Prove that the LMS weight update rule described in this...

Prove that the LMS weight update rule described in this chapter performs a gradient descent to minimize the squared error. In particular, define the squared error $E$ as in the text. Now calculate the derivative of $E$ with respect to the weight $w_i$, assuming that $\hat{V}(b)$ is a linear function as defined in the text. Gradient descent is achieved by updating each weight in proportion to $-\frac{\partial E}{\partial w_i}$. Therefore, you must show that the LMS training rule alters weights in this proportion for each training example it encounters.

Key Concepts

Least Mean Square (LMS) Algorithm

The LMS algorithm is a method used to adaptively adjust the weights of a linear model in order to minimize the prediction error. It achieves this by updating the weights in response to observed discrepancies between the target and predicted values, making it well-suited for real-time learning scenarios. This adaptive process is closely tied to incremental gradient descent, whereby the weights are moved in the direction that most reduces the error for each training example.

Squared Error Cost Function

The squared error cost function is a common metric used to quantify the performance of a linear predictor. It is computed as the square of the difference between the actual target value and the predicted output, summed or averaged over all training examples. This function is differentiable and convex, which makes it highly amenable to optimization techniques like gradient descent, ensuring that the global minimum can be found when minimizing the error.

Gradient Descent Method

Gradient descent is an iterative optimization technique used to minimize a cost function by moving in steps proportional to the negative of the gradient of the function with respect to the model parameters. The method hinges on the idea that the partial derivative of the cost function with respect to each parameter points in the direction of the steepest ascent, and thus, moving in the opposite direction leads to a reduction in error. It forms the backbone of many learning algorithms, including LMS.

Weight Update Rule

The weight update rule specifies how model parameters should be adjusted to minimize the cost function, typically by a fraction (learning rate) of the negative gradient of the cost function with respect to each weight. In the context of the LMS algorithm, this rule ensures that each weight is updated in proportion to both the input signal and the error (difference between the actual and predicted output). This rule embodies the core principle of gradient descent in the adaptation of linear models.

Linear Function Approximation

In linear function approximation, the predicted output is modeled as a weighted sum of the input features. This approach provides a simple yet powerful framework for mapping inputs to outputs in a linear fashion. The linear assumption facilitates analytical derivation of gradients, making it easier to apply optimization techniques like gradient descent, as the relationship between the weights and the predicted output remains straightforward and differentiable.

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