CS224d Assignment1 答案, Part(1/4)

下面是自己刷视频的过程中整理得到的作业答案。题目是2016春季学期发布的版本 (March - June, 2016)。这里只包含了需要推导的内容（万幸CSDN支持 LATEX ），代码实现稍后找个git上传上来。

Assignment1的答案一共被我分成了4部分，分别包含第1，2，3，4题。这部分包含第1题的答案。

1. Softmax (10 points)

(a). (5 points) Prove that softmax is invariant to constant offsets in the input, that is, for any input vector x and any constant c ,

softmax(x)=softmax(x+c)

where x+c means adding the constant c to every dimension of x . Remember that

softmax(x)i=exi∑jexj(1)

Note: In practice, we make use of this property and choose c=−maxixi when computing softmax probabilities for numerical stability (i.e. subtracting its maximum element from all elements of x ).

解：

softmax(x+c)i=exi+c∑jexj+c=ecexiec∑jexj=exi∑jexj=softmax(x)

综上，命题得证

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(b). 见代码，略

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