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--- 深度学习:反向传播算法 [2026/03/02 21:36] – [选择题答案] 张叶安
+++ 深度学习:反向传播算法 [2026/03/02 21:37] (当前版本) – [计算题答案] 张叶安
@@ 行 476: / 行 476: @@
 . **解答**：
-    **前向传播**：
+**前向传播**：
-    $$z = 0.5 \times 1 + (-0.3) \times 2 = 0.5 - 0.6 = -0.1$$
-    $$\hat{y} = \sigma(-0.1) = \frac{1}{1 + e^{0.1}} \approx \frac{1}{1.105} \approx 0.475$$
+$$z = 0.5 \times 1 + (-0.3) \times 2 = 0.5 - 0.6 = -0.1$$
+$$\hat{y} = \sigma(-0.1) = \frac{1}{1 + e^{0.1}} \approx \frac{1}{1.105} \approx 0.475$$
-    **反向传播**：
+**反向传播**：
-    $$\frac{\partial \mathcal{L}}{\partial \hat{y}} = -\frac{y}{\hat{y}} + \frac{1-y}{1-\hat{y}} = -\frac{1}{0.475} \approx -2.105$$
+$$\frac{\partial \mathcal{L}}{\partial \hat{y}} = -\frac{y}{\hat{y}} + \frac{1-y}{1-\hat{y}} = -\frac{1}{0.475} \approx -2.105$$
-    $$\frac{\partial \hat{y}}{\partial z} = \sigma(-0.1)(1-\sigma(-0.1)) = 0.475 \times 0.525 \approx 0.249$$
+$$\frac{\partial \hat{y}}{\partial z} = \sigma(-0.1)(1-\sigma(-0.1)) = 0.475 \times 0.525 \approx 0.249$$
-    $$\frac{\partial z}{\partial w_1} = x_1 = 1, \quad \frac{\partial z}{\partial w_2} = x_2 = 2$$
+$$\frac{\partial z}{\partial w_1} = x_1 = 1, \quad \frac{\partial z}{\partial w_2} = x_2 = 2$$
-    $$\frac{\partial \mathcal{L}}{\partial w_1} = -2.105 \times 0.249 \times 1 \approx -0.524$$
+$$\frac{\partial \mathcal{L}}{\partial w_1} = -2.105 \times 0.249 \times 1 \approx -0.524$$
-    $$\frac{\partial \mathcal{L}}{\partial w_2} = -2.105 \times 0.249 \times 2 \approx -1.048$$
+$$\frac{\partial \mathcal{L}}{\partial w_2} = -2.105 \times 0.249 \times 2 \approx -1.048$$
 . **解答**：
-    **计算一阶矩**：
+**计算一阶矩**：
-    $$\mathbf{m}_t = 0.9 \times [0.1, 0.1] + 0.1 \times [0.2, -0.3]$$
-    $$= [0.09, 0.09] + [0.02, -0.03] = [0.11, 0.06]$$
+$$\mathbf{m}_t = 0.9 \times [0.1, 0.1] + 0.1 \times [0.2, -0.3]$$
+$$= [0.09, 0.09] + [0.02, -0.03] = [0.11, 0.06]$$
-    **计算二阶矩**：
+**计算二阶矩**：
-    $$\mathbf{v}_t = 0.999 \times [0.01, 0.01] + 0.001 \times [0.04, 0.09]$$
-    $$= [0.00999, 0.00999] + [0.00004, 0.00009] = [0.01003, 0.01008]$$
+$$\mathbf{v}_t = 0.999 \times [0.01, 0.01] + 0.001 \times [0.04, 0.09]$$
+$$= [0.00999, 0.00999] + [0.00004, 0.00009] = [0.01003, 0.01008]$$
-    **偏差校正**：
+**偏差校正**：
-    $$1 - \beta_1^t = 1 - 0.9^{10} = 1 - 0.349 = 0.651$$
-    $$1 - \beta_2^t = 1 - 0.999^{10} = 1 - 0.990 = 0.010$$
+$$1 - \beta_1^t = 1 - 0.9^{10} = 1 - 0.349 = 0.651$$
+$$1 - \beta_2^t = 1 - 0.999^{10} = 1 - 0.990 = 0.010$$
-    $$\hat{\mathbf{m}}_t = \frac{[0.11, 0.06]}{0.651} \approx [0.169, 0.092]$$
+$$\hat{\mathbf{m}}_t = \frac{[0.11, 0.06]}{0.651} \approx [0.169, 0.092]$$
-    $$\hat{\mathbf{v}}_t = \frac{[0.01003, 0.01008]}{0.010} \approx [1.003, 1.008]$$
+$$\hat{\mathbf{v}}_t = \frac{[0.01003, 0.01008]}{0.010} \approx [1.003, 1.008]$$
----
-**本章完**

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