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Squared-loss empirical risk for linear prediction

For data

(X_{i}, Y_{i})

and predictions

h_{β} (X) = β^{T} X

, it is convenient to write

R_{n} (β) = \frac{1}{2 n} \sum_{i = 1}^{n} (β^{T} X_{i} - Y_{i})^{2} .

Its gradient is

\nabla R_{n} (β) = \frac{1}{n} \sum_{i = 1}^{n} (β^{T} X_{i} - Y_{i}) X_{i} .

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