神经网络反向传播中为何在导数前添加负号？

Great question—this is one of the most common "wait, why does that happen?" moments when learning neural networks. Let’s break this down clearly:

Why do we add the negative sign? #

The core idea here is gradient descent: our goal is to minimize the loss function ( L ). The gradient ( \frac{\partial L}{\partial \theta} ) (where ( \theta ) is any parameter like weights or biases) tells us the direction in which the loss increases most steeply. To minimize the loss, we need to update our parameters in the opposite direction of this gradient.

Think of it like hiking down a hill: if the gradient points uphill (toward higher loss), we need to step downhill (toward lower loss)—the negative sign flips our direction to do exactly that.

What if the gradient is already negative? Do we skip the sign? #

No—you always need the negative sign, regardless of the gradient’s sign. Here’s why:

• If ( \frac{\partial L}{\partial \theta} > 0 ): Increasing ( \theta ) makes the loss bigger. So we subtract a positive value (the gradient times learning rate) from ( \theta ) to decrease it, moving toward lower loss.
• If ( \frac{\partial L}{\partial \theta} < 0 ): Increasing ( \theta ) makes the loss smaller. The negative sign turns this negative gradient into a positive value, so we end up adding to ( \theta ) (since ( \theta = \theta - \alpha * (\text{negative number}) = \theta + \alpha * |\text{gradient}| )), which is exactly the direction we need to go.

Let’s use a concrete example to make this tangible:
Suppose our loss function is ( L = (w - 5)^2 ) (we want ( w = 5 ) to minimize loss).

• When ( w = 6 ): ( \frac{\partial L}{\partial w} = 2*(6-5) = 2 ) (positive). Update: ( w = 6 - \alpha*2 ) → moves closer to 5.
• When ( w = 4 ): ( \frac{\partial L}{\partial w} = 2*(4-5) = -2 ) (negative). Update: ( w = 4 - \alpha*(-2) = 4 + 2\alpha ) → also moves closer to 5.

If we skipped the negative sign when the gradient was negative, we’d do ( w = 4 + \alpha*(-2) = 4 - 2\alpha ), which takes us away from 5 and makes the loss worse.

The negative sign isn’t about "fixing" a positive or negative gradient—it’s a universal rule to ensure we always move toward minimizing the loss, no matter which way the gradient points.

Hope that clears things up! If you want to dive into the math of backprop in more detail, feel free to ask follow-ups.

内容的提问来源于stack exchange，提问作者taegyun kim