Assume that your hypothesis function for linear regression is of the form f(x) = w0 + w1x and that the current values of w0 and w1 are 1 and 2 respectively. Further assume that you are using a learning rate (alpha) of 0.001

What is the new w0 value associated with the point (1, 12), after one gradient update?

Question

Assume that your hypothesis function for linear regression is of the form f(x) = w0 + w1x and that the current values of w0 and w1 are 1 and 2 respectively. Further assume that you are using a learning rate (alpha) of 0.001

What is the new w0 value associated with the point (1, 12), after one gradient update?

BlackTom AI · Accepted Answer

To update w0 after observing the point (x, y) = (1, 12) with the current weights w0 = 1 and w1 = 2, we first compute the current hypothesis output h for x = 1: h = w0 + w1*x = 1 + 2*1 = 3.
Next, we consider the typical mean-squared-error (MSE) loss with a 1/2 factor on a single data point: L = (1/2)(h - y)^2. The gradient of L with respect to w0 is ∂L/∂w0 = (h - y) because ∂h/∂w0 = 1.
Compute the error term: h - y = 3 - 12 = -9.
The gradient descent update for w0 with learning rate alpha = 0.001 is w0_new = w0 - alpha * (∂L/∂w0) = 1 - 0.001 * (-9) = 1 + 0.009 = 1.009.
Thus, after one gradient update using this common formulation, the new w0 value becomes 1.009.
Note: if a different loss formulation were used (for example, omitting the 1/2 factor in L or using a different gradient convention), the numeric update could differ. However, with the standard MSE/1/2 convention, this is the result.

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