2.1- Multiple Inputs

2.1 Why Multiple Inputs?

A real problem usually depends on multiple factors.

Example:

x₁ = house size
x₂ = bedrooms

A neuron can use all of them.

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3.2 Multiple-Input Neuron

Single input:

$$y = wx+b$$

Multiple inputs:

$$\boxed{y=w_1x_1+w_2x_2+b}$$

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• Weight → controls how strongly an input affects the output.
• Positive weight → increasing input increases prediction.
• Negative weight → increasing input decreases prediction.
• Bias → shifts the output.

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3.4 General Formula

For 3 inputs:

$$y=w_1x_1+w_2x_2+w_3x_3+b$$

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Gradient descent still works:

$$parameter=parameter-\eta\times gradient$$

Each weight gets its own gradient.

For:

$$y=w_1x_1+w_2x_2+b$$

and:

$$L=(target-prediction)^2$$

we get:

$$\boxed{\frac{dL}{dw_1}=-2x_1(error)}$$ $$\boxed{\frac{dL}{dw_2}=-2x_2(error)}$$

In general:

$$\boxed{\frac{dL}{dw_i}=-2x_i(error)}$$

Bias remains:

$$\boxed{\frac{dL}{db}=-2(error)}$$

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