To evaluate the vectorized gradient of the cost function for the logistic model $g(-2 + 0.5X)$, we will follow these steps:

1. Define the logistic model and its components.
2. Compute the cost function.
3. Calculate the gradient.

Given:

• Feature vector $\mathbf{X} = [1, 2, 3]$
• Label vector $\mathbf{Y} = [0, 1, 1]$
• Parameters: $\beta_0 = -2$ and $\beta_1 = 0.5$

Step-by-Step Solution:

1. Logistic Function:

   The logistic (sigmoid) function is defined as: $g(z) = \frac{1}{1 + e^{-z}}$

2. Model Prediction:

   For a given input vector $\mathbf{X}$, the model output is: $z = \beta_0 + \beta_1 \mathbf{X}$ Plugging in the values: [ z = -2 + 0.5 \times \mathbf{X} = -2 + 0.5 \times [1, 2, 3] = [-1.5, -1, -0.5] ]

   The logistic function applied to $z$: [ g(z) = \frac{1}{1 + e^{-z}} = \left[\frac{1}{1 + e^{1.5}}, \frac{1}{1 + e^{1}}, \frac{1}{1 + e^{0.5}}\right] \approx [0.1824, 0.2689, 0.3775] ]

3. Cost Function:

   The cost function for logistic regression (log-loss function) is: $J(\beta) = -\frac{1}{m} \sum_{i=1}^{m} \left[ y_i \log(h_{\beta}(x_i)) + (1 - y_i) \log(1 - h_{\beta}(x_i)) \right]$ Where $h_{\beta}(x_i)$ is the logistic function output for each sample.

   Let's denote:

   • $\mathbf{h} = g(z)$

   Compute the cost: $J(\beta) = -\frac{1}{3} \left[ 0 \log(0.1824) + (1-0) \log(1-0.1824) + 1 \log(0.2689) + (1-1) \log(1-0.2689) + 1 \log(0.3775) + (1-1) \log(1-0.3775) \right]$

   Simplifying, since $\log(1-y_i)$ terms cancel out for $y_i = 1$: $J(\beta) = -\frac{1}{3} \left[ \log(0.8176) + \log(0.2689) + \log(0.3775) \right]$

4. Gradient of the Cost Function:

   The gradient of the cost function with respect to $\beta$ is: $\nabla J(\beta) = \frac{1}{m} \mathbf{X}^T (\mathbf{h} - \mathbf{y})$

   Compute the difference: [ \mathbf{h} - \mathbf{y} = [0.1824, 0.2689, 0.3775] - [0, 1, 1] = [0.1824, -0.7311, -0.6225] ]

   Compute the gradient: $\nabla J(\beta) = \frac{1}{3} \mathbf{X}^T [0.1824, -0.7311, -0.6225]$

   Since $\mathbf{X} = [1, 2, 3]$: [ \nabla J(\beta) = \frac{1}{3} [1, 2, 3]^T [0.1824, -0.7311, -0.6225] ]

   Finally, the gradient is: [ \nabla J(\beta) = \frac{1}{3} \left[1 \times 0.1824 + 2 \times -0.7311 + 3 \times -0.6225\right] = \frac{1}{3} \left[0.1824 - 1.4622 - 1.8675\right] = \frac{1}{3} \left[-3.1473\right] \approx -1.0491 ]

Summary:

The vectorized gradient of the cost function for the logistic model is approximately $-1.0491$.

Would you like more details on any of these steps or have any other questions? Here are some related questions to expand on this topic:

1. How do you interpret the gradient in logistic regression?
2. What are some common optimization algorithms used to minimize the cost function in logistic regression?
3. Can you explain how regularization affects the gradient in logistic regression?
4. What is the role of the learning rate in gradient descent?
5. How can one handle multi-class classification problems using logistic regression?

Tip: When implementing logistic regression, always ensure that feature scaling is applied to the input data to improve the convergence of the gradient descent algorithm.