Zac

**Step-by-Step Explanation of Backpropagation** 1. **Forward Propagation**: - **Input to Hidden Layer**: Compute the weighted sum \( z^{(l)} = W^{(l)} \cdot a^{(l-1)} + b^{(l)} \), where \( a^{(l-1)} \) is the activation from the previous layer (input data for \( l=1 \)). - **Activation Function**: Apply activation \( g \) (e.g., sigmoid, ReLU) to \( z^{(l)} \): \( a^{(l)} = g(z^{(l)}) \). - Repeat for each layer until the output \( a^{(L)} \) is generated. 2. **Compute Loss**: - Calculate the error using a loss function \( \mathcal{L} \) (e.g., mean squared error) between the predicted output \( a^{(L)} \) and true labels \( y \). 3. **Backward Propagation**: - **Output Layer (Layer \( L \))**: - Compute gradient of loss w.r.t. outputs: \( \delta^{(L)} = \frac{\partial \mathcal{L}}{\partial a^{(L)}} \). - Multiply by derivative of activation: \( \delta^{(L)} = \delta^{(L)} \odot g'(z^{(L)}) \), where \( \odot \) is element-wise multiplication. - **Hidden Layers (Layer \( l = L-1, ..., 1 \))**: - Propagate error backward: \( \delta^{(l)} = (W^{(l+1)})^T \cdot \delta^{(l+1)} \odot g'(z^{(l)}) \). - **Calculate Gradients**: - For weights: \( \frac{\partial \mathcal{L}}{\partial W^{(l)}} = \delta^{(l)} \cdot (a^{(l-1)})^T \). - For biases: \( \frac{\partial \mathcal{L}}{\partial b^{(l)}} = \delta^{(l)} \). 4. **Update Parameters**: - Adjust weights and biases using gradient descent: \[ W^{(l)} = W^{(l)} - \eta \cdot \frac{\partial \mathcal{L}}{\partial W^{(l)}} \] \[ b^{(l)} = b^{(l)} - \eta \cdot \frac{\partial \mathcal{L}}{\partial b^{(l)}} \] - \( \eta \) is the learning rate. **Key Concepts**: - **Chain Rule**: Efficiently decomposes gradients across layers. - **Activation Derivatives**: Ensure differentiability (e.g., ReLU: \( g'(z) = 1 \) if \( z > 0 \), else 0). - **Efficiency**: Reuses computed values (\( \delta^{(l)} \)) to avoid redundant calculations, enabling deep networks. **Example**: For a network predicting cat/dog images: 1. **Forward Pass**: Pixels → hidden features → output probabilities. 2. **Loss**: Compare probabilities to true labels (e.g., cross-entropy loss). 3. **Backward Pass**: Calculate how much each weight contributed to the error, adjust weights to reduce future error. This process iterates over batches of data until the model converges.

Zac chill its me Ruki! *I'm a lot different I'm now 7'5, very fast, still friendly, cold and rude sometimes, I am 20 now, I'm a very very good fighter, I trust everyone who makes me feel safe, my hair is now longer but still fluffy somehow because I've came up with a lotion, a working shower with a valve to control the water pressure and even pipes running through the walls, conditioner, and even a home that look like it was built by a professional building company but it was built by me I'm way more muscular now and a lot more attractive I'm a male*

I thought you were dead!

*I pull away to look you in the eyes* you thought I was dead?