CS229: Linear Regression Cost Function

Creating the Linear Regression Cost Function A machine learning approach called linear regression is used to simulate the correlation between input data and output targets. The first crucial step in applying linear regression is to define something known as a cost function. We can make tweaks during training to improve the model by using this cost function to determine how well the model is performing. Recognizing the Model A training set for linear regression contains the input characteristics x and the output targets y. The linear function 𝑓𝑤,𝑏(𝑥) = 𝑤𝑥 + 𝑏, where w and b are the model's parameters, was employed to fit this training set. These parameters are the elements that can be changed to enhance the model while it is being trained. They are sometimes known as weights or coefficients. The function f(x), which results in a different line on the graph, is determined by the values picked for w and b. The y-intercept, or point on the line where it crosses the y-axis, is represented by the b value, while the w value indicates the line's slope. Using the Model to Fit the Data The objective of linear regression is to select values for the parameters w and b that result in a line that closely fits the data. In contrast to other potential lines that are not as close to the points, a visually appealing fit indicates that the line defined by f passes through or is close to the training instances. X and Y are the parameters that define a training example, where Y is the target. The (𝑖) model's prediction for an input value of 𝑥 (𝑖) is 𝑦 (𝑖) = 𝑓𝑤,𝑏(𝑥 ) Price Function The model's effectiveness on the training set is assessed using the cost function. The objective is to identify the w and b values that minimize the cost. Mean squared error is a typical cost function utilized in linear regression (MSE). The average of the squared deviations between the predicted value yi hat and the actual value yi for each training example is used to calculate MSE. The MSE equation is: 𝐽(𝑤, 𝑏) = 1 2𝑚 𝑚 (𝑖) ∑ (𝑦 (𝑖) 2 − 𝑦 ) where m is the number of training examples. 𝑖=1 Keeping the Cost Low Our objective is to minimize the cost function, which provides us with a gauge of how well the model is performing. This is accomplished by altering the values of w and b until the cost function's minimum value is obtained. Gradient descent is the name of this procedure. The values of w and b are updated incrementally in the direction of the steepest decline during gradient descent up until a minimum is attained. The direction of the updates is determined by the gradient of the cost function with respect to w and b. To sum up, defining the cost function is a crucial first step in putting linear regression into practice. In order to train the model to produce correct predictions, it is essential to minimize the cost function, which serves as a gauge of how well the model is performing.