通过torch.topk导出梯度

Question

我想通过函数导出梯度torch.topk。

假设我的输入是一个向量 ,

然后通过参数矩阵进行变换 ,

并选择向量的前 k 个值 。

结果向量通过逐元素乘法进一步变换。

最后损失计算如下

。

我想知道，损失相对于 W 可微吗？形式上，我们可以计算以下梯度吗？

Answer 1

该topk()操作只是一个线性变换，用于选取张量的前k个元素。由于这是一种W @ X矩阵向量乘法运算，因此它也是可微分的。

示例：下面我以两种方式计算了流水线操作topk(Wx)，并显示两种方式产生的梯度是相同的。

In [1]: import torch

In [2]: x1 = torch.rand(6, requires_grad = True)

In [3]: W1 = torch.rand(6, 6, requires_grad = True)

In [4]: x1
Out[4]: tensor([0.1511, 0.5990, 0.6338, 0.5137, 0.5203, 0.0560], requires_grad=True)

In [5]: W1
Out[5]: 
tensor([[0.2541, 0.6699, 0.5311, 0.7801, 0.5042, 0.5475],
        [0.7523, 0.1331, 0.7670, 0.8132, 0.0524, 0.0269],
        [0.3974, 0.2880, 0.9142, 0.9906, 0.4401, 0.3984],
        [0.7956, 0.2071, 0.2209, 0.6192, 0.2054, 0.7693],
        [0.8587, 0.8415, 0.6033, 0.3812, 0.2498, 0.9813],
        [0.9033, 0.0417, 0.2272, 0.1576, 0.9087, 0.3284]], requires_grad=True)

In [6]: y1 = W1 @ x1

In [7]: y1
Out[7]: tensor([1.4699, 1.1260, 1.5721, 0.8523, 1.3969, 0.8776], grad_fn=<MvBackward>)

In [8]: yk, _ = torch.topk(y1, 3)

In [9]: yk
Out[9]: tensor([1.5721, 1.4699, 1.3969], grad_fn=<TopkBackward>)

In [10]: loss1 = (yk ** 2).sum()

In [11]: loss1.backward()

In [12]: W1.grad
Out[12]: 
tensor([[0.4442, 1.7609, 1.8633, 1.5102, 1.5296, 0.1646],
        [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000],
        [0.4751, 1.8833, 1.9928, 1.6152, 1.6359, 0.1760],
        [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000],
        [0.4222, 1.6734, 1.7706, 1.4352, 1.4535, 0.1564],
        [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]])

现在让我们评估同一组操作，但topk()显式地用作线性变换。请注意，构造的Wk矩阵通过乘法有选择地从 6 元素张量中挑选出前 k 个（此处为 3 个）元素。

In [13]: x2 = torch.tensor([0.1511, 0.5990, 0.6338, 0.5137, 0.5203, 0.0560], req
    ...: uires_grad=True)

In [14]: W2 = torch.tensor([[0.2541, 0.6699, 0.5311, 0.7801, 0.5042, 0.5475],
    ...:         [0.7523, 0.1331, 0.7670, 0.8132, 0.0524, 0.0269],
    ...:         [0.3974, 0.2880, 0.9142, 0.9906, 0.4401, 0.3984],
    ...:         [0.7956, 0.2071, 0.2209, 0.6192, 0.2054, 0.7693],
    ...:         [0.8587, 0.8415, 0.6033, 0.3812, 0.2498, 0.9813],
    ...:         [0.9033, 0.0417, 0.2272, 0.1576, 0.9087, 0.3284]], requires_gra
    ...: d=True)

In [15]: y2 = W2 @ x2

In [16]: y2
Out[16]: tensor([1.4700, 1.1260, 1.5721, 0.8523, 1.3969, 0.8776], grad_fn=<MvBackward>)

# Use the indices obtained earlier to construct the matrix
In [19]: _
Out[19]: tensor([2, 0, 4])

In [20]: k = 3

In [21]: Wk = torch.zeros(k, y2.shape[0])

In [22]: Wk[torch.arange(k), _] = 1

In [23]: Wk.requires_grad = True

In [24]: Wk
Out[24]: 
tensor([[0., 0., 1., 0., 0., 0.],
        [1., 0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 1., 0.]], requires_grad=True)


In [25]: yk2 = Wk @ y2

In [26]: yk2
Out[26]: tensor([1.5721, 1.4700, 1.3969], grad_fn=<MvBackward>)

In [27]: loss2 = (yk2 ** 2).sum()

In [28]: loss2.backward()

现在比较两种情况下获得的梯度：

In [29]: W2.grad
Out[29]: 
tensor([[0.4442, 1.7611, 1.8634, 1.5103, 1.5297, 0.1646],
        [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000],
        [0.4751, 1.8834, 1.9929, 1.6152, 1.6360, 0.1761],
        [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000],
        [0.4222, 1.6735, 1.7707, 1.4352, 1.4536, 0.1565],
        [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]])

In [30]: W1.grad
Out[30]: 
tensor([[0.4442, 1.7609, 1.8633, 1.5102, 1.5296, 0.1646],
        [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000],
        [0.4751, 1.8833, 1.9928, 1.6152, 1.6359, 0.1760],
        [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000],
        [0.4222, 1.6734, 1.7706, 1.4352, 1.4535, 0.1564],
        [0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000]])

In [31]: x1.grad
Out[31]: tensor([4.3955, 5.2256, 6.1213, 6.4732, 3.5637, 5.6037])

In [32]: x2.grad
Out[32]: tensor([4.3957, 5.2261, 6.1215, 6.4733, 3.5641, 5.6040])

正如您所看到的，除了一些浮点错误之外，结果是相同的，这些错误是在我复制 和 的值时引入的x1，但W1没有采用它们的完整精度。