3. 1+e" Logistic regression is a supervised learning algorithm used to estimate the model parameters from a training dataset D = {(x,y), i = 1, 2,...,m, y = {0, 1}}, such that the resulting hypothesis function h(x) = g(x) can predict the probability that a new input instance x belongs to the positive class (usually denoted as 1). To achieve this goal, logistic regression employs a logistic or sigmoid activation function g(z) =which transforms the linear combination of input features and model parameters to a probability value within the range (0,1). The probability serves as the basis for classifying new data instances. A common approach for logistic regression models is stochastic gradient ascent, which involves iteratively updating the model parameters based on the gradients of log-likelihood function. Stochastic gradient ascent aims to maximize the likelihood of observing the given data and refine the model's parameters accordingly. Please derive the following stochastic gradient ascent update rule for logistic regression models, 9, (+1) = 9, (*) + a(y) - hz(x)))x(),j = 0,1,...,n where a is the learning rate, y() is the binary label of the ith training instance *(¹) (either 0 or 1), ha (*) is the predicted probability of labeling the ith instance (as positive (class label is 1) based on the current values of the model parameters (t). 8, (c) is the current value of the parameter 0;. 0,(t+1) is the updated value of the parameter ej. Given a neural network, its structure is shown below. z," is the output of the linear part of jth neuron in layer l; a = g(2") is the output of the activation part of th neuron in layer I and g(2) is the activation function. x1 x2 za [1] Xn z al a2 릴리 [2] [3]

3. 1+e" Logistic regression is a supervised learning algorithm used to estimate the model parameters from a training dataset D = {(x,y), i = 1, 2,...,m, y = {0, 1}}, such that the resulting hypothesis function h(x) = g(x) can predict the probability that a new input instance x belongs to the positive class (usually denoted as 1). To achieve this goal, logistic regression employs a logistic or sigmoid activation function g(z) =which transforms the linear combination of input features and model parameters to a probability value within the range (0,1). The probability serves as the basis for classifying new data instances. A common approach for logistic regression models is stochastic gradient ascent, which involves iteratively updating the model parameters based on the gradients of log-likelihood function. Stochastic gradient ascent aims to maximize the likelihood of observing the given data and refine the model's parameters accordingly. Please derive the following stochastic gradient ascent update rule for logistic regression models, 9, (+1) = 9, () + a(y) - hz(x)))x(),j = 0,1,...,n where a is the learning rate, y() is the binary label of the ith training instance (¹) (either 0 or 1), ha (*) is the predicted probability of labeling the ith instance (as positive (class label is 1) based on the current values of the model parameters (t). 8, (c) is the current value of the parameter 0;. 0,(t+1) is the updated value of the parameter ej. Given a neural network, its structure is shown below. z," is the output of the linear part of jth neuron in layer l; a = g(2") is the output of the activation part of th neuron in layer I and g(2) is the activation function. x1 x2 za [1] Xn z al a2 릴리 [2] [3]