Answer:
To find the gradient of the logistic regression parameters using the given function h(x) = s(theta0 + theta1 * x), where s is the sigmoid function, we will first define the logistic regression loss function and then compute the derivatives step by step.
1. Logistic Regression Loss Function:
The logistic regression loss function is typically defined as the negative log-likelihood of the observed data. In this case, assuming we have a binary classification problem, it can be expressed as:
L(theta) = -[y * log(h(x)) + (1 - y) * log(1 - h(x))]
Where:
- L(theta) is the loss function
- y is the true label (either 0 or 1)
- h(x) is the predicted probability of the positive class (sigmoid function applied to the linear combination of parameters and features)
2. Derivative of Sigmoid Function:
The sigmoid function s(z) is defined as s(z) = 1 / (1 + e^(-z)). To compute the derivative, we can use the chain rule:
s'(z) = (1 / (1 + e^(-z))) * (1 - 1 / (1 + e^(-z)))
= s(z) * (1 - s(z))
3. Derivative of the Loss Function with Respect to theta0:
To find the derivative of the loss function with respect to theta0, we will use the chain rule:
dL(theta) / dtheta0 = -[y * (1 / h(x)) * h(x) * (1 - h(x)) * 1 + (1 - y) * (-1 / (1 - h(x))) * (-h(x)) * (1 - h(x)) * 1]
= -[y * (1 - h(x)) - (1 - y) * h(x)]
= h(x) - y
4. Derivative of the Loss Function with Respect to theta1:
To find the derivative of the loss function with respect to theta1, we will again use the chain rule:
dL(theta) / dtheta1 = -[y * (1 / h(x)) * h(x) * (1 - h(x)) * x + (1 - y) * (-1 / (1 - h(x))) * (-h(x)) * (1 - h(x)) * x]
= -[y * (1 - h(x)) - (1 - y) * h(x)] * x
= (h(x) - y) * x
These derivatives represent the gradients of the logistic regression parameters theta0 and theta1. By updating these parameters in the direction opposite to their gradients, we can iteratively optimize the logistic regression model using techniques like gradient descent.
Note: The chain rule is used to compute the derivatives by breaking down complex functions into simpler ones and applying the derivative rules for each component.
Step-by-step explanation: