1 := 4 * (x1 - 4)*(x1 - 4)*(x1 - 4)
        grad_x2 := 2 * (x2 - 3)
        grad_x3 := 16 * (x3 + 5)* (x3 + 5)* (x3 + 5)

        a := _get_argmin_newton(x1,x2,x3, grad_x1, grad_x2, grad_x3)

        fmt.Printf("grad_x1 = %f, grad_x2 = %f, grad_x3 = %f\n", grad_x1, grad_x2, grad_x3)

        x1 = x1 - a * grad_x1
        x2 = x2 - a * grad_x2
        x3 = x3 - a * grad_x3

        fmt.Printf("x1 = %f, x2 = %f, x3 = %f\n", x1, x2, x3)

    }
}