Regression and classification both belong to supervised learning. Regression is mostly based on the data which are continuous in nature whereas the classification algorithm is mostly based on the labels provided to the data set.
Regression is based on the paradigm of continuous prediction whereas classification is mostly used for predicting the probability of a particular object belonging to a class.
The outcome in the case of classification problem are mostly binary in nature, for example, whether the customer will buy a particular product or not, whether the particular student will pass or not, whether the particular interview will be cleared or not and many more.
Regression algorithms are mostly used for prediction of a continuous variable in the future time period based on certain inputs which can be continuous or categorical in nature.
In the case of classification algorithm the inputs and outputs are both categorical in nature and the probability of identifying a particular class is already defined in the classification problem. For example, the number of classes which will be produced in the classification algorithm is already predefined in the algorithm by the user.
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