Question22 Suppose that you want to build a gradient boosting with the following loss function: [math] Assuming that [math] is the actual output and [math] is the prediction for point [math] , which of the following statements is correct? Select one alternative: You train a weak learner and at each step, you estimate [math] You train a weak learner and at each step, you estimate [math] You train a weak learner and at each step, you estimate [math] You train a weak learner and at each step, you estimate [math] ResetMaximum marks: 2 Flag question undefined

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BlackTom AI · Accepted Answer

Let’s unpack the question step by step and then evaluate each answer choice in turn. The scenario describes building a gradient boosting model with a particular loss function, and at each boosting step we fit a weak learner to some target that is derived from the current residuals (the gradient or pseudo-residuals of the loss with respect to the model’s predictions). The correct form depends on the chosen loss, but a common pattern in gradient boosting is that the weak learner is trained to predict the negative gradient of the loss with respect to the current predictions f(x). In this specific setup, the loss appears to be a function of the residual (y_i − f_i), and the gradient (or pseudo-residual) would be derived from that loss function.

Option 1: You train a weak learner and at each step, you estimate
y_i − ŷ_i
(
( y_i − ŷ_i )^2 + 1
)^{1/2}

- This expression seems to present a form where the quantity being estimated involves the residual (y_i − ŷ_i) combined with a square root term. If the intended operation is division by the square root, then this would resemble a negative gradient like (y_i − ŷ_i) / sqrt((y_i − ŷ_i)^2 + 1). However, in the text shown, it’s not clearly separated by a division bar; rather, it appears as a product of the residual with a square-root term, which would be conceptually inconsistent with the typical negative gradient form. The key misconception here would be treating the square root term as a multiplier rather than as a normalizing denominator.

Option 2: You train a weak learner and at each step, you estimate
y_i − ŷ_i(
( y_i − ŷ_i )^2 + 1
)^{1/2}

- This option explicitly places the residual (y_i − ŷ_i) inside the parentheses that are then multiplied by the square-root term. In other words, it suggests the residual is being scaled by sqrt((y_i − ŷ_i)^2 + 1) rather than being divided by it. If the intended loss is differentiable in a way that yields a negative gradient proportional to (y_i − ŷ_i)/sqrt((y_i − ŷ_i)^2 + 1), then this option would again misrepresent the calculation by turning the division into a multiplication, which changes the scaling behavior and the training dynamics.

Option 3: You train a weak learner and at each step, you estimate
y_i − ŷ_i

- This option simply uses the plain residual as the target for the weak learner. While residuals are used in many additive modeling schemes, gradient boosting typically uses the negative gradient of the loss, which for many common non-quadratic losses is not simply (y_i − ŷ_i). Relying on the plain residual ignores the specific form of the chosen loss and can lead to incorrect updates unless the loss function reduces exactly to a squared-error form in which case the residual is appropriate. In short, this choice neglects the normalization or transformation implied by the loss’s curvature.

Option 4: You train a weak learner and at each step, you estimate
(
y_i − ŷ_i + 1
)^{1/2}

- This option proposes estimating the square root of (residual plus 1). This is not aligned with the standard gradient-boosting paradigm where the weak learner is trained to predict the (negative) gradient of the loss with respect to the current predictions. Taking a square root of a shifted residual alters the fundamental target and does not capture the directional information provided by the gradient. It introduces a nonlinear transformation of the residual that does not correspond to the gradient of the stated loss.

Overall, several options appear to misstate the intended negative gradient form by either misplacing division, multiplying by the square-root term, or replacing the gradient with an unrelated transformation of the residual. The correct approach in gradient boosting with a loss involving sqrt((y − f)^2 + 1) would typically yield a target proportional to (y_i − ŷ_i)/sqrt((y_i − ŷ_i)^2 + 1), i.e., the residual divided by the square root term, rather than a simple residual or a product with the square-root term. The remaining options diverge from that standard formulation in notable ways, such as omitting the division, multiplying by the normalizing factor, or applying a different transformation to the residual.

Note: If you confirm the exact intended loss function and its gradient, you can map the correct target precisely to one of these forms. The key diagnostic is whether the target involves a division by sqrt((y_i − ŷ_i)^2 + 1) (negative gradient form), rather than a plain residual, a product with the square root, or a different residual-based transformation.

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