OUTPUT ERROR
输出误差Output error
∂L/∂ŷ 表示输出再增一点，损失怎样变。∂L/∂ŷ asks how loss changes if output rises slightly.
01 · 一条装配线出了错01 · A production line misses target
最终尺寸偏了 0.268，前面每个旋钮都该被同样拧回去吗？The final measurement misses by 0.268. Should every upstream dial be corrected by the same amount?
02点输出误差，看责任信号逐层回流Click Output Error and Watch Responsibility Flow Back
唯一变量：反传阶段One variable: backward stage
反传阶段Backward stage
0/4
已回流层级levels returned
输出误差Output error
−0.268
∂L/∂ŷ
已得到梯度Gradients ready
0/6
条参数边parameter edges
灰/橙：前向数值gray/amber: forward values
红：输出责任red: output responsibility
青：已算出的参数梯度cyan: computed parameter gradient
03反向传播传的不是原数据Backprop Does Not Send the Original Data Back
error · local derivative · responsibility · gradientLOCAL RULE
局部导数Local derivative
每条边只报告自己的乘法关系。Each edge reports only its local multiplication rule.
RESPONSIBILITY
节点责任Node responsibility
上游责任乘本地导数，传给更前一层。Upstream responsibility times local derivative moves back.
PARAM GRAD
参数梯度Parameter gradient
每个权重得到自己对最终损失的影响。Each weight gets its own effect on final loss.
04最小机制：链式法则把局部关系串起来Minimal Mechanism: The Chain Rule Connects Local Relations
upstream × local = downstream responsibility∂L/∂w = ∂L/∂ŷ × ∂ŷ/∂w
本页输出层示例：∂L/∂v1 = (−0.268) × h1(0.680) =
−0.182。越往前，乘法链越长。Output-edge example: ∂L/∂v1 = (−0.268) × h1(0.680) = −0.182. Earlier
parameters have longer chains.
前向先保存数值Forward saves valuesx、h、ŷ 成为局部导数所需材料x, h, and ŷ feed local derivatives
误差从右向左Error moves right to left每站乘自己的局部导数each stop multiplies its local derivative
每条参数边得梯度Every parameter gets a gradient优化器再决定怎样更新optimizer later decides the update
05边界实验：把同一份误差平均甩给所有权重Boundary Test: Assign the Same Error to Every Weight
主动制造失败Create a failure跳过链式法则Skip the chain rule
故意把 −0.268 原样贴到六条边。画布会标红：不同输入值、权重和路径却被判成同等责任。Paste −0.268 onto all six edges. The graph turns red because different inputs, weights, and paths receive identical responsibility.
尚未执行。先完成一次正确反传。Not run yet. Run correct backprop first.
✗ 反向传播把输入数据倒着播放。Backprop replays input data backward.
→ ✓ 它传播损失对中间量的导数。It propagates derivatives of loss with respect to intermediates.
✗ 所有权重对错误负同样责任。Every weight shares equal responsibility.
→ ✓ 责任由路径上的局部导数连乘决定。Responsibility comes from products of local derivatives.
✗ 反传本身已经更新了参数。Backprop itself has updated the parameters.
→ ✓ 反传只算梯度，优化步骤另行使用它。Backprop computes gradients; optimization uses them later.
06 · 一句话带走06 · One line to keep