NDArray可以很方便的求解导数,比如下面的例子:(代码主要参考自https://zh.gluon.ai/chapter_crashcourse/autograd.html)
用代码实现如下:
1 import mxnet.ndarray as nd 2 import mxnet.autograd as ag 3 x = nd.array([[1,2],[3,4]]) 4 print(x) 5 x.attach_grad() #附加导数存放的空间 6 with ag.record(): 7 y = 2*x**2 8 y.backward() #求导 9 z = x.grad #将导数结果(也是一个矩阵)赋值给z 10 print(z) #打印结果
[[ 1. 2.] [ 3. 4.]] <NDArray 2x2 @cpu(0)> [[ 4. 8.] [ 12. 16.]] <NDArray 2x2 @cpu(0)>
对控制流求导
NDArray还能对诸如if的控制分支进行求导,比如下面这段代码:
1 def f(a): 2 if nd.sum(a).asscalar()<15: #如果矩阵a的元数和<15 3 b = a*2 #则所有元素*2 4 else: 5 b = a 6 return b
数学公式等价于:
这样就转换成本文最开头示例一样,变成单一函数求导,显然导数值就是x前的常数项,验证一下:
import mxnet.ndarray as nd import mxnet.autograd as ag def f(a): if nd.sum(a).asscalar()<15: #如果矩阵a的元数和<15 b = a*2 #则所有元素平方 else: b = a return b #注:1+2+3+4<15,所以进入b=a*2的分支 x = nd.array([[1,2],[3,4]]) print("x1=") print(x) x.attach_grad() with ag.record(): y = f(x) print("y1=") print(y) y.backward() #dy/dx = y/x 即:2 print("x1.grad=") print(x.grad) x = x*2 print("x2=") print(x) x.attach_grad() with ag.record(): y = f(x) print("y2=") print(y) y.backward() print("x2.grad=") print(x.grad)
x1= [[ 1. 2.] [ 3. 4.]] <NDArray 2x2 @cpu(0)>
y1= [[ 2. 4.] [ 6. 8.]] <NDArray 2x2 @cpu(0)>
x1.grad= [[ 2. 2.] [ 2. 2.]] <NDArray 2x2 @cpu(0)>
x2= [[ 2. 4.] [ 6. 8.]] <NDArray 2x2 @cpu(0)>
y2= [[ 2. 4.] [ 6. 8.]] <NDArray 2x2 @cpu(0)>
x2.grad= [[ 1. 1.] [ 1. 1.]] <NDArray 2x2 @cpu(0)>
头梯度
原文上讲得很含糊,其实所谓头梯度,就是一个求导结果前的乘法系数,见下面代码:
1 import mxnet.ndarray as nd 2 import mxnet.autograd as ag 3 4 x = nd.array([[1,2],[3,4]]) 5 print("x=") 6 print(x) 7 8 x.attach_grad() 9 with ag.record(): 10 y = 2*x*x 11 12 head = nd.array([[10, 1.], [.1, .01]]) #所谓的"头梯度" 13 print("head=") 14 print(head) 15 y.backward(head_gradient) #用头梯度求导 16 17 print("x.grad=") 18 print(x.grad) #打印结果
x= [[ 1. 2.] [ 3. 4.]] <NDArray 2x2 @cpu(0)>
head= [[ 10. 1. ] [ 0.1 0.01]] <NDArray 2x2 @cpu(0)>
x.grad= [[ 40. 8. ] [ 1.20000005 0.16 ]] <NDArray 2x2 @cpu(0)>
对比本文最开头的求导结果,上面的代码仅仅多了一个head矩阵,最终的结果,其实就是在常规求导结果的基础上,再乘上head矩阵(指:数乘而非叉乘)
链式法则
先复习下数学
注:最后一行中所有变量x,y,z都是向量(即:矩形),为了不让公式看上去很凌乱,就统一省掉了变量上的箭头。NDArray对复合函数求导时,已经自动应用了链式法则,见下面的示例代码:
1 import mxnet.ndarray as nd 2 import mxnet.autograd as ag 3 4 x = nd.array([[1,2],[3,4]]) 5 print("x=") 6 print(x) 7 8 x.attach_grad() 9 with ag.record(): 10 y = x**2 11 z = y**2 + y 12 13 z.backward() 14 15 print("x.grad=") 16 print(x.grad) #打印结果 17 18 print("w=") 19 w = 4*x**3 + 2*x 20 print(w) # 验证结果
x= [[ 1. 2.] [ 3. 4.]] <NDArray 2x2 @cpu(0)>
x.grad= [[ 6. 36.] [ 114. 264.]] <NDArray 2x2 @cpu(0)>
w= [[ 6. 36.] [ 114. 264.]] <NDArray 2x2 @cpu(0)>