Mean squared error (MSE) is the most common loss function in regression: it averages the square of the gap between each true value and its prediction. Squaring punishes big mistakes hard, keeps the function differentiable and gives gradient descent a clear, smooth target to minimise while training a model.
The linear function f(x) = ax + b is the simplest activation a neural network can use: in its identity form, f(x) = x, it is the standard choice for the output layer in regression, because it does not bound the range of possible values. In hidden layers it fails, because composing several linear functions collapses the whole network into one equivalent linear layer.
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