Derivatives are important for optimization because the zero derivatives might indicate a minimum, maximum, or saddle point. In calculus, the derivative of a function shows you how much a value changes when you modify its argument (or arguments). Gradient of a Function: Calculus Refresher In logistic regression, which is often used to solve classification problems, the functions □(□) and □(□) are defined as the following:Īgain, you need to find the weights □₀, □₁, …, □ᵣ, but this time they should minimize the cross-entropy function. In the case of binary outputs, it’s convenient to minimize the cross-entropy function that also depends on the actual outputs □ᵢ and the corresponding predictions □(□ᵢ): For example, you might try to predict whether an email is spam or not. In a classification problem, the outputs □ are categorical, often either 0 or 1. For example, in linear regression, you want to find the function □(□) = □₀ + □₁□₁ + ⋯ + □ᵣ□ᵣ, so you need to determine the weights □₀, □₁, …, □ᵣ that minimize SSR or MSE. SSR or MSE is minimized by adjusting the model parameters. A difference of zero indicates that the prediction is equal to the actual data. The lower the difference, the more accurate the prediction. Alternatively, you could use the mean squared error (MSE = SSR / □) instead of SSR.īoth SSR and MSE use the square of the difference between the actual and predicted outputs. In this type of problem, you want to minimize the sum of squared residuals (SSR), where SSR = Σᵢ(□ᵢ − □(□ᵢ))² for all observations □ = 1, …, □, where □ is the total number of observations. Your goal is to minimize the difference between the prediction □(□) and the actual data □. For example, you might want to predict an output such as a person’s salary given inputs like the person’s number of years at the company or level of education. You want to find a model that maps □ to a predicted response □(□) so that □(□) is as close as possible to □. In a regression problem, you typically have the vectors of input variables □ = (□₁, …, □ᵣ) and the actual outputs □. They tend to minimize the difference between actual and predicted outputs by adjusting the model parameters (like weights and biases for neural networks, decision rules for random forest or gradient boosting, and so on). Many machine learning methods solve optimization problems under the surface. The cost function, or loss function, is the function to be minimized (or maximized) by varying the decision variables. Remove ads Cost Function: The Goal of Optimization
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