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Rewards are a way of providing feedback to a reinforcement learning model on how good or bad it performed in each step. This is used by the model to learn and improve. We don't use rewards in supervised learning because we use loss function as the criteria to make the model understand its shortcomings and make improvements. You can think of loss as negative rewards. They both serve the same purpose.

Both refer to the same concept only. Fisher's linear discriminant is used to find the projection direction which maximizes the Fisher criterion (ie the ratio of between class variance to within class variance). The same is referred to in some informal instances as Maximum Fisher Discriminant.