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\newcommand{\foldsize}{B} \newcommand{\nriter}{R} [/math]

Consider a linear regression method that uses ERM to learn weights [math]\widehat{\weights}[/math] of a linear hypothesis map [math]h(\featurevec) =\weights^{T} \featurevec[/math]. The weights are learnt by minimizing the average squared error loss incurred by [math]h[/math] on a training set that is constituted by the data points [math]\big( \featurevec^{(\sampleidx)}, \truelabel^{(\sampleidx)} \big)[/math] for [math]\sampleidx=1,\ldots, 100[/math]. Someimtes it is useful to assign sample-weights [math]\sampleweight{\sampleidx}[/math] to the data points and learn [math]\widehat{\weights}[/math]. These sample-weights reflect varying levels of importance or relevance of different data points.

For simplicity we use the sample weights [math]\sampleweight{\sampleidx} = 2 \alpha \in [0,1][/math] for [math]\sampleidx=1,\ldots,50[/math] and [math]\sampleweight{\sampleidx} = 2(1 - \alpha)[/math] for [math]\sampleidx=51,\ldots,100[/math].

Can you find a closed-form expression (similar to equ_close_form_lin_reg) for the weights [math]\widehat{\weights}^{(\alpha)}[/math] that minimize the weighted average squared error

[[math]]f(\weights) \defeq (1/50)\sum_{\sampleidx=1}^{50} \alpha \big( \truelabel^{(\sampleidx)} - \weights^{T} \featurevec^{(\sampleidx)} \big)^{2} + (1/50)\sum_{\sampleidx=51}^{100} (1-\alpha) \big( \truelabel^{(\sampleidx)} - \weights^{T} \featurevec^{(\sampleidx)} \big)^{2}[[/math]]

for different [math]\alpha[/math]?