In the models we've seen so far, such as linear regression, we've talked about a criterion or objective function that the model must minimize while it is being trained. This criterion is also sometimes known as the cost function. For example, the least squares cost function for a model can be expressed as:

We've added a constant term of ½ in front of this for reasons that will become apparent shortly. We know from basic differentiation that when we are minimizing a function, multiplying the function by a constant factor does not alter the value of the minimum value of the function. In linear regression, just as with our perceptron model, our model's predicted is just the sum of a linear weighted combination of the input features. If we assume that our data is fixed and that the weights are variable and must be chosen so as to minimize our criterion, we can treat the cost function as being a function of the weights:

We have used the letter w to represent the model...