Deep Learning

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This is a learning note about deep learning.


Definition

The term, Deep Learning, refers to training Neural Networks, sometimes very large Neural Networks.

Logistic Regression

It tries to solve binary classification problem.

Given x, want

\[\begin{align*} \hat{y} = P(y = 1 | x) \end{align*}\]

Sigmoid function

\[\begin{align*} \sigma(z) = \frac{1}{1 + {e^{-z}}} \end{align*}\]

Output

\[\begin{align*} \hat{y} = \sigma({w^T} + b) \end{align*}\]

Loss (error) function

\[\begin{align*} L(\hat{y}, y) = -(y\log \hat{y} + (1-y)\log (1-\hat{y})) \end{align*}\]

Cost function

\[\begin{align*} J(w, b) = \frac{1}{m} \sum_{i=1}^m L(\hat{y}^{(i)}, y^{(i)}) \end{align*}\] \[\begin{align*} = -\frac{1}{m} \sum_{i=1}^m (y^{(i)}\log \hat{y}^{(i)} + (1-y^{(i)})\log (1-\hat{y}^{(i)})) \end{align*}\]

Gradient Descent

Modify the paramters w and b to reduce the loss L by utilizing derivatives with a Computation Graph.


Neural Network

Neural Network Representation

Input layer, hidden layer(s), output layer

Activation Function (binary classification)

If output is 0, 1, use sigmoid activation function, or tanh activation function for -1, 1.

For all other cases, use ReLU, or the rectified linear unit activation function.

Gradient descent for Neural Networks

  • Forward propagation
  • Backward propagation

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Published on April 01, 2019