Quick Check 17.3 – Data 100, Summer 2020

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(Fall 2019 Midterm 2) The learning rate can potentially affect which of the following? Select all that apply. Assume nothing about the function being minimized other than that its gradient exists. You may assume the learning rate is positive. *

The speed at which we converge to a minimum.

Whether gradient descent converges.

The direction in which the step is taken.

Whether gradient descent converges to a local minimum or a global minimum.

Required

Suppose we run gradient descent with a fixed learning rate of alpha = 0.1 to minimize the 2D function f(x, y) = 5 + x^2 + y^2 + 5xy. If our starting guess is x(0) = 1, y(0) = 2, what will be our next guess x(1), y(1)? The gradient of this function is:

x(1) = *

y(1) = *

Suppose we are performing gradient descent to minimize the empirical risk of a linear regression model y = theta_0 + theta_1*x1 + theta_2*(x1^2) + theta_3*x2 on a dataset with 100 observations. Let D be the number of components in the gradient, e.g. D = 2 for the equation in the previous question. What is D for the gradient used to optimize this linear regression model? *

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