A machine learning engineer is manually implementing the gradient descent algorithm to optimize a multivariate linear regression model. The objective is to minimize the Mean Squared Error (MSE) cost function by iteratively adjusting the model's parameters (weights). For each iteration of the algorithm, which of the following mathematical operations is most fundamental for determining the direction and magnitude of the update for a specific weight?
Calculating the Euclidean distance between the predicted and actual values.
Computing the second partial derivative (Hessian matrix) of the cost function.
Applying the chain rule to the model's activation function.
Calculating the partial derivative of the MSE cost function with respect to that specific weight.
The correct answer is to calculate the partial derivative of the MSE cost function with respect to that specific weight. In gradient descent, the goal is to minimize a cost function by adjusting model parameters. The gradient, which is a vector composed of the partial derivatives of the cost function with respect to each parameter, points in the direction of the steepest ascent of the cost function. Therefore, to minimize the cost, the algorithm updates the parameters by taking a step in the opposite direction of the gradient. The partial derivative for a specific weight tells us how a small change in that weight will affect the total error, thus defining the direction and contributing to the magnitude of the necessary update for that weight.
Computing the second partial derivative (Hessian matrix) is characteristic of second-order optimization methods, like Newton's method, which use curvature information to converge faster but are more computationally expensive. The question specifically asks about gradient descent, which is a first-order method.
Applying the chain rule is a necessary step in the process of deriving the partial derivative for complex functions (like in neural networks), but the fundamental quantity needed for the update step in gradient descent is the partial derivative itself.
Calculating the Euclidean distance between predicted and actual values is part of computing the overall MSE cost, not the update step. The partial derivative of this cost is what guides the optimization.
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